Celebratio Mathematica

A. A. Thompson: Property \( \mathrm{P} \) for certain classes of knots (genus one, band-connect sum). Ph.D. thesis, Rutgers University, 1986. Advised by M. G. Scharlemann and J. L. Shaneson. An article based on this dissertation was published in Topology 26:2 (1987). MR 2634826 phdthesis

Martin Scharlemann	Related	Volume
Julius L. Shaneson	Related

A. Thompson: “Property \( \mathrm{P} \) for the band-connect sum of two knots,” Topology 26 : 2 (1987), pp. 205–207. Based on the author’s 1986 PhD dissertation. MR 895572 Zbl 0628.57005 article

M. Scharlemann and A. Thompson: “Finding disjoint Seifert surfaces,” Bull. London Math. Soc. 20 : 1 (January 1988), pp. 61–64. MR 916076 Zbl 0654.57005 article

Given two Seifert surfaces \( S \) and \( T \) for a knot \( K \), there is a sequence of Seifert surfaces \( S = S_0 \), \( S_1,\dots \), \( S_n=T \) such that for each \( i \), \( 1\leq i\leq n \), \( S_i \) is disjoint from \( S_{i-1} \). The standard proof (see, for example [4]), which is useful in showing that any two Seifert matrices of \( K \) are \( S \)-equivalent, puts no limit on the genus of the intermediate Seifert surfaces \( S_1,\dots \), \( S_{n-1} \). Here we present a simple proof that

if \( S \) and \( T \) are of minimal genus, then we may take all \( S_i \) to be of minimal genus, and
for \( S \) an arbitrary Seifert surface, there is a sequence \( S = S_0 \), \( S_1,\dots \), \( S_n \) or Seifert surfaces such that \[ \operatorname{genus}(S_{i-1}) > \operatorname{genus}(S_i) ,\] \( S_n \) is of minimal genus, and for each \( i \), \( 1\leq i \leq n \), \[ S_i \cap S_{i-1} = \emptyset .\]

M. Scharlemann and A. Thompson: “Unknotting number, genus, and companion tori,” Math. Ann. 280 : 2 (March 1988), pp. 191–205. MR 929535 Zbl 0616.57003 article

In [Scharlemann 1985a] a complicated combinatorial argument showed that the band sum of knots is unknotted if and only if the band sum is a connected sum of unknots. This argument has since been dramatically simplified [Thompson 1987] and extended [Gabai 1987; S3, Sect. 8] using the newly developed machinery of Gabai. In [Scharlemann 1985b] a similar but more complicated combinatorial argument demonstrated that unknotting number one knots are prime. It seems natural to ask whether the Gabai machinery can simplify the proof of this old conjecture as well.

In fact the Gabai machine reveals a connection between the unknotting number of a knot, its genus, and the position of its companion tori. In Sect. 3 we show (roughly) that, if a single crossing change made to a knot \( K \) reduces its genus by more than one, then any companion torus to \( K \) can be made disjoint from the crossing. In particular, of \( K \) were composite and of unknotting number one, then the swallow-follow companion torus would remain as a companion to the unknot, which is impossible. Therefore no composite knot has unknotting number one.

This argument exploits the drop (by at least two) in the genus of a composite knot when a crossing change unknots it. It is natural to ask whether, in general, the genus of a knot drops (or at least does not rise) as it is unknotted by crossing changes. Knots exist for which a crossing change both lowers the unknotting number and raises the genus. A specific example (due to Chuck Livingston) is given in the appendix. Boileau and Murakami have shown us others. In Sect. 1 we give a general construction, again using the Gabai machine, which seems to produce myriads of examples.

Section 2 is a technical section which readies the Gabai machine (as presented in [Scharlemann 1989]) for use in Sect. 3. In Sect. 4 we view crossing changes as a special case of a more general operation, that of attaching an \( n \)-half-twisted band, and discuss how the main results of Sect. 3 generalize.

