TIGHT FIBRED KNOTS WITHOUT L-SPACE SURGERIES

Filip Misev,Gilberto Spano

doi:10.1017/s0017089520000543

Abstract

AbstractWe show that there exist infinitely many knots of every fixed genus $g\geq 2$ which do not admit surgery to an L-space, despite resembling algebraic knots and L-space knots in general: they are algebraically concordant to the torus knot T(2, 2g + 1) of the same genus and they are fibred and strongly quasipositive.

Highlights

Introduction and statement of resultAlgebraic knots, which include torus knots, are L-space knots: they admit Dehn surgeries to L-spaces, certain 3-manifolds generalising lens spaces which are defined in terms of Heegaard Floer homology [7].The first author recently described a method to construct infinite families of knots of any fixed genus g 2 which all have the same Seifert form as the torus knot T(2, 2g + 1) of the same genus, and which are all fibred, hyperbolic and strongly quasipositive
This is described in the article [10], where a specific family of pairwise distinct knots Kg,n, n ∈ N, with these properties is constructed for every fixed genus g 2
For every integer g 2, there exists an infinite family of pairwise distinct genus g knots Kg,n, n ∈ N, with the following properties

Summary

Introduction

Introduction and statement of resultAlgebraic knots, which include torus knots, are L-space knots: they admit Dehn surgeries to L-spaces, certain 3-manifolds generalising lens spaces which are defined in terms of Heegaard Floer homology [7].The first author recently described a method to construct infinite families of knots of any fixed genus g 2 which all have the same Seifert form as the torus knot T(2, 2g + 1) of the same genus, and which are all fibred, hyperbolic and strongly quasipositive. (1) Kg,n is algebraically concordant to the torus knot T(2, 2g + 1) (2) Kg,n is fibred, hyperbolic and strongly quasipositive (3) Kg,n does not admit any nontrivial Dehn surgery to a Heegaard Floer L-space Hedden proved that the open book associated to an L-space knot (or to its mirror) supports the tight contact structure of S3, under Giroux’ correspondence (see [8, Theorem 1.2, Proposition 2.1] and Ozsvath-Szabo [16, Corollary 1.6]).

Results

Conclusion