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 (1988), pp. 191–205. MR 929535 Zbl 0616.57003 article

M. Scharlemann and A. Thompson: “Link genus and the Conway moves,” Comment. Math. Helv. 64 : 4 (December 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 \( (\mathrm{surface})\times I \) are standard,” Math. Ann. 295 : 3 (January 1993), pp. 549–564. MR 1204837 Zbl 0814.57010 article

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
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 (Knoxville, TN, 18–25 May 1992). 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
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.GT/9910099 article

Abigail A. Thompson	Related
Hiroshi Goda	Related

M. Scharlemann and A. Thompson: “Unknotting tunnels and Seifert surfaces,” Proc. London Math. Soc. (3) 87 : 2 (September 2003), pp. 523–544. MR 1990938 Zbl 1047.57008 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 \( \operatorname{genus}(K) > 1 \). 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 (August 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 and Topology Monographs 7. Geometry & Topology Publications (Coventry, UK), 2004. MR 2172481 Zbl 1207.57016 incollection

Andrew J. Casson	Related
Abigail A. Thompson	Related
Yo'av Rieck	Related
Cameron McAllan Gordon	Related	Volume

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

Suppose \( F \) is a compact orientable surface, \( K \) is a knot in \( F\times I \), and \[ (F\times I)_{\mathrm{surg}} \] is the 3-manifold obtained by some nontrivial surgery on \( K \). If \( F\times\{0\} \) compresses in \( (F\times I)_{\mathrm{surg}} \), 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 2R and slice-ribbon conjectures,” Geom. Topol. 14 : 4 (2010), pp. 2305–2347. MR 2740649 Zbl 1214.57008 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
Abigail A. Thompson	Related

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