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Computing knot Floer homology in cyclic branched covers. (English) Zbl 1160.57010
Any knot \(K\) in the three-sphere can be represented by a grid diagram, and it was shown by C. Manolescu, P. Ozsváth and S. Sarkar [A combinatorial description of knot Floer homology, arXiv:math/0607691] that these diagrams can be used to compute the Heegaard Floer knot homology of \(K\) combinatorially. In this paper, the author shows that a Heegaard diagram for \(K\) gives rise to a Heegaard diagram for the preimage \(\tilde{K}\) in \(\Sigma_m(K)\), the \(m\)-fold cyclic cover of \(S^3\) branched along \(K\). Starting with a grid diagram for \(K\), this leads to a combinatorial computation of the knot Floer homology of \(\tilde{K}\). The resulting computations for many three-bridge knots with crossing number up to 11 are included. The paper concludes with some observations and a conjecture that if the knot Floer homology of \(K\) is thin (supported on a single diagonal) then it is isomorphic to the knot Floer homology of \(\tilde{K}\) in \(\Sigma_2(K)\) in a certain (canonical) \(\mathrm{spin}^c\) structure. The special case of the conjecture for two-bridge knots was proved by J. E. Grigsby [Algebr. Geom. Topol. 6, 1355–1398 (2006; Zbl 1133.57006)].

MSC:
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57R58 Floer homology
57M12 Low-dimensional topology of special (e.g., branched) coverings
Software:
HFK; Gridlink
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References:
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