A colored Khovanov bicomplex

Volume 103 / 2014

Noboru Ito Banach Center Publications 103 (2014), 111-143 MSC: 57M27, 57M25. DOI: 10.4064/bc103-0-5


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 by 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).


  • Noboru ItoWaseda Institute for Advanced Study 1-6-1
    Nishi Waseda, Shinjuku-ku
    Tokyo 169-8050, Japan

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