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**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 PoincarĂ© 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 \(N>2\).

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 PoincarĂ© 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 \(N>2\).

Reviewer: Cynthia L. Curtis (Ewing)

### MSC:

57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |