Alternating numbers of torus knots with small braid index

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 Ozsváth, Stipsicz, and Szabó. 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.

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References in corpus (2)

• Optimal Cobordisms between Torus Knots
• Khovanov homology: torsion and thickness

Cited by in corpus (3)

• Knot Floer homology and the unknotting number
• The upsilon invariant at 1 of 3-braid knots
• On the Turaev genus of torus knots