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).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc103-0-5,
author = {Noboru Ito},
title = {A colored Khovanov bicomplex},
journal = {Banach Center Publications},
volume = {102},
year = {2014},
pages = {111-143},
zbl = {1337.57038},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc103-0-5}
}
Noboru Ito. A colored Khovanov bicomplex. Banach Center Publications, Tome 102 (2014) pp. 111-143. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc103-0-5/