Celebratio Mathematica

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.

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Xingru Zhang	Related

M. Boileau, S. Boyer, and C. M. Gordon: “On definite strongly quasipositive links and L-space branched covers,” Adv. Math. 357 (December 2019). article no. 106828, 63pp. MR 4016557 Zbl 07130222 ArXiv 1811.08862 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 (June 2019), pp. 536–576. Zbl 1422.57012 ArXiv 1710.07658 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

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