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The knot Floer cube of resolutions and the composition product

Columbia University, USA

https://doi.org/10.1142/S0218216520500066Cited by:0 (Source: Crossref)

Abstract

We examine the relationship between the oriented cube of resolutions for knot Floer homology and HOMFLY-PT homology. By using a filtration induced by additional basepoints on the Heegaard diagram for a knot $K$ , we see that the filtered complex decomposes as a direct sum of HOMFLY-PT complexes of various subdiagrams. Applying Jaeger’s composition product formula for knot polynomials, we deduce that the graded Euler characteristic of this direct sum is the HOMFLY-PT polynomial of $K$ .

Keywords:

AMSC: 57M25

References

1. N. Dowlin, Knot Floer homology and Khovanov–Rozansky homology for singular links, Algebraic Geom. Topol. 18(7) (2018) 3839–3885. Crossref, Web of Science, Google Scholar
2. N. Dunfield, S. Gukov and J. Rasmussen, The superpolynomial for knot homologies, Experimental Math. 15(2) (2006) 129–160. Crossref, Web of Science, Google Scholar
3. A. Gilmore, Framed graphs and the non-local ideal in the knot Floer cube of resolutions, Algebraic Geom. Topol. 15 (2015) 1239–1302. Crossref, Web of Science, Google Scholar
4. F. Jaeger, Composition product and models for the HOMFLY polynomial, Enseign. Math. 35 (1989) 323–361. Google Scholar
5. M. Khovanov and L. Rozansky, Matrix factorizations and link homology, Fund. Math. 199 (2008) 1–99. Crossref, Web of Science, Google Scholar
6. M. Khovanov and L. Rozansky, Matrix factorizations and link homology II, Geom. Topol. 12 (2008) 1387–1425. Crossref, Web of Science, Google Scholar
7. D. Krasner, Integral HOMFLY-PT and sl(n)-link homology, Int. J. Math. Math. Sci. 2010 (2010) 25 pp. Crossref, Google Scholar
8. R. Lipshitz, A cylindrical reformulation of Heegaard Floer homology, Geom. Topol. 10(2) (2006) 955–1096. Crossref, Web of Science, Google Scholar
9. C. Manolescu, An untwisted cube of resolutions for knot Floer homology, Quantum Topol. 5 (2014) 185–223. Crossref, Google Scholar
10. P. Ozsváth and Z. Szabó, Holomorphic discs and knot invariants, Adv. Math. 186 (2004) 58–116. Crossref, Web of Science, Google Scholar
11. P. Ozsváth and Z. Szabó, Holomorphic discs, link invariants, and the multi-variable Alexander polynomial, Algebraic Geom. Topol. 8 (2008) 615–692. Crossref, Web of Science, Google Scholar
12. P. Ozsváth and Z. Szabó, A cube of resolutions for knot floer homology, J. Topol. 2(4) (2009) 865–910. Crossref, Web of Science, Google Scholar
13. J. Rasmussen, Floer homology and knot complements, PhD Thesis, Harvard University (2003). Google Scholar
14. J. Rasmussen, Some differentials on Khovanov-Rozansky homology, Geom. Topol. 19(6) (2015) 3031–3104. Crossref, Web of Science, Google Scholar
15. E. Wagner, Khovanov-Rozansky graph homology and composition product, J. Knot Theory Ramifications, 17(12) (2008) 1549–1559. Link, Web of Science, Google Scholar