Celebratio Mathematica

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

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

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
Abigail A. Thompson	Related	Volume
Yoav Moriah	Related
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

Abigail A. Thompson	Related	Volume
Klaus Johannson	Related

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

Abigail A. Thompson	Related	Volume
Hiroshi Goda	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

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
Abigail A. Thompson	Related	Volume
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@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

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.

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.

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