Asymptotic behavior of unknotting numbers of links in a twist family

Abstract

By twisting a given link $L$ along an unknotted circle $c$, we obtain an infinite family of links $\{ L_n \}$. We introduce the ``stable unknotting number'' which describes the asymptotic behavior of unknotting numbers of links in the twist family. We show the stable unknotting number for any twist family of links depends only on the winding number of $L$ about $c$ (the minimum geometric intersection number of $L$ with a Seifert surface of $c$) and is independent of the wrapping number of $L$ about $c$ (the minimum geometric intersection number of $L$ with a disk bounded by $c$). Thus there are twist families for which the discrepancy between the wrapping number and the stable unknotting number is arbitrarily large.