Celebratio Mathematica

C. M. Gordon: Knots and embeddings. Ph.D. thesis, University of Cambridge, 1970. Advised by J. F. P. Hudson. phdthesis

C. M. Gordon: “\( h \)-cobordisms of pairs,” Proc. Cambridge Philos. Soc. 68 : 3 (November 1970), pp. 641–652. MR 270382 Zbl 0203.56203 article

C. M. Gordon: “A short proof of a theorem of Plans on the homology of the branched cyclic coverings of a knot,” Bull. Am. Math. Soc. 77 : 1 (January 1971), pp. 85–87. MR 267567 Zbl 0206.52301 article

C. M. Gordon: “Twist-spun torus knots,” Proc. Am. Math. Soc. 32 : 1 (1972), pp. 319–322. MR 288752 Zbl 0231.55006 article

C. M. Gordon: “Knots whose branched cyclic coverings have periodic homology,” Trans. Am. Math. Soc. 168 (1972), pp. 357–370. MR 295327 Zbl 0238.55001 article

C. M. Gordon: “Embedding piecewise linear manifolds with boundary,” Proc. Cambridge Philos. Soc. 72 : 1 (July 1972), pp. 21–25. MR 295359 Zbl 0236.57009 article

C. M. Gordon and W. Heil: “Simply-connected branched coverings of \( S^3 \),” Proc. Am. Math. Soc. 35 : 1 (September 1972), pp. 287–288. MR 296930 Zbl 0257.55002 article

C. M. Gordon: “Some higher-dimensional knots with the same homotopy groups,” Quart. J. Math. Oxford Ser. (2) 24 : 1 (1973), pp. 411–422. MR 326746 Zbl 0263.57005 article

A knot \( \kappa \) of \( S^{n-2} \) in \( S^n \) is a pair \( (S^n,S^{n-2}) \) determined by a smooth emedding of \( S^{n-2} \) in \( S^n \). The complement of \( \kappa \) is \( S^n\setminus S^{n-2} \) and by \( \pi_1(\kappa) \) we shall mean \( \pi_1(S^n\setminus S^{n-2}) \). If \( \kappa_1 \) and \( \kappa_2 \) are two knots of \( S^{n-2} \) in \( S^n \), then \( \kappa_1\#\kappa_2 \) will denote their connected sum. In this paper we shall prove:

For \( n\geq 4 \), there exist infinitely many examples of triples \( \{\kappa_1 \), \( \kappa_2 \), \( \kappa_3\} \) of knots of \( S^{n-2} \) in \( S^n \) such that

\( X_i \), the complement of \( \kappa_i \), is fibred over \( S^1 \) with fibre \( F \) (\( i=1 \), 2, 3),
\( \pi_1(\kappa_i)\cong \pi_1(F)\times\mathbf{Z} \) (\( i=1 \), 2, 3),
no two of the six groups \( \pi_q(\kappa_i\#\kappa_j) \) (\( i\leq j \), \( i,j = 1 \), 2, 3) are isomorphic
no two of \( X_1, X_2, X_3, F\times S^1 \) are homotopy equivalent.

In particular, for \( n\geq 4 \) there exist knots of \( S^{n-2} \) in \( S^n \) with the same homotopy groups whose complements are not homotopy equivalent. Moreover, the homology invariants of their infinite cyclic coverings are all trivial.

C. M. Gordon: “On the higher-dimensional Smith conjecture,” Proc. London Math. Soc. (3) 29 : 1 (July 1974), pp. 98–110. MR 356073 Zbl 0287.57019 article

The classical Smith conjecture [Smith 1939], which states that no periodic transformation of \( S^3 \) can have a tame knotted \( S^1 \) as its fixed-point set, is still unresolved, although Waldhausen has proved [1969] that it is true for transformations of even period. Its higher-dimensional analogues are false, however, as Giffen has shown [1966] by considering certain branched cyclic coverings of twist-spun knots [Zeeman 1965], the only case left unsettled in [1966] being that of transformations of even period on \( S^4 \). In the present paper we show that the analogue of the Smith conjecture is false in this case also, by describing a method (unrelated to Giffen’s) for constructing counterexamples for any period in any dimension greater than or equal to 4 (Theorem 1). (Theorem 1 has also been announced by Sumners [1973], who uses yet another method.) Next we give a simpler proof of Giffen’s main result (Theorem 2), which, like his, uses the nice fibration properties of twist-spun knots pointed out by Zeeman [1965], but which avoids explicit reference to branched cyclic coverings. Furthermore, by combining twist-spinning with spinning, we obtain knots in dimensions greater than or equal to 5 each of which is the fixed-point set of a particularly nice transformation of period \( m \) for all \( m \) not divisible by some given prime \( p \) (Theorem 3), and in Theorem 4, we show that this is the best possible. Theorem 4 states that if a knot is the fixed-point set of a transformation of period \( p \) with certain homotopy properties for all primes \( p \), then its complement is homotopy-equivalent to \( S^1 \). This generalizes a result of Hsiang [1964] about \( S^1 \)-actions on \( S^n \). Finally, and in contrast to Theorem 4, we give examples of knots in dimensions greater than or equal to 5 each of which is the fixed-point set of a transformation of period \( p \) for all primes \( p \) (Theorem 5).

J. H. Conway and C. M. Gordon: “A group to classify knots,” Bull. London Math. Soc. 7 : 1 (March 1975), pp. 84–86. MR 375282 Zbl 0304.55001 article

C. M. Gordon and D. W. Sumners: “Knotted ball pairs whose product with an interval is unknotted,” Math. Ann. 217 : 1 (February 1975), pp. 47–52. MR 380816 Zbl 0293.57010 article

Let \( B^n \) denote the smooth \( n \)-ball, and \( g:B^n\to B^{n+2} \) be a smooth proper embedding, that is, \( g^{-1}(\partial B^{n+2}) = \partial B^n \). Then \( (B^{n+2},gB^n) \) is a smooth ball pair, possibly knotted. By taking the cartesian product of \( (B^{n+2},gB^n) \) with an interval \( I \), we obtain the ball pair \[ (B^{n+3},g^{\prime} B^{n+1}) = (B^{n+2},gB^n)\times I ,\] with \( g^{\prime} = g\times id \). In this paper we construct, for all \( n\geq 2 \), infinitely many distinct examples of smooth knotted ball pairs whose product with an interval yields the unknotted ball pair. This improves the result of Kato [1969], who produced such examples for \( n\geq 4 \). The examples obtained here are in the lowest possible dimension (\( n = 2 \)), because if \( (B^3,gB^1)\times I \) is unknotted, then \( (B^3, gB^1) \) must itself be unknotted, because the complement is a homotopy circle.

We give two different methods for constructing examples in the lowest-dimensional situation \( (B^4,gB^2) \), and produce the higher-dimensional examples by \( P \)-spinning each of the \( (B^4,gB^2) \) about an unknotted face. The first of the methods is a simple handlebody construction, which is really the analogue for pairs of the Mazur manifold phenomenon [Zeeman 1962]. The second method is derived from a more general unknotting result, a corollary of which is the following:

The untwisted double of any slice knot bounds a smooth ball pair \( (B^4,gB^2) \) such that \( (B^4,gB^2)\times I \) is unknotted.

C. M. Gordon and W. Heil: “Cyclic normal subgroups of fundamental groups of 3-manifolds,” Topology 14 : 4 (1975), pp. 305–309. MR 400232 Zbl 0331.57001 article

Let \( M \) be a compact, connected, orientable 3-manifold. If \( M \) is a Seifert fibre space (other than \( S^3 \)), then \( \pi_1(M) \) has a non-trivial cyclic normal subgroup, namely that generated by the class of an ordinary fibre. If the orbit-surface (Zerlegungsfläche) of the fibring is orientable, then this subgroup is central. Conversely, Waldhausen has shown [1967] that if \( M \) is irreducible and sufficiently large (in the sense of [Waldhausen 1968]), and if \( \pi_1(M) \) has a non-trivial centre, then \( M \) has a Seifert fibring with orientable orbit-surface. By an argument based on Waldhausen’s we prove the following (which was announced in [Heil 1973]).

Let \( M \) be a compact, connected, orientable, irreducible and sufficiently large 3-manifold. If \( \pi_1(M) \) contains a (non-trivial) cyclic normal subgroup, then either

\( M \) is a Seifert fibre space or
there exists a closed surface \( F \) in \( M \) which separates \( M \) into two twisted line bundles over a non-orientable surface \( G \).