M. Scharlemann and A. Thompson: “Link genus and the Conway moves,” Comment. Math. Helv. 64 : 4 (1989), pp. 527–535. MR 1022995 Zbl 0693.57004 article

J. Hass and A. Thompson: “A necessary and sufficient condition for a 3-manifold to have Heegaard genus one,” Proc. Am. Math. Soc. 107 : 4 (December 1989), pp. 1107–1110. MR 984792 Zbl 0694.57006 article

A. Thompson: “Knots with unknotting number one are determined by their complements,” Topology 28 : 2 (1989), pp. 225–230. MR 1003584 Zbl 0677.57006 article

W. Menasco and A. Thompson: “Compressing handlebodies with holes,” Topology 28 : 4 (1989), pp. 485–494. MR 1030989 Zbl 0683.57004 article

A. Thompson: “Thurston norm minimizing surfaces and skein trees for links in \( S^3 \),” Proc. Am. Math. Soc. 106 : 4 (August 1989), pp. 1085–1090. MR 969321 Zbl 0686.57004 article

M. Scharlemann and A. Thompson: “Detecting unknotted graphs in 3-space,” J. Diff. Geom. 34 : 2 (1991), pp. 539–560. MR 1131443 Zbl 0751.05033 article

A. Thompson: “A polynomial invariant of graphs in 3-manifolds,” Topology 31 : 3 (July 1992), pp. 657–665. MR 1174264 Zbl 0773.57002 article

In the last few years there has been a great deal of interest in new polynomial invariants for links in \( S^3 \), inspired by the work of Vaughan Jones. His paper [1985], introducing a new link polynomial, led to the definition of several other new polynomials for links. It is interesting to attempt to generalize these to obtain a polynomial invariant of graphs in \( S^3 \). This has been done (for example, see [Kauffman 1989]) to some extent, but thus far the polynomial invariants for graphs in \( S^3 \) have placed strong restrictions on either the type of graph considered or the type of isotopy under which the polynomial is invariant. In contrast to the situation for graphs imbedded in \( S^3 \), there are very satisfactory polynomials for abstract graphs, such as the Tutte polynomial [Tutte 1954]. These give detailed information about the intrinsic properties of the graphs.

The philosophy behind these polynomials is that very little information about a graph \( G \) is lost if one considers, instead of \( G \), the simpler graphs \( G/e \) and \( G - e \), where \( e \) is some edge of \( G \), \( G/e \) is obtained from \( G \) by identifying \( e \) to a point, and \( G - e \) is obtained from \( G \) by removing \( e \) from \( G \). Given a graph \( G \) imbedded in \( S^3 \), one can try a direct application of this approach. Certainly if \( e \) is an edge of \( G \) with distinct endpoints, it seems that most of the information about \( G \) and its imbedding in \( S^3 \) is retained if we instead consider the graphs \( G/e \) and \( G - e \). However there arises the question of how to interpret \( G/e \) when the edge \( e \) has both endpoints on the same vertex of \( G \). In this paper we suggest a solution to this difficulty. We thus obtain a polynomial associated to any finite graph in \( S^3 \), invariant under any isotopy of \( S^3 \).

In §1 we define a polynomial invariant for pairs \( (M,A) \) where \( M \) is a 3-manifold and \( A \) is a finite set \( \{a_1,\dots \), \( a_n\} \) of disjoint arcs properly imbedded in \( M \). This suggests many possible polynomial invariants for graphs in the 3-sphere. In §2 we choose one of these and explore some of its properties. In §3 we give an example of another polynomial invariant for graphs in \( S^3 \) to illustrate the range of possibilities available.

M. Scharlemann and A. Thompson: “Heegaard splittings of \( (\textrm{surface})\times I \) are standard,” Math. Ann. 295 : 3 (1993), pp. 549–564. MR 1204837 Zbl 0814.57010 article

Frohman and Hass have shown [1989] that genus three Heegaard splittings of the 3-torus are standard. Boileau and Otal [1990] generalize this result to show that all Heegaard splittings of the 3-torus are standard. A crucial part of Boileau–Otal’s argument is to show that all Heegaard splittings of a torus crossed with an interval are standard. We generalize this part of their paper to prove that all Heegaard splittings of a closed orientable genus \( g \) surface crossed with an interval are standard. Many of our arguments are based on theirs; we differ substantially in Sect. 4, which allows us to obtain the more general result.

