Celebratio Mathematica — Gordon — Working with Cameron

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

Works connected to Michel Charles Boileau

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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