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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
##### Keywords:
knot Floer homology; branched cover
##### Software:
 [1] J A Baldwin, W D Gillam, Computations of Heegaard Floer knot homology · Zbl 1288.57010 [2] M Culler, Gridlink: a tool for knot theorists [3] R Diestel, Graph theory, Graduate Texts in Mathematics 173, Springer (2005) · Zbl 1074.05001 [4] R H Fox, A quick trip through knot theory, Prentice-Hall (1962) 120 · Zbl 1246.57002 [5] C M Gordon, Some aspects of classical knot theory, Lecture Notes in Math. 685, Springer (1978) 1 · Zbl 0386.57002 [6] J E Grigsby, Combinatorial description of knot Floer homology of cyclic branched covers · Zbl 1179.57022 [7] J E Grigsby, Knot Floer homology in cyclic branched covers, Algebr. Geom. Topol. 6 (2006) 1355 · Zbl 1133.57006 [8] J Grigsby, D Ruberman, S Strle, Knot concordance and Heegaard Floer homology invariants in branched covers · Zbl 1149.57007 [9] D A Lee, R Lipshitz, Covering spaces and $$\mathbbQ$$-gradings on Heegaard Floer homology · Zbl 1153.53061 [10] R Lipshitz, A cylindrical reformulation of Heegaard Floer homology, Geom. Topol. 10 (2006) 955 · Zbl 1130.57035 [11] C Manolescu, P Ozsváth, On the Khovanov and knot Floer homologies of quasi-alternating links · Zbl 1195.57032 [12] C Manolescu, P Ozsváth, S Sarkar, A combinatorial description of knot Floer homology · Zbl 1179.57022 [13] C Manolescu, P Ozsváth, Z Szabó, D Thurston, On combinatorial link Floer homology · Zbl 1155.57030 [14] P Ozsváth, Z Szabó, Heegaard Floer homology and alternating knots, Geom. Topol. 7 (2003) 225 · Zbl 1130.57303 [15] P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58 · Zbl 1062.57019 [16] P Ozsváth, Z Szabó, Knots with unknotting number one and Heegaard Floer homology, Topology 44 (2005) 705 · Zbl 1083.57013 [17] J A Rasmussen, Floer homology of surgeries on two-bridge knots, Algebr. Geom. Topol. 2 (2002) 757 · Zbl 1013.57020 [18] J A Rasmussen, Floer homology and knot complements, PhD thesis, Harvard University (2003) [19] D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish (1990) · Zbl 0854.57002 [20] S Sarkar, J Wang, A combinatorial description of some Heegaard Floer homologies