Computing cobordism maps in link Floer homology and the reduced Khovanov TQFT
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We study the maps induced on link Floer homology by elementary decorated link cobordisms. We compute these for births, deaths, stabilizations, and destabilizations, and show that saddle cobordisms can be computed in terms of maps in a decorated skein exact triangle that extends the oriented skein exact triangle in knot Floer homology. In particular, we completely determine the Alexander and Maslov grading shifts. As a corollary, we compute the maps induced by elementary cobordisms between unlinks. We show that these give rise to a \((1+1)\)-dimensional TQFT that coincides with the reduced Khovanov TQFT. Hence, when applied to the cube of resolutions of a marked link diagram, it gives the complex defining the reduced Khovanov homology of the knot. Finally, we define a spectral sequence from (reduced) Khovanov homology using these cobordism maps, and we prove that it is an invariant of the (marked) link.
KeywordsKnot cobordism Heegaard Floer homology Khovanov homology Spectral sequence TQFT
Mathematics Subject Classification57M27 57R58
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We thank John Baldwin, Matt Hedden, Tom Hockenhull, Joan Licata, Andrew Lobb, Ciprian Manolescu, Tom Mrowka, Jacob Rasmussen, Ian Zemke, and the anonymous referee for their comments and suggestions. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 674978). The first author was supported by a Royal Society Research Fellowship. The second author was supported by an EPSRC Doctoral Training Award and LMS Grant PMG 16-17 07. The first author would also like to thank the Isaac Newton Institute for its hospitality.
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