C. M. Gordon: “Knots, homology spheres, and contractible 4-manifolds,” Topology 14 : 2 (June 1975), pp. 151–172. MR 402762 Zbl 0304.57003 article

C. M. Gordon: “A note on spun knots,” Proc. Am. Math. Soc. 58 : 1 (July 1976), pp. 361–362. MR 413119 Zbl 0334.57010 article

C. M. Gordon: “Knots in the 4-sphere,” Comment. Math. Helv. 51 (December 1976), pp. 585–596. MR 440561 Zbl 0346.55004 article

For convenience we work in the PL category. A knot of \( S^{n-1} \) in \( S^{n+1} \) is a locally flat submanifold of \( S^{n+1} \) homeomorphic to \( S^{n-1} \). The closure of the complement of a regular neighbourhood of a knot is its exterior. Two knots are equivalent if there is a homeomorphism of \( S^{n+1} \) taking one to the other. Equivalent knots therefore have homeomorphic exteriors. As far as the converse is concerned, it is known that for \( n\geq 3 \) there are at most two inequivalent knots with a given exterior [Gluck 1962; Browder 1967; Kato 1969; Lashof and Shaneson 1969; Swarup 1971]. Recently, Cappell and Shaneson have shown that for \( n = 4, 5 \), inequivalent knots with homeomorphic exteriors do exist [1976]. Their examples are certain knots whose exteriors fibre over \( S^1 \) with fibre \( T^n \)-open disc, where \( T^n \) is the \( n \)-dimensional torus (compare [Cappell 1970]). Since this approach uses the generalized Poincaré conjecture, however, in the case \( n = 3 \) it only yields knots in homotopy 4-spheres.

In the present paper, we use twist-spun knots to prove

There exist inequivalent knots \( K_1 \), \( K_2,\dots \), \( K_1^* \), \( K_2^*,\dots \) of \( S^2 \) in \( S^4 \), such that \( K_i \) and \( K_i^* \) have homeomorphic exteriors (\( i = 1 \), \( 2,\dots \)).

In the course of the proof, we show that removing a regular neighbourhood of a twist-spun knot in \( S^4 \) and sewing it back differently always gives \( S^4 \). In particular, this answers a question of Zeeman [1965, p. 494, problem 1], and enables us to give some new counterexamples to the 4-dimensional Smith conjecture [Giffen 1966; Gordon 1974; Sumners 1975].

C. M. Gordon: “Uncountably many stably trivial strings in codimension two,” Quart. J. Math. Oxford Ser. (2) 28 : 4 (December 1977), pp. 369–379. MR 470963 Zbl 0379.57001 article

Let \( R^n \) denote \( n \)-dimensional euclidean space. An \( n \)-string is a pair \( (R^{n+2},Y) \), where \( Y \) is a (locally flat) submanifold and closed subset of \( R^{n+2} \) which is homeomorphic to \( R^n \). Two \( n \)-strings are equivalent if they are homeomorphic as pairs. An \( n \)-string \( (R^{n+2},Y) \) is trivial if it is equivalent to \( (R^{n+2},R^n) \), where \( R^n\subset R^{n+2} \) is the standard inclusion; it is stably trivial if the \( (n{+}1) \)-string \[ (R^{n+2},Y)\times R^1 \] is trivial. (These definitions may be interpreted in either the smooth, piecewise-linear, or topological category.) The extent to which the triviality or stable triviality of an \( n \)-string is determined by the (proper) homotopy of its complement is studied in [Stallings 1963] and [Ichiraku and Kato 1972].

The main purpose of this note is to prove the following (which was announced in [Gordon and Sumners 1975]).

For each \( n\geq 1 \) there exist uncountably many inequivalent stably trivial \( n \)-strings.

C. M. Gordon and R. A. Litherland: “On the signature of a link,” Invent. Math. 47 : 1 (February 1978), pp. 53–69. MR 500905 Zbl 0391.57004 article

A. J. Casson and C. M. Gordon: “On slice knots in dimension three,” pp. 39–53 in Algebraic and geometric topology (Stanford, CA, 2–21 August 1976), Part 2. Edited by R. J. Milgram. Proceedings of Symposia in Pure Mathematics 32. American Mathematical Society (Providence, RI), 1978. MR 520521 Zbl 0394.57008 incollection

Under the equivalence relation of concordance (sometimes called cobordism), smooth knots in the 3-sphere \( S^3 \) form an abelian group with respect to connected sum [Fox and Milnor 1966]. The knots \( K \) representing the zero class are precisely those which are slice, that is, satisfy \[ (S^3,K) = \partial(B^4,D) \] for some smooth 2-disc \( D \) in the 4-ball \( B^4 \), Now associated with any knot \( K \) and a Seifert surface \( V \) spanning \( K \), is a bilinear Seifert pairing \[ \theta_V: H_1(V)\times H_1(V)\to \mathbf{Z} \] [Seifert 1935; Levine 1969]. We say that \( K \) is algebraically slice if \( \theta_V \) vanishes on a subgroup of \( H_1(V) \) whose rank is \( \frac{1}{2} \)rank \( H_1(V) \) (this condition is independent of the choice of \( V \)). It is known that a necessary condition for \( K \) to be slice is that it be algebraically slice. Moreover, in higher (odd) dimensions analogous definitions may be made, and there the conditions are equivalent [Levine 1969]. We shall show that this is not the case in dimension 3.

Andrew J. Casson	Related
Richard James Milgram	Related

C. M. Gordon: “Some aspects of classical knot theory,” pp. 1–60 in Knot theory (Plans-sur-Bex, Switzerland, 1977). Edited by J. C. Hausmann. Lecture Notes in Mathematics 685. Springer (Berlin), 1978. This volume is dedicated to the memory of Christos Demetriou Papakyriakopoulos, 1914–1976. MR 521730 Zbl 0386.57002 incollection

Man’s fascination with knots has a long history, but they do not appear to have been considered from the mathematical point of view until the 19th century. Even then, the unavailability of appropriate methods meant that initial progress was, in a sense, slow, and at the beginning of the present century rigorous proofs had still not appeared. The arrival of algebraic-topological methods soon changed this, however, and the subject is now a highly-developed one, drawing on both algebra and geometry, and providing an opportunity for interplay between them.

The aim of the present article is to survey some topics in this theory of knotted circles in the 3-sphere. Completeness has not been attempted, nor is it necessarily the case that the topics chosen for discussion and the results mentioned are those that the author considers the most important: non-mathematical factors also contributed to the form of the article.

Christos Dimitriou Papakyriakopoulos	Related
Jean-Claude Hausmann	Related

C. M. Gordon: “Problems,” pp. 309–311 in Knot theory (Plans-sur-Bex, Switzerland, 1977). Edited by J. C. Hausmann. Lecture Notes in Mathematics 685. Springer (Berlin), 1978. This volume is dedicated to the memory of Christos Demetriou Papakyriakopoulos, 1914–1976. Zbl 0386.57010 incollection

Christos Dimitriou Papakyriakopoulos	Related
Jean-Claude Hausmann	Related

C. M. Gordon and R. A. Litherland: “On a theorem of Murasugi,” Pac. J. Math. 82 : 1 (1979), pp. 69–74. MR 549833 Zbl 0391.57003 article

Let \( l = k_1\cup k_2 \) be a 2-component link in \( S^3 \), with \( k_2 \) unknotted. The 2-fold cover of \( S^3 \) branched over \( k_2 \) is again \( S^3 \); let \( k_1^{(2)} \) be the inverse image of \( k_1 \), and suppose that \( k_1^{(2)} \) is connected. How are the signatures \( \sigma(k_1) \), \( \sigma(k_1^{(2)}) \) of the knots \( k_1 \) and \( k_1^{(2)} \) related? This question was considered (from a slightly different point of view) by Murasugi, who gave the following answer [Topology, 9 (1970), 283–298].

Theorem 1 (Murasugi). \[ \sigma(k_1^{(2)}) = \sigma(k_1) + \xi(l). \] Recall [Muasugi 1970] that the invariant \( \xi(l) \) is defined by first orienting \( l \), giving, an oriented link \( \overline{l} \), say, and then setting \[ \xi(l) = \sigma(l) + Lk(\overline{k_1},\overline{k_2}) ,\] where \( \sigma \) denotes signature and \( Lk \) linking number.

In the present note we shall give an alternative, more conceptual, proof of Theorem 1, and in fact obtain it as a special case of a considerably more general result.

