Arc index and Turaev genus: a new conjecture for links

Speaker(s)

Alvaro Del Valle Vilchez

Affiliation

Universidad de Sevilla

Language of the talk

English

Date

Dec. 10, 2025, 10:30 a.m.

Room

room 4070

Seminar

Seminar Algebraic Topology

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The Turaev genus g_T is a numerical invariant of knots and links that measures how far a link is from being alternating (that is, from admitting an alternating diagram). Determining whether a link is alternating is, in general, not straightforward. In this work we compare the Turaev genus with another numerical invariant, the arc index α. In particular, we conjecture that

c(L) + 2 - α(L) ≥ 2 g_T(L)

for any prime, non-split link L, where c(L) is the number of crossings of L. We present various techniques to verify the conjecture for relevant families of links: adequate links, closures of positive 3-braids, torus links, and Kanenobu knots. This is joint work with Adam M. Lowrance.

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