Alternating numbers of torus knots with small braid index

TL;DR

This paper precisely determines the alternating number of torus knots with braid index up to 4 by combining invariants and bounds, providing a complete characterization for these knots.

Contribution

It introduces a new bound for braid index 4 and uses the upsilon-invariant to exactly compute the alternating number for these knots.

Findings

The alternating number for torus knots with braid index 4 is exactly determined.

A new bound for braid index 4 knots is established.

The upsilon-invariant effectively provides a lower bound for the alternating number.

Abstract

We calculate the alternating number of torus knots with braid index 4 and less. For the lower bound, we use the upsilon-invariant recently introduced by Ozsv\'ath, Stipsicz, and Szab\'o. For the upper bound, we use a known bound for braid index $3$ and a new bound for braid index $4$. Both bounds coincide, so that we obtain a sharp result.