An estimate of the Rasmussen invariant for links and the determination for certain links
The Rasmussen invariant is a knot concordance invariant given by \textit{J. Rasmussen} [Invent. Math. 182, No. 2, 419--447 (2010; Zbl 1211.57009)]. The Rasmussen invariant was extended to a link invariant by \textit{A. Beliakova} and \textit{S. Wehrli} [Can. J. Math. 60, No. 6, 1240--1266 (2008; Zbl 1171.57010)]. This invariant is also called the Rasmussen invariant and denoted by \(s\). In the paper under review, the author gives an estimate of the Rasmussen invariant for links. Let \(L\) be a link and \(D_{L}\) be a diagram of \(L\). Let \(D^{0+}_L\) be the diagram obtained by eliminating all negative crossings of \(D_L\) in the same manner as in the Seifert algorithm. Denote by \(l_{0}(D_{L})\) the number of the split components of the link represented by \(D_{L}^{0+}\). Then, the author proves the following: \(s(L) \geq w(D_{L})-O(D_{L})+2l_{0}(D_{L})-1\), where \(w(D_{L})\) is the writhe of \(D_{L}\) and \(O(D_{L})\) is the number of Seifert circles of \(D_{L}\) (Theorem 1.5). This result was independently given by \textit{A. Lobb} [Compos. Math. 147, No. 2, 661--668 (2011; Zbl 1220.57007)]. \textit{T. Abe} [Proc. Am. Math. Soc. 139, No. 7, 2647--2656 (2011; Zbl 1226.57004)] has shown that equality holds if \(D_{L}\) is a homogeneous diagram (cf. Remark 6.3). Moreover, the author computes the Ozsváth-Szabó \(\tau\)-invariants and the Rasmussen invariants for certain pretzel knots.
scientific article; zbMATH DE number 6522013
scientific article