C. M. Gordon: “Homology of groups of surfaces in the 4-sphere,” Math. Proc. Cambridge Philos. Soc. 89 : 1 (January 1981), pp. 113–117. MR 591977 Zbl 0454.57021 article

C. M. Gordon, R. A. Litherland, and K. Murasugi: “Signatures of covering links,” Can. J. Math. 33 : 2 (April 1981), pp. 381–394. MR 617628 Zbl 0469.57004 article

Richard Andrew Litherland	Related
Kunio Murasugi	Related

C. M. Gordon: “Ribbon concordance of knots in the 3-sphere,” Math. Ann. 257 : 2 (October 1981), pp. 157–170. MR 634459 Zbl 0451.57001 article

C. M. Gordon: “Dehn surgery and satellite knots,” Trans. Am. Math. Soc. 275 : 2 (1983), pp. 687–708. MR 682725 Zbl 0519.57005 article

J. H. Conway and C. M. Gordon: “Knots and links in spatial graphs,” J. Graph Theory 7 : 4 (1983), pp. 445–453. MR 722061 Zbl 0524.05028 article

A. J. Casson and C. M. Gordon: “A loop theorem for duality spaces and fibred ribbon knots,” Invent. Math. 74 : 1 (February 1983), pp. 119–137. MR 722728 Zbl 0538.57003 article

Four-manifold theory (Durham, NH, July 4–10, 1982). Edited by C. Gordon and R. Kirby. Contemporary Mathematics 35. American Mathematical Society (Providence, RI), 1984. MR 780574

C. M. Gordon and R. A. Litherland: “Incompressible surfaces in branched coverings,” Chapter 7, pp. 139–152 in The Smith conjecture (New York, 6–7 April 1979). Edited by J. W. Morgan and H. Bass. Pure and Applied Mathematics 112. Academic Press (Orlando, FL), 1984. MR 758466 Zbl 0599.57006 incollection

Our primary aim is to show that if \( \widetilde{\Sigma} \) is the \( n \)-fold cyclic branched covering, \( n > 1 \), of a prime knot \( K \) in a homotopy 3-sphere \( \Sigma \), such that \( \Sigma - K \) is irreducible and contains a nonperipheral incompressible surface, then \( \widetilde{\Sigma} \) is not a homotopy sphere. This will be achieved by using the equivariant loop theorem to show that \( \widetilde{\Sigma} \) contains an incompressible surface of positive genus. Actually, we shall work in the more general context of a regular branched covering of a link in an arbitrary closed, orientable 3-manifold, as this is needed for the study of noncyclic finite group actions on homotopy 3-spheres. Our main result is Theorem 1, which asserts that, under certain circumstances, an incompressible surface in the complement of a link gives rise to one in any regular branched covering of the link. The special case of incompressible tori is considered in Theorem 2; here the proof actually uses the Smith conjecture. Interpreting Theorem 2 in terms of hyperbolic structures, using Thurston’s uniformization theorem for Haken manifolds (see Thurston [1979] and Chapter V, this volume by Morgan) we obtain Corollary 2.1, which states that if a regular branched covering of a link is hyperbolic, then so is the complement of the link.

These results are stated in Section 2, which also contains the necessary definitions and terminology. The proofs of Theorems 1 and 2 and Corollary 2.1 are given in Section 3. Finally, in Section 4, we give an elementary proof of the equivariant loop theorem for involutions (Theorem 3). A similar proof has been given by Kim and Tollefson [1980].

John Willard Morgan	Related
Richard Andrew Litherland	Related
Hyman Bass	Related

C. M. Gordon and R. A. Litherland: “Incompressible planar surfaces in 3-manifolds,” Topology Appl. 18 : 2–3 (December 1984), pp. 121–144. MR 769286 Zbl 0554.57010 article

M. Culler, C. M. Gordon, J. Luecke, and P. B. Shalen: “Dehn surgery on knots,” Bull. Am. Math. Soc. (N.S.) 13 : 1 (July 1985), pp. 43–45. MR 788388 Zbl 0571.57008 article

John Edwin Luecke	Related
Peter B. Shalen	Related
Marc Edward Culler	Related

C. M. Gordon and J. M. Montesinos: “Fibred knots and disks with clasps,” Math. Ann. 275 : 3 (September 1986), pp. 405–408. MR 858286 Zbl 0578.57004 article

C. M. Gordon: “On the \( G \)-signature theorem in dimension four,” pp. 159–180 in À la recherche de la topologie perdue [In search of the lost topology]. Edited by L. Guillou and A. Marin. Progress in Mathematics 62. Birkhäuser (Boston), 1986. MR 900251 incollection

Alexis Marin	Related
Lucien Guillou	Related

A. J. Casson and C. M. Gordon: “Cobordism of classical knots,” pp. 181–199 in À la recherche de la topologie perdue [In search of the lost topology]. Edited by L. Guillou and A. Marin. Progress in Mathematics 62. Birkhäuser (Boston), 1986. With an appendix by P. M. Gilmer. MR 900252 incollection

Andrew J. Casson	Related
Alexis Marin	Related
Lucien Guillou	Related
Patrick M. Gilmer	Related

M. Culler, C. M. Gordon, J. Luecke, and P. B. Shalen: “Dehn surgery on knots,” Ann. of Math. (2) 125 : 2 (1987), pp. 237–300. MR 881270 Zbl 0633.57006 article

John Edwin Luecke	Related
Marc Edward Culler	Related

C. M. Gordon and J. Luecke: “Only integral Dehn surgeries can yield reducible manifolds,” Math. Proc. Cambridge Philos. Soc. 102 : 1 (July 1987), pp. 97–101. MR 886439 Zbl 0655.57500 article

Let \( K \) be a knot in \( S^3 \), and let \( K(a/b) \) denote the closed oriented 3-manifold obtained by \( a/b \)-Dehn surgery on \( K \). (We parametrize the Dehn surgeries on \( K \) by \( \mathbb{Q}\cap \{1/0\} \) as in [Rolfsen 1976]; in particular, \( K(1/0) = S^3 \).) If \( K \) is a \( (p,q) \)-cable knot, then \( K(pq) \) is always reducible (see Section 3), and it is conjectured by González-Acuña and Short in [1986] that these are the only examples where Dehn surgery on a knot in \( S^3 \) yields a reducible manifold. One of the results in this direction proved in [Rolfsen 1976] is that if \( \pi_1(K(a/b)) \) is a non-trivial free product then \( |b|\leq 5 \). We show that this may be sharpened to the assertion in the title.

A. J. Casson and C. M. Gordon: “Reducing Heegaard splittings,” Topology Appl. 27 : 3 (December 1987), pp. 275–283. MR 918537 Zbl 0632.57010 article

C. M. Gordon: “On primitive sets of loops in the boundary of a handlebody,” Topology Appl. 27 : 3 (1987), pp. 285–299. MR 918538 Zbl 0634.57007 article

M. Culler, C. M. Gordon, J. Luecke, and P. B. Shalen: “Correction: ‘Dehn surgery on knots’,” Ann. Math. (2) 127 : 3 (1988), pp. 663. Correction to an article published in Ann. Math. 125:2 (1987). MR 942524 Zbl 0645.57006 article

John Edwin Luecke	Related
Peter B. Shalen	Related
Marc Edward Culler	Related

C. M. Gordon and J. Luecke: “Knots are determined by their complements,” J. Am. Math. Soc. 2 : 2 (1989), pp. 371–415. A much shorter version of this article was published in Bull. Am. Math. Soc. 20:1 (1989). MR 965210 Zbl 0678.57005 article

C. M. Gordon and J. Luecke: “Knots are determined by their complements,” Bull. Am. Math. Soc. (N.S.) 20 : 1 (January 1989), pp. 83–87. A much longer version of this article was published in J. Am. Math. Soc. 2:2 (1989). MR 972070 Zbl 0672.57009 article

C. M. Gordon: “Combinatorial methods in knot theory,” pp. 1–23 in Algebra and topology 1990 (Taejon, South Korea, 8–11 August 1990). Edited by S. H. Bae and G. T. Jin. Korea Advanced Institute of Science and Technology, 1990. MR 1098718 Zbl 0743.57002 incollection

Sunghan Bae	Related
Gyo Taek Jin	Related

C. M. Gordon: “Dehn surgery on knots,” pp. 631–642 in Proceedings of the International Congress of Mathematicians (Kyoto, 21–29 August 1990), vol. 1. Edited by I. Satake. Springer (Tokyo), 1991. MR 1159250 Zbl 0743.57008 incollection

