## Khovanov-Rozansky homology of two-bridge knots and links.(English)Zbl 1125.57004

In [Matrix factorizations and link homology, preprint, arXiv:math.QA/0401268], M. Khovanov and L. Rozansky introduced a family of link invariants generalizing the Jones polynomial homology of M. Khovanov’s paper [Duke Math. J. 101, 359–426 (2000; Zbl 0960.57005)].
In the paper reviewed here, the author seeks to compute the reduced version of the homologies, which are invariants of an oriented link with a marked component, for 2-bridge knots and links. The invariants are bigraded homology groups $$HKR^{I,j}_N$$, where $$N$$ is a positive integer. The homology when $$N=1$$ is the same for all links, and many computer calculations have been made for the theory when $$N=2$$. In the current paper, the author examines the theory for 2-bridge knots when $$N>4$$. In this case, the homology groups are determined by the HOMFLY polynomial and the signature of the knot. More specifically, 2-bridge knots are $$N$$-thin for $$N>4$$, meaning that the Poincaré polynomial satisfies a particular equation involving the HOMFLY polynomial and the signature. In contrast, the author gives an example of an alternating knot which is not $$N$$-thin for any $$N>2$$.

### MSC:

 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 57M25 Knots and links in the $$3$$-sphere (MSC2010)

Zbl 0960.57005
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