The paper is organized as follows: Section 1 begins with a discussion of compression bodies and their spines. In Sect. 2 we discuss Heegaard splittings and state the main theorem. The proof of the main theorem begins in Sect. 3, where we prove a lemma which splits the remainder of the proof into two cases. These cases are considered in Sect. 4 and Sect. 5. In Sect. 6 we exploit the main theorem to generalize a theorem of Frohman.

M. Scharlemann and A. Thompson: “Thin position and Heegaard splittings of the 3-sphere,” J. Diff. Geom. 39 : 2 (1994), pp. 343–357. MR 1267894 Zbl 0820.57005 article

We present here a simplified proof of the theorem, originally due to Waldhausen [1968], that a Heegaard splitting of \( S^3 \) is determined solely by its genus. The proof combines Gabai’s powerful idea of “thin position” [1987] with Johannson’s [1991] elementary proof of Haken’s theorem [1968] (Heegaard splittings of reducible 3-manifolds are reducible). In §3.1, 3.2 & 3.8 we borrow from Otal [1991] the idea of viewing the Heegaard splitting as a graph in 3-space in which we seek an unknotted cycle.

Along the way we show also that Heegaard splittings of boundary reducible 3-manifolds are boundary reducible [Casson and Gordon 1987, 1.2], obtain some (apparently new) characterizations of graphs in 3-space with boundary-reducible complement, and recapture a critical lemma of [Menasco and Thompson 1989].

M. Scharlemann and A. Thompson: “Thin position for 3-manifolds,” pp. 231–238 in 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 1282766 Zbl 0818.57013 incollection

Cameron McAllan Gordon	Related	Volume
Yoav Moriah	Related
Martin Scharlemann	Related	Volume
Dov Bronislaw Wajnryb	Related

M. Scharlemann and A. Thompson: “Pushing arcs and graphs around in handlebodies,” pp. 163–171 in Low-dimensional topology. Edited by K. Johannson. Conference Proceedings and Lecture Notes in Geometry and Topology 3. International Press (Cambridge, MA), 1994. MR 1316180 Zbl 0868.57024 incollection

Martin Scharlemann	Related	Volume
Klaus Johannson	Related

A. Thompson: “Thin position and the recognition problem for \( S^3 \),” Math. Res. Lett. 1 : 5 (1994), pp. 613–630. MR 1295555 Zbl 0849.57009 article

A. Thompson: “A note on Murasugi sums,” Pac. J. Math. 163 : 2 (April 1994), pp. 393–395. MR 1262303 Zbl 0809.57003 article

A. Thompson: “Book Review: G. Hemion, ‘The classification of knots and 3-dimensional spaces’,” Bull. Am. Math. Soc. 31 : 2 (October 1994), pp. 252–254. Book by Geoffrey Hemion (Oxford Univ. Press, 1992). MR 1568144 article

J. Hass and A. Thompson: “Neon bulbs and the unknotting of arcs in manifolds,” J. Knot Theor. Ramif. 6 : 2 (April 1997), pp. 235–242. MR 1452439 Zbl 0886.57003 article

A. Thompson: “Thin position and bridge number for knots in the 3-sphere,” Topology 36 : 2 (March 1997), pp. 505–507. MR 1415602 Zbl 0867.57009 article

The bridge number of a knot in \( S^3 \), introduced by Schubert [1954] is a classical and well-understood knot invariant. The concept of thin position for a knot was developed fairly recently by Gabai [1987]. It has proved to be an extraordinarily useful notion, playing a key role in Gabai’s proof of property \( R \) as well as Gordon–Luecke’s solution of the knot complement problem [1989]. The purpose of this paper is to examine the relation between bridge number and thin position. We show that either a knot in thin position is also in the position which realizes its bridge number or the knot has an incompressible meridianal planar surface properly imbedded in its complement. The second possibility implies that the knot has a generalized tangle decomposition along an incompressible punctured 2-sphere. Using results from [Culler et al. 1987; Gordon and Reid 1995], we note that if thin position for a knot \( K \) is not bridge position, then there exists a closed incompressible surface in the complement of the knot and that the tunnel number of the knot is strictly greater than one.

