Heegaard Floer homology of surgeries on two-bridge links

Abstract

We give an $O(p^2)$ time algorithm to compute the generalized Heegaard Floer complexes $A_{s_1,s_2}^{-}(\overrightarrow{L})$'s for a two-bridge link $\overrightarrow{L}=b(p,q)$ by using nice diagrams. Using the link surgery formula of Manolescu-Ozsváth, we also show that ${\mathbf{HF}}^{-}$ and their $d$-invariants of all integer surgeries on two-bridge links are determined by $A_{s_1,s_2}^{-}(\overrightarrow{L})$'s. We obtain a polynomial time algorithm to compute ${\mathbf{HF}}^{-}$ of all the surgeries on two-bridge links, with $\mathbb{Z}/2\mathbb{Z}$ coefficients. In addition, we calculate some examples explicitly: ${\mathbf{HF}}^{-}$ and the $d$-invariants of all integer surgeries on a family of hyperbolic two-bridge links including the Whitehead link.

58 pages, 14 figures. Abstract and introduction revised; Theorem 1.1, Theorem 5.12 (now 5.13) adapted; Section 5.1 adapted; Section 5.6 and Example 5.16 added; sign errors fixed; typos in statement of Proposition 6.8 (now 6.9) corrected and computations simplified; Definition 2.2 (now 2.3) corrected. arXiv admin note: text overlap with arXiv:1011.1317 by other authors

References in corpus (3)

• Knot Floer homology detects fibred knots
• Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology
• Knot Floer homology of (1,1)-knots

Cited by in corpus (4)

• $L$-space surgeries on links
• Immersed concordances of links and Heegaard Floer homology
• Invariants of knotted surfaces from link homology and bridge trisections
• Contact (+1)-surgeries along Legendrian Two-component Links