Heegaard Floer homology of surgeries on two-bridge links

Liu, Yajing

doi:10.2140/agt.2022.22.473

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Heegaard Floer homology of surgeries on two-bridge links

Yajing Liu

Algebraic & Geometric Topology 22 (2022) 473–557

DOI: 10.2140/agt.2022.22.473

Abstract

Heegaard Floer homology is combinatorially computable, but it is still unknown for many $3$ –manifolds, especially for ${HF}^{-}$ of hyperbolic manifolds. We conduct some computations by utilizing the link surgery formula of Manolescu and Ozsváth. Using nice diagrams and algebraic methods, we show that ${H F}^{-}$ and $d$ –invariants of surgeries on a $2$ –bridge link $L$ are determined only by the homotopy type of the $A_{s_{1}, s_{2}}^{-} (L)$ , where the maps in surgery complex by counting triangles are algebraically determined. Thus, this leads to a practical polynomial time algorithm. Using this algorithm, we calculate some examples explicitly: ${H F}^{-}$ and the $d$ –invariants of all integer surgeries on a family of hyperbolic two-bridge links including the Whitehead link.

Keywords

Heegaard Floer homology, link surgery, 2-bridge link

Mathematical Subject Classification 2010

Primary: 57M27, 57R58

References

Publication

Received: 26 February 2014

Revised: 8 July 2014

Accepted: 19 July 2020

Published: 3 August 2022

Authors

Yajing Liu
Department of Mathematics UCLA Los Angeles, CA United States

Volume 22, issue 2 (2022)