C. Adams, J. Hass, and A. Thompson: How to ace calculus: The streetwise guide. W. H. Freeman (New York), 1998. book

Joel Hass	Related	Volume
Colin C. Adams	Related

A. Thompson: “Algorithmic recognition of 3-manifolds,” Bull. Am. Math. Soc. (N.S.) 35 : 1 (1998), pp. 57–66. MR 1487190 Zbl 0890.57027 article

A. Thompson: “The disjoint curve property and genus 2 manifolds,” Topology Appl. 97 : 3 (November 1999), pp. 273–279. MR 1711418 Zbl 0935.57022 article

H. Goda, M. Scharlemann, and A. Thompson: “Levelling an unknotting tunnel,” Geom. Topol. 4 (2000), pp. 243–275. MR 1778174 Zbl 0958.57007 ArXiv math/9910099 article

Martin Scharlemann	Related	Volume
Hiroshi Goda	Related

C. Adams, J. Hass, and A. Thompson: How to ace the rest of calculus: The streetwise guide. W. H. Freeman (New York), 2001. book

Joel Hass	Related	Volume
Colin C. Adams	Related

M. Scharlemann and A. Thompson: “Unknotting tunnels and Seifert surfaces,” Proc. London Math. Soc. (3) 87 : 2 (2003), pp. 523–544. MR 1990938 Zbl 1047.57008 ArXiv math/0010212 article

Let \( K \) be a knot with an unknotting tunnel \( \gamma \) and suppose that \( K \) is not a 2-bridge knot. There is an invariant \[ \rho = p/q\in\mathbb{Q}/2\mathbb{Z} ,\] with \( p \) odd, defined for the pair \( (K,\gamma) \).

The invariant \( \rho \) has interesting geometric properties. It is often straightforward to calculate; for example, for \( K \) a torus knot and \( \gamma \) an annulus-spanning arc, \[ \rho(K,\gamma) = 1 .\] Although \( \rho \) is defined abstractly, it is naturally revealed when \( K\cup\gamma \) is put in thin position. If \( \rho\neq 1 \) then there is a minimal-genus Seifert surface \( F \) for \( K \) such that the tunnel \( \gamma \) can be slid and isotoped to lie on \( F \). One consequence is that if \[ \rho(K,\gamma)\neq 1 \] then \( K > 1 \). This confirms a conjecture of Goda and Teragaito for pairs \( (K,\gamma) \) with \[ \rho(K,\gamma)\neq 1 .\]

M. Scharlemann and A. Thompson: “Thinning genus two Heegaard spines in \( S^3 \),” J. Knot Theor. Ramif. 12 : 5 (2003), pp. 683–708. MR 1999638 Zbl 1048.57002 article

“2003 Satter Prize,” Notices Am. Math. Soc. 50 : 4 (April 2003), pp. 474–475. article

M. Scharlemann and A. Thompson: “On the additivity of knot width,” pp. 135–144 in Proceedings of the Casson Fest (Fayetteville, AR, 10–12 April 2003 and Austin, TX, 19–21 May 2003). Edited by C. Gordon and Y. Rieck. Geometry & Topology Monographs 7. Geometry & Topology Publications (Coventry, UK), 2004. Based on the 28th University of Arkansas spring lecture series in the mathematical sciences and a conference on the topology of manifolds of dimensions 3 and 4. This paper was “Dedicated to Andrew Casson, a mathematician’s mathematician.”. MR 2172481 Zbl 1207.57016 ArXiv math/0403326 incollection

