Abstract
Although computing the Khovanov homology of links is common in literature, no general formulae have been given for all of them. We give the graded Euler characteristic and the Khovanov homology of the 2-strand braid link ,, and the 3-strand braid .
Highlights
Khovanov homology is an invariant for oriented links which was introduced by Mikhail Khovanov in 2000 as a categorification of the Jones polynomial [1]
Khovanov assigned a bigraded chain complex Ci, j ( L) to the oriented link diagram L whose differential was graded of bidegree (1, 0) and whose homotopy type depended only on the isotopy class of L
This paper is concerned with the link invariants: the Khovanov homology and the Jones polynomial
Summary
Khovanov homology is an invariant for oriented links which was introduced by Mikhail Khovanov in 2000 as a categorification of the Jones polynomial [1]. Khovanov assigned a bigraded chain complex Ci, j ( L) to the oriented link diagram L whose differential was graded of bidegree (1, 0) and whose homotopy type depended only on the isotopy class of L. The bigraded homology group H i, j ( D) of the chain complex Ci, j ( D) provides an invariant of oriented links, known as Khovanov homology. Khovanov’s construction is combinatorial from which Khovanov homology is algorithmically computable, we shall follow rather a simple way of Bar-Natan’s, which he introduced in [2] to compute the Khovanov homology
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