Celebratio Mathematica

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

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.

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

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

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

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

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

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

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

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 and Y. Hu: “Taut foliations in branched cyclic covers and left-orderable groups,” Trans. Amer. Math. Soc. 372 : 11 (2019), pp. 7921–7957. MR 4029686 Zbl 1445.57017 article

Steven Patrick Boyer	Related
Ying Hu	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

S. Boyer, C. M. Gordon, and Y. Hu: Cyclic branched covers of Seifert links and properties related to the \( ADE \) link conjecture. Preprint, 2024. ArXiv 2402.15914 techreport

year	title	people

			clear

Cameron McAllan Gordon

Working with Cameron: from exceptional surgeries to the L-space conjecture

Filter the Bibliography List