Andrew J. Casson	Related
Yo'av Rieck	Related
Cameron McAllan Gordon	Related	Volume
Martin Scharlemann	Related	Volume

@incollection {key2172481m,
	AUTHOR = {Scharlemann, Martin and Thompson, Abigail},
	TITLE = {On the additivity of knot width},
	BOOKTITLE = {Proceedings of the {C}asson {F}est},
	EDITOR = {Gordon, Cameron and Rieck, Yoav},
	SERIES = {Geometry \& Topology Monographs},
	NUMBER = {7},
	PUBLISHER = {Geometry \& Topology Publications},
	ADDRESS = {Coventry, UK},
	YEAR = {2004},
	PAGES = {135--144},
	DOI = {10.2140/gtm.2004.7.135},
	NOTE = {(Fayetteville, AR, 10--12 April 2003
		 and Austin, TX, 19--21 May 2003). Based
		 on the 28th University of Arkansas spring
		 lecture series in the mathematical sciences
		 and a conference on the topology of
		 manifolds of dimensions 3 and 4. This
		 paper was ``Dedicated to Andrew Casson,
		 a mathematician's mathematician''. ArXiv:math/0403326.
		 MR:2172481. Zbl:1207.57016.},
	ISSN = {1464-8989},
}

M. Scharlemann and A. Thompson: “Surfaces, submanifolds, and aligned Fox reimbedding in non-Haken 3-manifolds,” Proc. Am. Math. Soc. 133 : 6 (2005), pp. 1573–1580. MR 2120271 Zbl 1071.57015 ArXiv math/0308011 article

A. Thompson: “Invariants of curves in \( \mathbb{R}P^2 \) and \( \mathbb{R}^2 \),” Geom. Topol. 6 (2006), pp. 2175–2186. MR 2263063 Zbl 1128.53010 ArXiv math/0602003 article

J. Hass, J. H. Rubinstein, and A. Thompson: “Knots and \( k \)-width,” Geom. Dedicata 143 : 7 (December 2009), pp. 7–18. MR 2576289 Zbl 1189.57005 ArXiv math/0604256 article

J. Hyam Rubinstein	Related	Volume
Joel Hass	Related	Volume

M. Scharlemann and A. A. Thompson: “Surgery on a knot in (surface \( \times I \)),” Algebr. Geom. Topol. 9 : 3 (2009), pp. 1825–1835. MR 2550096 Zbl 1197.57011 ArXiv 0807.0405 article

Suppose \( F \) is a compact orientable surface, \( K \) is a knot in \( F\times I \), and \( (F\times I)_{\textrm{surg}} \) is the 3-manifold obtained by some nontrivial surgery on \( K \). If \( F\times\{0\} \) compresses in \( (F\times I)_{\textrm{surg}} \), then there is an annulus in \( F\times I \) with one end \( K \) and the other end an essential simple closed curve in \( F\times\{0\} \). Moreover, the end of the annulus at \( K \) determines the surgery slope.

An application: Suppose \( M \) is a compact orientable 3-manifold that fibers over the circle. If surgery on \( K\subset M \) yields a reducible manifold, then either

the projection \( K\subset M\to S^1 \) has nontrivial winding number,
\( K \) lies in a ball,
\( K \) lies in a fiber, or
\( K \) is cabled.

J. Hass, A. Thompson, and W. Thurston: “Stabilization of Heegaard splittings,” Geom. Topol. 13 : 4 (2009), pp. 2029–2050. MR 2507114 Zbl 1177.57018 ArXiv 0802.2145 article

Joel Hass	Related	Volume
William Paul Thurston	Related

M. Scharlemann and A. Thompson: Fibered knots and Property 2R. Preprint, 2009. ArXiv 0901.2319 techreport

R. E. Gompf, M. Scharlemann, and A. Thompson: “Fibered knots and potential counterexamples to the property \( 2{R} \) and slice-ribbon conjectures,” Geom. Topol. 14 : 4 (2010), pp. 2305–2347. MR 2740649 Zbl 1214.57008 ArXiv 1103.1601 article

If there are any 2-component counterexamples to the Generalized Property R Conjecture, a least genus component of all such counterexamples cannot be a fibered knot. Furthermore, the monodromy of a fibered component of any such counterexample has unexpected restrictions.