C. M. Gordon: “On spines of 3-manifolds with boundary,” J. Austral. Math. Soc. Ser. A 55 : 1 (1993), pp. 132–136. MR 1231699 Zbl 0809.57008 article

C. M. Gordon and J. Luecke: “Links with unlinking number one are prime,” Proc. Am. Math. Soc. 120 : 4 (1994), pp. 1271–1274. MR 1195721 Zbl 0818.57008 article

Geometric topology (Haifa, Israel, 10–16 June 1992). Edited by C. Gordon, Y. Moriah, and B. Wajnryb. Contemporary Mathematics 164. American Mathematical Society (Providence, RI), 1994. MR 1282749 Zbl 0794.00024 book

Dov Bronislaw Wajnryb	Related
Yoav Moriah	Related

C. M. Gordon: “Some embedding theorems and undecidability questions for groups,” pp. 105–110 in Combinatorial and geometric group theory (Edinburgh, 1993). Edited by A. J. Duncan, N. D. Gilbert, and J. Howie. London Math. Soc. Lecture Note Ser. 204. Cambridge Univ. Press, 1995. MR 1320278 Zbl 0843.20027 incollection

Nicholas David Gilbert	Related
Andrew James Duncan	Related
James Howie	Related

C. M. Gordon and A. W. Reid: “Tangle decompositions of tunnel number one knots and links,” J. Knot Theory Ramif. 4 : 3 (1995), pp. 389–409. MR 1347361 Zbl 0841.57012 article

C. M. Gordon and J. Luecke: “Dehn surgeries on knots creating essential tori, I,” Comm. Anal. Geom. 3 : 3–4 (1995), pp. 597–644. MR 1371211 Zbl 0865.57015 article

C. M. Gordon and J. Luecke: “Reducible manifolds and Dehn surgery,” Topology 35 : 2 (April 1996), pp. 385–409. MR 1380506 Zbl 0859.57016 article

“Mathematics people,” Notices Am. Math. Soc. 46 : 6 (June–July 1996). Announces Gordon’s award of a Guggenheim Fellowship. article

C. M. Gordon: “Combinatorial methods in Dehn surgery,” pp. 263–290 in Lectures at Knots ’96 (Tokyo, 22–31 July 1996). Edited by S. Suzuki. Series on Knots and Everything 15. World Scientific (River Edge, NJ), 1997. MR 1474525 Zbl 0940.57022 ArXiv math/9704223 incollection

This is an expository paper, in which we give a summary of some of the joint work of John Luecke and the author on Dehn surgery. We consider the situation where we have two Dehn fillings \( M(\alpha) \) and \( M(\beta) \) on a given 3-manifold \( M \), each containing a surface that is either essential or a Heegaard surface. We show how a combinatorial analysis of the graphs of intersection of the two corresponding punctured surfaces in \( M \) enables one to find faces of these graphs which give useful topological information about \( M(\alpha) \) and \( M(\beta) \), and hence, in certain cases, good upper bounds on the intersection number \( \Delta(\alpha,\beta) \) of the two filling slopes.

C. M. Gordon: “Boundary slopes of punctured tori in 3-manifolds,” Trans. Am. Math. Soc. 350 : 5 (1998), pp. 1713–1790. MR 1390037 Zbl 0896.57011 article

Let \( M \) be an irreducible 3-manifold with a torus boundary component \( T \), and suppose that \( r \), \( s \) are the boundary slopes on \( T \) of essential punctured tori in \( M \), with their boundaries on \( T \). We show that the intersection number \( \Delta(r,s) \) of \( r \) and \( s \) is at most 8. Moreover, apart from exactly four explicit manifolds \( M \), which contain pairs of essential punctured tori realizing \( \Delta(r,s) = 8 \), 8, 7 and 6 respectively, we have \( \Delta(r,s)\leq 5 \). It follows immediately that if \( M \) is atoroidal, while the manifolds \( M(r) \), \( M(s) \) obtained by \( r \)-and \( s \)-Dehn filling on \( M \) are toroidal, then \( \Delta(r,s)\leq 8 \), and \( \Delta(r,s)\leq 5 \) unless \( M \) is one of the four examples mentioned above.

Let \( \mathcal{H}_0 \) be the class of 3-manifolds \( M \) such that \( M \) is irreducible, atoroidal, and not a Seifert fibre space. By considering spheres, disks and annuli in addition to tori, we prove the following. Suppose that \( M\in\mathcal{H}_0 \), where \( \partial M \) has a torus component \( T \), and \( \partial M - T \neq \varnothing \). Let \( r \), \( s \) be slopes on \( T \) such that \( M(r) \), \( M(s)\notin\mathcal{H}_0 \). Then \( \Delta(r,s)\leq 5 \). The exterior of the Whitehead sister link shows that this bound is best possible.

C. M. Gordon: “Dehn filling: A survey,” pp. 129–144 in Knot theory (Warsaw, 13 July–17 August 1995). Edited by V. F. R. Jones, J. Kania-Bartoszyńska, J. Przytycki, P. Traczyk, and V. G. Turaev. Banach Center Publications 42. Polish Academy of Sciencies, Institute of Mathematics (Warsaw), 1998. MR 1634453 Zbl 0916.57016 incollection

In this paper we give a brief survey of the present state of knowledge on exceptional Dehn fillings on 3-manifolds with torus boundary.

For our discussion, it is necessary to first give a quick overview of what is presently known, and what is conjectured, about the structure of 3-manifolds. This is done in Section 2. In Section 3 we summarize the known bounds on the distances between various kinds of exceptional Dehn fillings, and compare these with the distances that arise in known examples. In Section 4 we make some remarks on the special case of complements of knots in the 3-sphere.

We have chosen to phrase questions as conjectures; this gives them a certain edge and perhaps increases the likelihood that someone will try to (dis)prove them. Incidentally, no particular claim is made for unattributed conjectures; most of them are lore to the appropriate folk.

Related survey articles are [Gordon 1991] and [Luecke 1995].

Vaughan F. R. Jones	Related	Volume
Józef Henryk Przytycki	Related
Joanna Kania-Bartoszyńska	Related
Paweł Traczyk	Related
Vladimir G. Turaev	Related

C. M. Gordon: “Toroidal Dehn surgeries on knots in lens spaces,” Math. Proc. Cambridge Philos. Soc. 125 : 3 (1999), pp. 433–440. A corrigendum to this article was published in Math. Proc. Cambridge Philos. Soc. 128:2 (2000). MR 1656809 Zbl 0926.57014 article

Let \( M \) be a compact, orientable 3-manifold with \( \partial M \) a torus. If \( r \) is a slope on \( \partial M \) (the isotopy class of an unoriented essential simple loop), then we can form the closed 3-manifold \( M(r) \) by gluing a solid torus \( V_r \) to \( M \) along their boundaries in such a way that \( r \) bounds a disc in \( V_r \). We say that \( M(r) \) is obtained from \( M \) by \( r \)-Dehn filling.

Assume now that \( M \) contains no essential sphere, disc, torus or annulus. Then, by Thurston’s Geometrization Theorem for Haken manifolds [Thurston 1978, 1982], \( M \) is hyperbolic, in the sense that \( \operatorname{int} M \) has a complete hyperbolic structure of finite volume. Furthermore, \( M(r) \) is hyperbolic for all but finitely many \( r \) [Thurston 1978, 1982] and the precise nature of the set of exceptional slopes \[ E(M) = \{r: M(r) \text{ is not hyperbolic}\} \] has been the subject of a considerable amount of investigation. The maximal observed value of \( e(M) = |E(M)| \) (the cardinality of \( E(M) \)) is 10, realized, apparently uniquely, by the exterior of the figure eight knot [Thurston 1978].

Let \( \Delta(r_1,r_2) \) denote as usual the minimal geometric intersection number of two slopes \( r_1 \) and \( r_2 \). If \( \mathscr{S} \) is any set of slopes, then clearly any upper bound for \[ \Delta(\mathscr{S}) = \max\{\Delta(r_1,r_2): r_1, r_2 \in \mathscr{S}\} \] gives one for \( |\mathscr{S}| \). For example, one can check (using [Gordon and Litherland 1984, Lemma 2.1]) that for \( 1\leq \Delta(\mathscr{S})\leq 10 \), the maximum values of \( |\mathscr{S}| \) are as given in Table 1.

