A colored Khovanov bicomplex

Noboru Ito

doi:10.4064/bc103-0-5

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arXiv · 2004.08181

A colored Khovanov bicomplex

Noboru Ito

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Abstract

In this note, we prove the existence of a tri-graded Khovanov-type bicomplex (Theorem 1.2). The graded Euler characteristic of the total complex associated with this bicomplex is the colored Jones polynomial of a link. The first grading of the bicomplex is a homological one derived from cabling of the link (i.e., replacing a strand of the link with several parallel strands); the second grading is related to the homological grading of ordinary Khovanov homology; finally, the third grading is preserved by the differentials, and corresponds to the degree of the variable in the colored Jones polynomial. In particular, we introduce a way to take a small cabling link diagram directly from a big cabling link diagram (Theorem 3.2). This is a refined version of arXiv:0907.5247.

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Noboru Ito. 2020-04-17. A colored Khovanov bicomplex. https://doi.org/10.4064/bc103-0-5

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