The simplest plausible counterexample to the Generalized Property R Conjecture could be a 2-component link containing the square knot. We characterize all two-component links that contain the square knot and which surger to \( \#_2(S^1\times S^2) \). We exhibit a family of such links that are probably counterexamples to Generalized Property R. These links can be used to generate slice knots that are not known to be ribbon.

Robert E. Gompf	Related
Martin Scharlemann	Related	Volume

J. Hass and A. Thompson: “Is it knotted?,” pp. 129–135 in Expeditions in mathematics. Edited by T. Shubin, D. F. Hayes, and G. L. Alexanderson. MAA Spectrum 68. Mathematical Association of America (Washington, DC), 2011. incollection

Tatiana Shubin	Related
Gerald Lee Alexanderson	Related
Joel Hass	Related	Volume
David Frank Hayes	Related

J. Johnson and A. Thompson: “On tunnel number one knots that are not \( (1,n) \),” J. Knot Theor. Ramif. 20 : 4 (2011), pp. 609–615. MR 2796230 Zbl 1218.57008 ArXiv math/0606226 article

J. H. Rubinstein and A. Thompson: “3-manifolds with Heegaard splittings of distance two,” pp. 341–346 in Geometry and topology down under. Edited by C. D. Hodgson, W. H. Jaco, M. G. Scharlemann, and S. Tillman. Contemporary Mathematics 597. American Mathematial Society (Providence, RI), 2013. MR 3186682 incollection

It is well-known that the distance of a Heegard splitting, as defined by Hempel [2001], is related to global properties of the underlying 3-manifold. For example, Casson–Gordon [1987] show that existence of a splitting of distance one implies that the manifold is Haken, i.e. has an embedded incompressible surface. Distance zero splittings are either stabilized, or the underlying manifold is reducible. In this paper, we study the case of distance two splittings. These are interesting as many known examples of splittings of distance greater than one are actually of distance two. Our aim is to describe how simple conditions on a distance two splitting imply interesting properties of the underlying manifold.

J. Hyam Rubinstein	Related	Volume
Martin Scharlemann	Related	Volume
William H. Jaco	Related
Craig David Hodgson	Related
Stephan Tillmann	Related

A. Thompson: “Does diversity trump ability? An example of the misuse of mathematics in the social sciences,” Notices Am. Math. Soc. 61 : 9 (2014), pp. 1024–1030. MR 3241558 Zbl 1338.91122 article

A. Thompson: Finding geodesics in a triangulated 2-sphere. Preprint, August 2014. ArXiv 1408.5949 techreport

A. Thompson: “Tori and Heegaard splittings,” Ill. J. Math. 60 : 1 (2016), pp. 141–148. MR 3665175 Zbl 1367.57009 ArXiv 1603.08154 article

In Studies in modern topology (1968) 39–98 Prentice Hall, Haken showed that the Heegaard splittings of reducible 3-manifolds are reducible, that is, a reducing 2-sphere can be found which intersects the Heegaard surface in a single simple closed curve. When the genus of the “interesting” surface increases from zero, more complicated phenomena occur. Kobayashi (Osaka J. Math. 24 (1987) 173–215) showed that if a 3-manifold \( M^3 \) contains an essential torus \( T \), then it contains one which can be isotoped to intersect a (strongly irreducible) Heegaard splitting surface \( F \) in a collection of simple closed curves which are essential in \( T \) and in \( F \). In general, there is no global bound on the number of curves in this collection. We show that given a 3-manifold \( M \), a minimal genus, strongly irreducible Heegaard surface \( F \) for \( M \), and an essential torus \( T \), we can either restrict the number of curves of intersection of \( T \) with \( F \) (to four), find a different essential surface and minimal genus Heegaard splitting with at most four essential curves of intersection, find a thinner decomposition of \( M \), or produce a small Seifert-fibered piece of \( M \).