In particular, any upper bound for \( \Delta(M) = \Delta(E(M)) \) gives a corresponding bound for \( e(M) \). (The maximal observed value of \( \Delta(M) \) is 8, realized by the figure eight knot exterior and the figure eight sister manifold [Thurston 1978; Hodgson and Weeks 2000].)

If \( M(r) \) is not hyperbolic, then it is either reducible (contains an essential sphere), toroidal (contains an essential torus), a small Seifert fibre space (one with base \( S^2 \) and at most three singular fibres), or a counterexample to the Geometrization Conjecture [Thurston 1978, 1982]. A survey of the presently known upper bounds on the distances \( \Delta(r_1,r_2) \) between various classes of exceptional slopes \( r_1 \) and \( r_2 \), and the maximal values realized by known examples, is given in [Gordon 1998]. (See also [Wu 1998] for a discussion of the additional cases that arise when \( M \) has more than one boundary component.) In the present note we prove the following theorem, which deals with one further pair of possibilities.

C. M. Gordon and Y.-Q. Wu: “Toroidal and annular Dehn fillings,” Proc. London Math. Soc. (3) 78 : 3 (1999), pp. 662–700. MR 1674841 Zbl 1024.57020 ArXiv math/9705221 article

C. M. Gordon: “3-dimensional topology up to 1960,” pp. 449–489 in History of topology. Edited by I. M. James. North-Holland (Amsterdam), 1999. MR 1674921 Zbl 0956.57005 incollection

In this paper we discuss the development of 3-dimensional topology, from its beginnings in the 1880’s, up until roughly 1960. The decision to stop at 1960 was more or less arbitrary, and indeed we will sometimes briefly describe developments beyond that date. Our account is very much in the nature of a survey of the literature (an internal history, if you will), an approach which is feasible because the literature is so finite. (This continues to be true through the 1960’s, when the number of people working in 3-dimensional topology was still relatively small. During the last twenty years or so, not only has the actual literature grown tremendously, but the number of major themes in the subject has also increased.)

C. M. Gordon and J. Luecke: “Toroidal and boundary-reducing Dehn fillings,” Topology Appl. 93 : 1 (April 1999), pp. 77–90. MR 1684214 Zbl 0926.57019 article

C. M. Gordon: “Small surfaces and Dehn filling,” pp. 177–199 in Proceedings of the Kirbyfest (Berkeley, CA, 22–26 June 1998). Edited by J. Hass and M. Scharlemann. Geometry & Topology Monographs 2. Geometry & Topology Publications (Coventry, UK), 1999. Dedicated to Rob Kirby on the occasion of his 60th birthday. MR 1734408 Zbl 0948.57014 ArXiv math/9911251 incollection

Joel Hass	Related	Volume
Martin Scharlemann	Related	Volume
Robion C. Kirby	Related	Volume

C. M. Gordon, Y.-Q. Wu, and X. Zhang: “Non-integral toroidal surgery on hyperbolic knots in \( S^3 \),” Proc. Am. Math. Soc. 128 : 6 (2000), pp. 1869–1879. MR 1644022 Zbl 0947.57022 ArXiv math/9704222 article

Ying-Qing Wu	Related
Xingru Zhang	Related

C. M. Gordon: “Corrigendum: ‘Toroidal Dehn surgeries on knots in lens spaces’,” Math. Proc. Cambridge Philos. Soc. 128 : 2 (2000), pp. 381. Corrigendum to an article published in Math. Proc. Cambridge Philos. Soc. 125:3 (1999). MR 1735297 Zbl 0953.57011 article

C. M. Gordon and Y.-Q. Wu: “Annular and boundary reducing Dehn fillings,” Topology 39 : 3 (May 2000), pp. 531–548. MR 1746907 Zbl 0944.57014 ArXiv math/9810126 article

Let \( M \) be a simple 3-manifold, i.e. one that contains no essential sphere, disk, annulus or torus, with a torus boundary component \( \partial_0 M \). One is interested in obtaining upper bounds for the distance (intersection number) \( \Delta(\alpha,\beta) \) between slopes \( \alpha \), \( \beta \) on \( \delta_0 M \) such that Dehn filling \( M \) along \( \alpha \), \( \beta \) produces manifolds \( M(\alpha) \), \( M(\beta) \) that are not simple. There are ten cases, according to whether \( M(\alpha) \) (\( M(\beta) \)) contains an essential sphere, disk, annulus or torus. Here we show that if \( M(\alpha) \) contains an essential annulus and \( M(\beta) \) contains an essential disk then \( \Delta(\alpha,\beta)\leq 2 \). This completes the determination of upper bounds for \( \Delta(\alpha,\beta) \) in all ten cases.

C. M. Gordon and J. Luecke: “Dehn surgeries on knots creating essential tori, II,” Comm. Anal. Geom. 8 : 4 (2000), pp. 671–725. MR 1792371 Zbl 0970.57010 article

C. M. Gordon and Y.-Q. Wu: “Annular Dehn fillings,” Comment. Math. Helv. 75 : 3 (2000), pp. 430–456. MR 1793797 Zbl 0964.57020 ArXiv math/0010327 article

Knots in Hellas ’98: Proceedings of the international conference on knot theory an its ramifications (Delphi, Greece, 7–15 August 1998), vol. 1. Edited by C. M. Gordon, V. F. R. Jones, L. H. Kauffman, S. Lambropoulou, and J. H. Przytycki. Series on Knots and Everything 24. World Scientific (River Edge, NJ), 2000. Volume 2 was published as J. Knot Theory Ramif. 10:2 (2001). Volume 3 was published as J. Knot Theory Ramif. 10:5 (2001). MR 1865695 Zbl 0959.00034 book

Vaughan F. R. Jones	Related	Volume
Józef Henryk Przytycki	Related
Louis H. Kauffman	Related
Sofia S. F. Lambropoulou	Related

S. Boyer, C. M. Gordon, and X. Zhang: “Dehn fillings of large hyperbolic 3-manifolds,” J. Diff. Geom. 58 : 2 (2001), pp. 263–308. MR 1913944 Zbl 1042.57007 article

Let \( M \) be a compact, connected, orientable, hyperbolic 3-manifold whose boundary is a torus and which contains an essential closed surface \( S \). It is conjectured that 5 is an upper bound for the distance between two slopes on \( \partial M \) whose associated fillings are not hyperbolic manifolds. In this paper we verify the conjecture when the first Betti number of \( M \) is at least 2 by showing that given a pseudo-Anosov mapping class \( f \) of a surface and an essential simple closed curve \( \gamma \) in the surface, then 5 is an upper bound for the diameter of the set of integers \( n \) for which the composition of \( f \) with the \( n \)th power of a Dehn twist along \( \gamma \) is not pseudo-Anosov. This sharpens an inequality of Albert Fathi. For large manifolds \( M \) of first Betti number 1 we obtain partial results. Set \[ \mathcal{C}(S) = \bigl\{\text{slopes } r \mid \ker(\pi_1(S)\to \pi_1(M(r))) \neq \{1\} \bigr\}. \] A singular slope for \( S \) is a slope \( r_0\in \mathcal(S) \) such that any other slope in \( \mathcal{C}(S) \) is at most distance 1 from \( r_0 \). We prove that the distance between two exceptional filling slopes is at most 5 if either (i) there is a closed essential surface \( S \) in \( M \) with \( \mathcal{C}(S) \) finite, or (ii) there are singular slopes \( r_1\neq r_2 \) for closed essential surfaces \( S_1 \), \( S_2 \) in \( M \).

