Turaev genus and arc index: a new conjecture for links (Álvaro Del Valle, Universidad de Sevilla)

23.04.2026 14:15

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 $\alpha$. In particular, we conjecture that $c(L) + 2 - \alpha(L) \geq 2 g_T(L)$ 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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