Alternating numbers of torus knots with small braid index
Peter Feller, Simon Pohlmann, and Raphael Zentner

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 and a new bound for braid index . Both bounds coincide, so that we obtain a sharp result.
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