Braids with as many full twists as strands realize the braid index

Abstract

We characterize the fractional Dehn twist coefficient of a braid in terms of a slope of the homogenization of the Upsilon function, where Upsilon is the function‐valued concordance homomorphism defined by Ozsváth, Stipsicz, and Szabó. We use this characterization to prove that n‐braids with fractional Dehn twist coefficient larger than n−1 realize the braid index of their closure. As a consequence, we are able to prove a conjecture of Malyutin and Netsvetaev stating that n‐times twisted braids realize the braid index of their closure. We provide examples that address the optimality of our results. The paper ends with an appendix about the homogenization of knot concordance homomorphisms.

Examples of non-minimal open books with high fractional Dehn twist coefficient

“…. We were able to answer Question 3 in the positive for n = 3 and, for general n, with the stronger assumption |ω(β)| > n − 1; see [FH19]. Hence, we feel justified to ask Question 1.…”

The upsilon invariant at 1 of 3–braid knots

“…Figure 3: A schematic of a cobordism between K D O and the connected sum of torus knots J " D T 2; P r i D1 p i C" p # T 2;q 1 C" 1 # T 2;q 2 C" 2 # # T 2;q r C" r realized by r 1 C " saddle moves. so (18) and (19)…”

“…Remark 6.2 The fractional Dehn twist coefficient was computed for 3-braids in Murasugi normal form (see Definition 4.15) in [29, Proposition 6.6].In the proof of Corollary 6.1, we will use that the fractional Dehn twist coefficient of any 3-braid is completely determined by the writhe wr. / and the homogenized upsilon invariant Q of : we have, by[19, Theorem 1.3],(32)!. / D Q .…”