In 2004, J. Rasmussen [Invent. Math. 182, No. 2, 419–447 (2010; Zbl 1211.57009)] discovered a concordance invariant for knots which is based on M. Khovanov’s work [Duke Math. J. 101, No. 3, 359–426 (2000; Zbl 0960.57005)] and Lee’s work [E. S. Lee, Adv. Math. 197, No. 2, 554–586 (2005; Zbl 1080.57015)]. This invariant is called Rasmussen’s \(s\)-invariant. A. Beliakova and S. Wehrli [Can. J. Math. 60, No. 6, 1240–1266 (2008; Zbl 1171.57010)] defined a natural generalization of Rasmussen’s \(s\)-invariant to oriented links, which is also called the \(s\)-invariant. On the other hand, for any oriented link \(L\), J. Pardon [Algebr. Geom. Topol. 12, No. 2, 1081–1098 (2012; Zbl 1263.57007)] defined another generalization \(d_L: \mathbb{Z}\times \mathbb{Z} \to \mathbb{Z}_{\geq0}\). In particular, for a knot \(K\), \(d_K(0, s) \neq 0\) if and only if \(s = s(K) \pm 1\), where \(s(K)\) is the \(s\)-invariant of \(K\).
A. Lobb [Compos. Math. 147, No. 2, 661–668 (2011; Zbl 1220.57007)] and T. Kawamura [Topology Appl. 196, Part B, Article ID 5451, 558–574 (2015; Zbl 1330.57008)] discovered a lower bound \(V(D)\) and an upper bound \(U(D)\) for the \(s\)-invariant of a knot \(K\), which depend on a knot diagram \(D\) of \(K\). Their result also holds for an oriented link \(L\) and a connected link diagram \(D\) of \(L\). Their lower bound does not work if the diagram \(D\) is not connected. However, by a property of the \(s\)-invariant, we can show that \(2-2n+V(D)\) also gives a lower bound of \(s(L)\), where \(n\) is the number of the connected components of \(D\).
In the paper under review, the author introduces a new class of links called pseudo-thin links and gives some properties of these links. An oriented non-split link \(L\) is pseudo-thin if \(d_L(0, \cdot)\) is supported in two points. An oriented split link is pseudo-thin if each of its split components is pseudo-thin. For example, positive links and quasi-alternating links are pseudo-thin. The author describes a relation between the \(s\)-invariant of a pseudo-thin link and that of its mirror image (Proposition 3.6). The author also studies strong concordance for pseudo-thin links (Section 5.2), which is inspired by Pardon’s work on strong concordance for homologically thin links. Furthermore, the \(s\)-invariants and the slice genera for some links are also computed.
In this paper, the author also shows that for any pseudo-thin link \(L\) and any link diagram \(D\) of \(L\), we obtain \(s(L)\geq 2-2r+V (D)\), where \(r\) is the number of the split components of \(L\) (Theorem 3.13). The reviewer thinks that this inequality might also hold for any non pseudo-thin link as mentioned above.