Volume 20, issue 4 (2020)

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$\tau$–invariants for knots in rational homology spheres

Katherine Raoux

Algebraic & Geometric Topology 20 (2020) 1601–1640
Abstract

Ozsváth and Szabó used the knot filtration on $\stackrel{̂}{CF}\left({S}^{3}\right)$ to define the $\tau$–invariant for knots in the $3$–sphere. We generalize their construction and define a collection of $\tau$–invariants associated to a knot $K$ in a rational homology sphere $Y\phantom{\rule{-0.17em}{0ex}}$. We then show that some of these invariants provide lower bounds for the genus of a surface with boundary $K$ properly embedded in a negative-definite $4$–manifold with boundary $Y\phantom{\rule{-0.17em}{0ex}}$.

Keywords
Heegaard Floer, knot invariants, genus bound, rational homology spheres
Primary: 57M27
Secondary: 57R58
Publication
Received: 12 December 2016
Revised: 20 May 2019
Accepted: 8 November 2019
Published: 20 July 2020
Authors
 Katherine Raoux Department of Mathematics Michigan State University East Lansing, MI United States