A. Thompson: Dehn surgery on complicated fibered knots in the 3-sphere. Preprint, 2016. ArXiv 1604.04902 techreport

J. Hass, A. Thompson, and A. Tsvietkova: “The number of surfaces of fixed genus in an alternating link complement,” Int. Math. Res. Not. 2017 : 6 (March 2017), pp. 1611–1622. MR 3658176 ArXiv 1508.03680 article

Joel Hass	Related	Volume
Anastasiia Tsvietkova	Related

R. Kirby and A. Thompson: “A new invariant of 4-manifolds,” Proc. Natl. Acad. Sci. USA 115 : 43 (2018), pp. 10857–10860. MR 3871787 article

R. Kirby and A. Thompson: “Trisections and link surgeries,” New Zealand J. Math. 52 (2021 [2021–2022]), pp. 145–152. MR 4374440 article

J. Hass, A. Thompson, and A. Tsvietkova: “Alternating links have at most polynomially many Seifert surfaces of fixed genus,” Indiana Univ. Math. J. 70 : 2 (2021), pp. 525–534. MR 4257618 article

Joel Hass	Related	Volume
Anastasiia Tsvietkova	Related

J. Hass, A. Thompson, and A. Tsvietkova: “Tangle decompositions of alternating link complements,” Illinois J. Math. 65 : 3 (2021), pp. 533–545. MR 4312193 article

Decomposing knots and links into tangles is a useful technique for understanding their properties. The notion of prime tangles was introduced by Kirby and Lickorish; Lickorish proved that by summing prime tangles one obtains a prime link. In a similar spirit, summing two prime alternating tangles will produce a prime alternating link if summed correctly with respect to the alternating property. Given a prime alternating link, we seek to understand whether it can be decomposed into two prime tangles, each of which is alternating. We refine results of Menasco and Thistlethwaite to show that if such a decomposition exists, either it is visible in an alternating link diagram or the link is of a particular form, which we call a pseudo-Montesinos link.

Joel Hass	Related	Volume
Anastasiia Tsvietkova	Related

J. Hass, A. Thompson, and W. Thurston: “Stabilization of Heegaard splittings,” pp. 375–396 in Collected works of William P. Thurston with commentary, II: 3-manifolds, complexity and geometric group theory. American Mathematical Society (Providence, RI), 2022. MR 4556478 incollection

Joel Hass	Related	Volume
William Paul Thurston	Related

J. Meier, A. Thompson, and A. Zupan: “Cubic graphs induced by bridge trisections,” Math. Res. Lett. 30 : 4 (2023), pp. 1207–1231. MR 4728442 article

Every embedded surface \( \mathcal{K} \) in the 4-sphere admits a bridge trisection, a decomposition of \( (S^4, \mathcal{K}) \) into three simple pieces. In this case, the surface is determined by an embedded 1-complex, called the 1-skeleton of the bridge trisection. As an abstract graph, the 1-skeleton is a cubic graph \( \Gamma \) that inherits a natural Tait coloring, a 3-coloring of the edge set of \( \Gamma \) such that each vertex is incident to edges of all three colors. In this paper, we reverse this association: We prove that every Tait-colored cubic graph is isomorphic to the 1-skeleton of a bridge trisection corresponding to an unknotted surface. When the surface is nonorientable, we show that such an embedding exists for every possible normal Euler number. As a corollary, every tri-plane diagram for a knotted surface can be converted to a tri-plane diagram for an unknotted surface via crossing changes and interior Reidemeister moves. Tools used to prove the main theorem include two new operations on bridge trisections, crosscap summation and tubing, which may be of independent interest.

Jeffrey Meier	Related
Alexander Martin Zupan	Related

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