Steven Patrick Boyer	Related
Xingru Zhang	Related

Knots 2000 Korea (Volume 1) (Yongpyong, South Korea, 31 July–5 August 2000), published as J. Knot Theor. Ramif. 11 : 3. Issue edited by J. S. Birman, C. M. Gordon, G. T. Jin, L. H. Kauffman, A. Kawauchi, K. H. Ko, J. P. Levine, and Y. Matsumoto. World Scientific (Singapore), 2002. Zbl 0995.00510 book

Ki Hyoung Ko	Related
Yukio Matsumoto	Related
Akio Kawauchi	Related
Louis H. Kauffman	Related
Joan S. Birman	Related	Volume
Jerome Paul Levine	Related
Gyo Taek Jin	Related

Knots 2000 Korea (Volume 2) (Yongpyong, South Korea, 31 July–5 August 2000), published as J. Knot Theory Ramif. 11 : 4. Issue edited by J. S. Birman, C. M. Gordon, G. T. Jin, L. H. Kauffman, A. Kawauchi, K. H. Ko, J. P. Levine, and Y. Matsumoto. World Scientific (Singapore), June 2002. MR 1915489 Zbl 0995.00511 book

Ki Hyoung Ko	Related
Yukio Matsumoto	Related
Akio Kawauchi	Related
Louis H. Kauffman	Related
Joan S. Birman	Related	Volume
Jerome Paul Levine	Related
Gyo Taek Jin	Related

C. M. Gordon: “Links and their complements,” pp. 71–82 in Topology and geometry: Commemorating SISTAG (Singapore, 2–6 July 2001). Edited by A. J. Berrick, M. C. Leung, and X. Xu. Contemporary Mathematics 314. American Mathematical Society (Providence, RI), 2002. MR 1941623 Zbl 1027.57006 incollection

Xingwang Xu	Related
A. J. Berrick	Related
Man-Cheung Leung	Related

Knots 2000 Korea (Volume 3) (Yongpyong, South Korea, 31 July–5 August 2000), published as J. Knot Theor. Ramif. 11 : 6. Issue edited by J. S. Birman, C. M. Gordon, G. T. Jin, L. H. Kauffman, A. Kawauchi, K. H. Ko, J. P. Levine, and Y. Matsumoto. World Scientific (Singapore), September 2002. Zbl 1018.00508 book

Ki Hyoung Ko	Related
Yukio Matsumoto	Related
Akio Kawauchi	Related
Louis H. Kauffman	Related
Joan S. Birman	Related	Volume
Jerome Paul Levine	Related
Gyo Taek Jin	Related

C. M. Gordon: “On the reversibility of twist-spun knots,” J. Knot Theory Ramif. 12 : 7 (2003), pp. 893–897. MR 2017959 Zbl 1048.57004 article

C. M. Gordon: “Dehn filling,” pp. 41–59 in Low dimensional topology (Beijing, 1998–1999). Edited by B. Li, S. Wang, and X. Zhao. New Studies in Advanced Mathematics 3. International Press (Somerville, MA), 2003. MR 2052245 Zbl 1044.57005 incollection

Xuezhi Zhao	Related
Benghe Li	Related
Shicheng Wang	Related

C. M. Gordon, D. D. Long, and A. W. Reid: “Surface subgroups of Coxeter and Artin groups,” J. Pure Appl. Algebra 189 : 1–3 (May 2004), pp. 135–148. MR 2038569 Zbl 1057.20031 article

Darren David Long	Related
Alan William Reid	Related

C. M. Gordon and J. Luecke: “Non-integral toroidal Dehn surgeries,” Comm. Anal. Geom. 12 : 1–2 (2004), pp. 417–485. MR 2074884 Zbl 1062.57006 article

If we perform a non-trivial Dehn surgery on a hyperbolic knot in the 3- sphere, the result is usually a hyperbolic 3-manifold. However, there are exceptions: there are hyperbolic knots with surgeries that give lens spaces [Berge], small Seifert fiber spaces [Bleiler and Hodgson 1992; Dean 2003; Eudave-Muñoz 2002; Mattman et al. 2006], and toroidal manifolds, that is, manifolds containing (embedded) incompressible tori [Eudave-Muñoz 1997, 2002]. In particular, Eudave-Muñoz [1997] has explicitly described an infinite family of hyperbolic knots \( k(l,m,n,p) \), each of which has a specific half-integral toroidal surgery. (These are the only known examples of non-trivial, non-integral, non-hyperbolic surgeries on hyperbolic knots.) Here we show that these knots are the only hyperbolic knots with non-integral toroidal surgeries.

Proceedings of the Casson Fest (Fayetteville, AR, 10–12 April 2003 and Austin, TX, 19–21 May 2003), published as Geom. Topol. Monogr. 7. Issue edited by C. Gordon and Y. Rieck. Geometry & Topology Publications (Coventry, UK), 2004. MR 2169266 Zbl 1066.57002 book

Andrew J. Casson	Related
Yo'av Rieck	Related

C. M. Gordon: “Artin groups, 3-manifolds and coherence,” pp. 193–198 in FICOFEST: A conference in low dimensional topology (Mérida, Mexico, 9–13 December 2002), published as Bol. Soc. Mat. Mexicana (3) 10 : Special issue. Birkhäuser (Basel), 2004. Dedicated to Fico on the occasion of his 60th birthday. MR 2199348 Zbl 1100.57001 incollection

Topologie (Oberwolfach, Germany, 5–11 September 2004), published as Oberwolfach Rep. 1 : 4. Issue edited by C. Gordon, W. Lück, and R. Oliver. 2004. Report 44. Zbl 1078.55500 book

Robert Oliver	Related
Wolfgang Lück	Related

C. M. Gordon and J. Luecke: “Knots with unknotting number 1 and essential Conway spheres,” Algebr. Geom. Topol. 6 (2006), pp. 2051–2116. MR 2263059 Zbl 1129.57009 ArXiv math.GT/0601265 article

For a knot \( K \) in \( S^3 \), let \( \mathbf{T}(K) \) be the characteristic toric sub-orbifold of the orbifold \( (S^3,K) \) as defined by Bonahon–Siebenmann. If \( K \) has unknotting number one, we show that an unknotting arc for \( K \) can always be found which is disjoint from \( \mathbf{T}(K) \), unless either \( K \) is an EM-knot (of Eudave-Muñoz) or \( (S^3,K) \) contains an EM-tangle after cutting along \( \mathbf{T}(K) \). As a consequence, we describe exactly which large algebraic knots (ie, algebraic in the sense of Conway and containing an essential Conway sphere) have unknotting number one and give a practical procedure for deciding this (as well as determining an unknotting crossing). Among the knots up to 11 crossings in Conway’s table which are obviously large algebraic by virtue of their description in the Conway notation, we determine which have unknotting number one. Combined with the work of Ozsváth–Szabó, this determines the knots with 10 or fewer crossings that have unknotting number one. We show that an alternating, large algebraic knot with unknotting number one can always be unknotted in an alternating diagram.

As part of the above work, we determine the hyperbolic knots in a solid torus which admit a non-integral, toroidal Dehn surgery. Finally, we show that having unknotting number one is invariant under mutation.

Topologie (Oberwolfach, Germany, 17–23 September 2006), published as Oberwolfach Rep. 3 : 4. Issue edited by C. Gordon, W. Lück, and R. Oliver. 2006. Zbl 1177.57002 book

Robert Oliver	Related
Wolfgang Lück	Related

Workshop on Heegaard splittings (Haifa, Israel, 11–16 July 2005). Edited by C. Gordon and Y. Moriah. Geometry & Topology Monographs 12. Geometry & Topology Publications (Coventry, UK), 2007. MR 2404079 Zbl 1133.57002 book

C. M. Gordon: “Problems,” pp. 401–411 in Workshop on Heegaard splittings (Haifa, Israel, 11–16 July 2005). Edited by C. Gordon and Y. Moriah. Geometry & Topology Monographs 12. Geometry & Topology Publications (Coventry, UK), 2007. MR 2408256 Zbl 1135.57302 ArXiv 0904.0242 incollection

C. M. Gordon and Y.-Q. Wu: Toroidal Dehn fillings on hyperbolic 3-manifolds, vol. 194. Memoirs of the American Mathematical Society 909. American Mathematical Society (Providence, RI), 2008. MR 2419168 Zbl 1166.57014 ArXiv math/0512038 book

Topologie (Oberwolfach, Germany, 14–20 September 2008), published as Oberwolfach Rep. 5 : 4. Issue edited by C. Gordon, R. Oliver, and T. Schick. 2008. Zbl 1177.55004 book

Thomas Schick	Related
Robert Oliver	Related

S. Boyer, C. M. Gordon, and X. Zhang: “Reducible and finite Dehn fillings,” J. Lond. Math. Soc. (2) 79 : 1 (2009), pp. 72–84. MR 2472134 Zbl 1162.57015 ArXiv 0710.3786 article

Steven Patrick Boyer	Related
Xingru Zhang	Related

C. Gordon: “Dehn surgery and 3-manifolds,” pp. 21–71 in Low dimensional topology. Edited by T. S. Mrowka and P. S. Ozsváth. IAS/Park City Mathematics Series 15. American Mathematical Society (Providence, RI), 2009. MR 2503492 Zbl 1194.57003 incollection

These notes are somewhat expanded versions of the six lectures given at the 2006 Park City Mathematics Institute Graduate Summer School. The main focus of the lectures was exceptional Dehn surgeries on knots, and, more generally, exceptional Dehn fillings on hyperbolic 3-manifolds.

