#### Volume 22, issue 2 (2022)

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Heegaard Floer homology of surgeries on two-bridge links

### Yajing Liu

Algebraic & Geometric Topology 22 (2022) 473–557
##### Abstract

Heegaard Floer homology is combinatorially computable, but it is still unknown for many $3$–manifolds, especially for ${\mathrm{HF}}^{-}$ 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 ${HF}^{-}$ and $d$–invariants of surgeries on a $2$–bridge link $L$ are determined only by the homotopy type of the ${A}_{{s}_{1},{s}_{2}}^{-}\left(L\right)$, 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: ${HF}^{-}$ and the $d$–invariants of all integer surgeries on a family of hyperbolic two-bridge links including the Whitehead link.