A colored Khovanov bicomplex

Noboru Ito

Banach Center Publications (2014)

  • Volume: 103, Issue: 1, page 111-143
  • ISSN: 0137-6934

Abstract

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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).

How to cite

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Noboru Ito. "A colored Khovanov bicomplex." Banach Center Publications 103.1 (2014): 111-143. <http://eudml.org/doc/281994>.

@article{NoboruIto2014,
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 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).},
author = {Noboru Ito},
journal = {Banach Center Publications},
language = {eng},
number = {1},
pages = {111-143},
title = {A colored Khovanov bicomplex},
url = {http://eudml.org/doc/281994},
volume = {103},
year = {2014},
}

TY - JOUR
AU - Noboru Ito
TI - A colored Khovanov bicomplex
JO - Banach Center Publications
PY - 2014
VL - 103
IS - 1
SP - 111
EP - 143
AB - 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).
LA - eng
UR - http://eudml.org/doc/281994
ER -

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