Alternation 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.

Khovanov width and dealternation number of positive braid links

“…Furthermore, we take J to be the positive torus knot T (2, 4(m − 1) + 1). Note that, by Proposition 7, there exists a cobordism of genus (7) g…”

“…The latter follows for example from the fact that all Whitehead doubles W (K) of a knot K satisfy alt(W (K)) ≤ 1, while w Kh (W (K)) ≤ dalt(W (K)) + 2 can be arbitrarily large. While the lower bound given by w Kh no longer holds for alt, the lower bound given by |τ + υ| still holds; compare [7,Corollary 3].…”

The Turaev genus of torus knots

“…An immediate consequence of the definition is that alt(L) ≤ dalt(L) for any link L. Feller, Pohlmann, and Zentner [FPZ15] computed the alternation number of torus knots on four or fewer strands, and Baader, Feller, Lewark, and Zentner [BFLZ16] gave bounds on the alternation and dealternating number of some families of torus links on six or fewer strands. Our arguments in Section 4 resemble those in [FPZ15]. See [Low15] for more comparisons between Turaev genus, dealternating number, alternation number, and other related invariants.…”

Knot concordances and alternating knots

“…In an earlier paper, Abe [1] showed that the difference between the Rasmussen invariant and the signature of a knot provides a bound on the alternation number of knot, alt(K), the minimum number of crossing changes required to convert K into an alternating knot. Extending these results, Feller-Pohlmann-Zentner [6] used Υ K (t) to find another lower bound on the alternation number. The work here is built from the key observations of [1,6].…”

“…Extending these results, Feller-Pohlmann-Zentner [6] used Υ K (t) to find another lower bound on the alternation number. The work here is built from the key observations of [1,6].…”