In Lecture 1 we describe the crude classification of 3-manifolds that comes from cutting them along essential surfaces of non-negative Euler characteristic, and say what this means for exteriors of knots. In Lecture 2 we discuss Dehn surgery on knots, and in particular describe a construction, framed surgery on knots on surfaces, which is the source of many examples of exceptional Dehn surgeries. Lecture 3 summarizes some facts and conjectures about exceptional Dehn surgeries on knots. In Lecture 4 we introduce rational tangle fillings on tangles; these induce Dehn fillings on the double branched cover of the tangle. Tangle fillings have the advantage that they are easy to visualize, and although they impose a symmetry on the manifold in question, nevertheless it turns out that many examples of exceptional Dehn fillings arise in this way. Lecture 5 gives more examples of exceptional Dehn fillings derived from tangles. In Lecture 6 we discuss some classification results about exceptional Dehn fillings; many of these take the form that a hyperbolic 3-manifold has a pair of non-hyperbolic Dehn fillings of a particular kind if and only if it is the double branched cover of one of a certain explicit family of tangles. We conclude with a sketch of the proof of one of these classification results, describing in particular how one shows that the fillings under consideration arise from tangle fillings.

Peter Steven Ozsváth	Related
Tomasz Stanislaw Mrowka	Related

C. Gordon and H. Wilton: “On surface subgroups of doubles of free groups,” J. Lond. Math. Soc. (2) 82 : 1 (August 2010), pp. 17–31. MR 2669638 Zbl 1205.20049 ArXiv 0902.3693 article

F. González-Acuña, C. M. Gordon, and J. Simon: “Unsolvable problems about higher-dimensional knots and related groups,” Enseign. Math. (2) 56 : 1–2 (2010), pp. 143–171. MR 2674857 Zbl 1213.57004 ArXiv 0908.4009 article

Francisco Javier González-Acuña	Related
Jonathan Kalman Simon	Related

C. Gordon and F. Rodriguez-Villegas: “On the divisibility of \( \#\mathrm{Hom}(\Gamma,G) \) by \( |G| \),” J. Algebra 350 : 1 (January 2012), pp. 300–307. MR 2859888 Zbl 1260.20066 ArXiv 1105.6066 article

C. M. Gordon: “Exceptional Dehn filling,” pp. 124–134 in Introductory lectures on knot theory (Trieste, Italy, 11–29 May 2009). Edited by L. H. Kauffman, S. Lambropoulou, S. Jablan, and J. H. Przytycki. Series on Knots and Everything 46. World Scientific (Hackensack, NJ), 2012. MR 2885233 Zbl 1256.57016 incollection

Slavik Vlado Jablan	Related
Józef Henryk Przytycki	Related
Louis H. Kauffman	Related
Sofia S. F. Lambropoulou	Related

S. Boyer, C. M. Gordon, and X. Zhang: “Characteristic submanifold theory and toroidal Dehn filling,” Adv. Math. 230 : 4–6 (July–August 2012), pp. 1673–1737. MR 2927352 Zbl 1248.57004 ArXiv 1104.3321 article

The exceptional Dehn filling conjecture of the second author concerning the relationship between exceptional slopes \( \alpha \) and \( \beta \) on the boundary of a hyperbolic knot manifold \( M \) has been verified in all cases other than small Seifert filling slopes. In this paper, we verify it when \( \alpha \) is a small Seifert filling slope and \( \beta \) is a toroidal filling slope in the generic case where \( M \) admits no punctured-torus fiber or semi-fiber, and there is no incompressible torus in \( M(\beta) \) which intersects \( \partial M \) in one or two components. Under these hypotheses we show that \( \Delta(\alpha,\beta)\leq 5 \). Our proof is based on an analysis of the relationship between the topology of \( M \), the combinatorics of the intersection graph of an immersed disk or torus in \( M(\alpha) \) , and the two sequences of characteristic subsurfaces associated to an essential punctured torus properly embedded in \( M \).

Steven Patrick Boyer	Related
Xingru Zhang	Related

D. Calegari and C. Gordon: “Knots with small rational genus,” Comment. Math. Helv. 88 : 1 (2013), pp. 85–130. MR 3008914 Zbl 1275.57019 ArXiv 0912.1843 article

If \( K \) is a rationally null-homologous knot in a 3-manifold \( M \), the rational genus of \( K \) is the infimum of \( -\chi(S)/2p \) over all embedded orientable surfaces \( S \) in the complement of \( K \) whose boundary wraps \( p \) times around \( K \) for some \( p \) (hereafter: \( S \) is a \( p \)-Seifert surface for \( K \)). Knots with very small rational genus can be constructed by “generic” Dehn filling, and are therefore extremely plentiful. In this paper we show that knots with rational genus less than \( 1/402 \) are all geometric — i.e. they may be isotoped into a special form with respect to the geometric decomposition of \( M \) — and give a complete classification. Our arguments are a mixture of hyperbolic geometry, combinatorics, and a careful study of the interaction of small \( p \)-Seifert surfaces with essential subsurfaces in \( M \) of non-negative Euler characteristic.

S. Boyer, C. M. Gordon, and L. Watson: “On L-spaces and left-orderable fundamental groups,” Math. Ann. 356 : 4 (2013), pp. 1213–1245. MR 3072799 Zbl 1279.57008 ArXiv 1107.5016 article

Examples suggest that there is a correspondence between L-spaces and three-manifolds whose fundamental groups cannot be left-ordered. In this paper we establish the equivalence of these conditions for several large classes of manifolds. In particular, we prove that they are equivalent for any closed, connected, orientable, geometric three-manifold that is non-hyperbolic, a family which includes all closed, connected, orientable Seifert fibred spaces. We also show that they are equivalent for the twofold branched covers of non-split alternating links. To do this we prove that the fundamental group of the twofold branched cover of an alternating link is left-orderable if and only if it is a trivial link with two or more components. We also show that this places strong restrictions on the representations of the fundamental group of an alternating knot complement with values in \( \operatorname{Homeo_+}(S^1) \).

Liam Watson	Related
Steven Patrick Boyer	Related

K. L. Baker, C. Gordon, and J. Luecke: “Obtaining genus 2 Heegaard splittings from Dehn surgery,” Algebr. Geom. Topol. 13 : 5 (2013), pp. 2471–2634. MR 3116298 Zbl 1294.57013 article

John Edwin Luecke	Related
Kenneth Lee Baker	Related

S. Boyer, C. M. Gordon, and X. Zhang: “Dehn fillings of knot manifolds containing essential once-punctured tori,” Trans. Am. Math. Soc. 366 : 1 (2014), pp. 341–393. MR 3118399 Zbl 1290.57005 ArXiv 1109.5151 article

In this paper we study exceptional Dehn fillings on hyperbolic knot manifolds which contain an essential once-punctured torus. Let \( M \) be such a knot manifold and let \( \beta \) be the boundary slope of such an essential once-punctured torus. We prove that if Dehn filling \( M \) with slope \( \alpha \) produces a Seifert fibred manifold, then \( \Delta(\alpha,\beta )\leq 5 \). Furthermore we classify the triples \( (M;\alpha,\beta) \) when \( \Delta(\alpha,\beta)\geq 4 \). More precisely, when \( \Delta(\alpha,\beta)=5 \), then \( M \) is the (unique) manifold \( \textit{Wh}(-3/2) \) obtained by Dehn filling one boundary component of the Whitehead link exterior with slope \( -3/2 \), and \( (\alpha,\beta) \) is the pair of slopes \( (-5,0) \). Further, \( \Delta(\alpha,\beta)=4 \) if and only if \( (M;\alpha,\beta) \) is the triple \[ \Bigl(\textit{Wh}\bigl(\tfrac{-2n\pm 1}{n}\bigr);-4,0\Bigr) \] for some integer \( n \) with \( |n| > 1 \). Combining this with known results, we classify all hyperbolic knot manifolds \( M \) and pairs of slopes \( (\beta,\gamma) \) on \( \partial M \) where \( \beta \) is the boundary slope of an essential once-punctured torus in \( M \) and \( \gamma \) is an exceptional filling slope of distance 4 or more from \( \beta \). Refined results in the special case of hyperbolic genus one knot exteriors in \( S^3 \) are also given.

