The algebraic crossing number and the braid index of knots and links

TL;DR

This paper investigates the relationship between the algebraic crossing number and the braid index of knots and links, providing new examples and conditions under which the conjecture about their uniqueness holds.

Contribution

It demonstrates that the conjecture can hold even when the Morton-Franks-Williams inequality is not sharp, and extends the conjecture's validity to cable knots and connected sums.

Findings

Infinitely many knots and links satisfy the conjecture despite non-sharp inequality.

The conjecture's truth for certain knots implies its truth for their cable and connected sum.

Provides conditions under which the inequality's sharpness is not necessary for the conjecture.

Abstract

It has been conjectured that the algebraic crossing number of a link is uniquely determined in minimal braid representation. This conjecture is true for many classes of knots and links. The Morton-Franks-Williams inequality gives a lower bound for braid index. And sharpness of the inequality on a knot type implies the truth of the conjecture for the knot type. We prove that there are infinitely many examples of knots and links for which the inequality is not sharp but the conjecture is still true. We also show that if the conjecture is true for K and L, then it is also true for the (p,q)-cable of K and for the connect sum of K and L.