Relations among the lowest degree of the Jones polynomial and geometric invariants for a closed positive braid (Q1601060)

The Jones polynomial is an ambient isotopy invariant of oriented links which can be defined easily by skein relations. This invariant introduced by Jones in 1984, takes its values in the ring \(\mathbb Z[v^{\pm 1/2}]\). If \(L\) is an oriented link with \(r\) components, we define the unknotting number of \(L\), \(u(L)\), as the minimal number of crossing changes needed to create the trivial link with \(r\) components. The main result in the paper under review is a relationship between the lowest degree of the Jones polynomial \(V(L,t)\) and the unknotting number of \(L\). More precisely, it is shown that if \(L\) is a closed positive braid link with \(r\) components then the lowest degree of \(V(L,t)\) is equal to the unknotting number \(u(L)\) minus \(\frac{r-1}{2}\). Furthermore, relationships between the lowest degree of \(V(L,t)\) and the slice Euler characteristic are obtained.