Steven Patrick Boyer	Related
Xingru Zhang	Related

C. Gordon and T. Lidman: “Taut foliations, left-orderability, and cyclic branched covers,” Acta Math. Vietnam. 39 : 4 (December 2014), pp. 599–635. A corrigendum to this article was published in Acta Math. Vietnam. 42:4 (2014). MR 3292587 Zbl 1310.57023 ArXiv 1406.6718 article

Y. Koda and M. Ozawa: “Essential surfaces of non-negative Euler characteristic in genus two handlebody exteriors,” Trans. Am. Math. Soc. 367 : 4 (2015), pp. 2875–2904. With an appendix by Cameron Gordon. MR 3301885 Zbl 1310.57015 ArXiv 1212.5928 article

Makoto Ozawa	Related
Yuya Koda	Related

K. L. Baker, C. Gordon, and J. Luecke: “Bridge number, Heegaard genus and non-integral Dehn surgery,” Trans. Am. Math. Soc. 367 : 8 (2015), pp. 5753–5830. MR 3347189 Zbl 1329.57017 ArXiv 1202.0263 article

John Edwin Luecke	Related
Kenneth Lee Baker	Related

K. L. Baker, C. Gordon, and J. Luecke: “Bridge number and integral Dehn surgery,” Algebr. Geom. Topol. 16 : 1 (2016), pp. 1–40. MR 3470696 Zbl 1339.57005 ArXiv 1303.7018 article

In a 3-manifold \( M \), let \( K \) be a knot and \( \hat{R} \) be an annulus which meets \( K \) transversely. We define the notion of the pair \( (R,\hat{K}) \) being caught by a surface \( Q \) in the exterior of the link \( K\cup\partial\hat{R} \). For a caught pair \( (\hat{R},K) \), we consider the knot \( K^n \) gotten by twisting \( K \) \( n \) times along \( \hat{R} \) and give a lower bound on the bridge number of \( K^n \) with respect to Heegaard splittings of \( M \)–as a function of \( n \), the genus of the splitting, and the catching surface \( Q \). As a result, the bridge number of \( K^n \) tends to infinity with \( n \). In application, we look at a family of knots \( \{K^n\} \) found by Teragaito that live in a small Seifert fiber space \( M \) and where each \( K^n \) admits a Dehn surgery giving \( S^3 \). We show that the bridge number of \( K^n \) with respect to any genus 2 Heegaard splitting of \( M \) tends to infinity with \( n \). This contrasts with other work of the authors as well as with the conjectured picture for knots in lens spaces that admit Dehn surgeries giving \( S^3 \).

John Edwin Luecke	Related
Kenneth Lee Baker	Related

C. M. Gordon: “Riley’s conjecture on \( \mathrm{SL}(2,\mathbb{R}) \) representations of 2-bridge knots,” J. Knot Theory Ramif. 26 : 2 (2017). article no. 1740003, 6pp. MR 3604485 Zbl 1362.57006 ArXiv 1602.02787 article

C. M. Gordon: “On embedding infinite cyclic covers in compact 3-manifolds,” Sci. China Math. 60 : 9 (2017), pp. 1575–1578. MR 3689185 Zbl 1383.57006 article

C. Gordon and T. Lidman: “Corrigendum: ‘Taut foliations, left-orderability, and cyclic branched covers’,” Acta Math. Vietnam. 42 : 4 (December 2017), pp. 775–776. Corrigendum to an article published in Acta Math. Vietnam. 39:4 (2014). MR 3708042 Zbl 1422.57005 article

C. Gordon and T. Lidman: “Knot contact homology detects cabled, composite, and torus knots,” Proc. Am. Math. Soc. 145 : 12 (2017), pp. 5405–5412. MR 3717966 Zbl 1381.57005 ArXiv 1509.01642 article

Knots, low-dimensional topology and applications: Knots in Hellas (Olympia, Greece, 17–23 July 2016). Edited by C. C. Adams, C. M. Gordon, V. F. R. Jones, L. H. Kauffman, S. Lambropoulou, K. C. Millett, J. H. Przytycki, R. Ricca, and R. Sazdanovic. Springer Proceedings in Mathematics & Statistics 284. Springer (Cham, Switzerland), 2019. MR 3986037 Zbl 1419.57001 book

Renzo Luigi Ricca	Related
Vaughan F. R. Jones	Related	Volume
Radmila Sazdanović	Related
Józef Henryk Przytycki	Related
Kenneth C. Millett	Related
Louis H. Kauffman	Related
Colin C. Adams	Related
Sofia S. F. Lambropoulou	Related

M. Boileau, S. Boyer, and C. M. Gordon: “On definite strongly quasipositive links and L-space branched covers,” Adv. Math. 357 (2019), pp. 106828, 63. MR 4016557 Zbl 1432.57005 article

We investigate the problem of characterising the family of strongly quasipositive links which have definite symmetrised Seifert forms and apply our results to the problem of determining when such a link can have an L-space cyclic branched cover. In particular, we show that if \[ \delta_n = \sigma_1\sigma_2\cdots\sigma_{n-1} \] is the dual Garside element and \( b = \delta_n^k P \in B_n \) is a strongly quasipositive braid whose braid closure \( \hat{b} \) is definite, then \( k\geq 2 \) implies that \( \hat{b} \) is one of the torus links \( T(2,q) \), \( T(3,4) \), \( T(3,5) \) or pretzel links \( P(-2,2,m) \), \( P(-2,3,4) \). Applying [Boileau et al. 2019, Theorem 1.1] we deduce that if one of the standard cyclic branched covers of \( \hat{b} \) is an L-space, then \( \hat{b} \) is one of these links. We show by example that there are strongly quasipositive braids \( \delta_n P \) whose closures are definite but not one of these torus or pretzel links. We also determine the family of definite strongly quasipositive 3-braids and show that their closures coincide with the family of strongly quasipositive 3-braids with an L-space branched cover.

Steven Patrick Boyer	Related
Michel Charles Boileau	Related

M. Boileau, S. Boyer, and C. M. Gordon: “Branched covers of quasi-positive links and L-spaces,” J. Topol. 12 : 2 (2019), pp. 536–576. MR 4072174 Zbl 1422.57012 article

Let \( L \) be an oriented link such that \( \Sigma_n(L) \), the \( n \)-fold cyclic cover of \( S^3 \) branched over \( L \), is an L-space for some \( n\geq 2 \). We show that if either \( L \) is a strongly quasi-positive link other than one with Alexander polynomial a multiple of \[ (t-1)^{2g(L)+(|L|-1)} ,\] or \( L \) is a quasi-positive link other than one with Alexander polynomial divisible by \[ (t-1)^{2g_4(L)+(|L|-1)} ,\] then there is an integer \( n(L) \), determined by the Alexander polynomial of \( L \) in the first case and the Alexander polynomial of \( L \) and the smooth 4-genus of \( L \), \( g_4(L) \), in the second, such that \( n\leq n(L) \). If \( K \) is a strongly quasi-positive knot with monic Alexander polynomial such as an L-space knot, we show that \( \Sigma_n(K) \) is not an L-space for \( n\geq 6 \), and that the Alexander polynomial of \( K \) is a non-trivial product of cyclotomic polynomials if \( \Sigma_n(K) \) is an L-space for some \( n = 2 \), 3, 4, 5. Our results allow us to calculate the smooth and topological 4-ball genera of, for instance, quasi-alternating quasi-positive links. They also allow us to classify strongly quasi-positive alternating links and 3-strand pretzel links.

Steven Patrick Boyer	Related
Michel Charles Boileau	Related

S. Boyer, C. M. Gordon, and Y. Hu: Slope detection and toroidal 3-manifolds. Preprint, 2021. Zbl 0587.57008 ArXiv 2106.14378 techreport

Steven Patrick Boyer	Related
Ying Hu	Related

S. Boyer, C. M. Gordon, and Y. Hu: Recalibrating \( \mathbb{R} \)-order trees and \( \mathrm{Homeo}_+(S^1) \)-representations of link groups. Preprint, 2023. ArXiv 2306.10357 techreport

Steven Patrick Boyer	Related
Ying Hu	Related

S. Boyer, C. M. Gordon, and Y. Hu: JSJ decompositions of knot exteriors, Dehn surgery and the \( L \)-space conjecture. Preprint, 2023. Zbl 1058.57004 ArXiv 2307.06815 techreport

Steven Patrick Boyer	Related
Ying Hu	Related

S. Boyer, C. M. Gordon, and X. Zhang: Dehn fillings of knot manifolds containing essential twice-punctured tori. Mem. Amer. Math. Soc. 295. 2024. ArXiv 2004.04219 book

Steven Patrick Boyer	Related
Xingru Zhang	Related

Cameron McAllan Gordon

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