Asymptotic behavior of unknotting numbers of links in a twist family

TL;DR

This paper studies the asymptotic behavior of unknotting numbers in twist families obtained by twisting a link $L$ about a disjoint unknot $c$, introducing the stable unknotting number $u_s(L_n)=\\lim_{n\\to\\infty} u(L_n)/n$ and showing it depends only on the winding number $\\omega$ via $u_s(L_n)=\\frac{\\omega(\\\\omega-1)}{2}$, independent of the wrapping number $\\eta$. It proves the result by combining a linear lower bound from the smooth slice genus $g_4$ with an upper bound derived from a cobordism argument and the Rasmussen $s$-invariant, and it establishes an equivalence: $u_s(L_n)=0$ iff $\\omega \\le 1$ iff $u(L_n)$ is bounded. The paper also shows that the discrepancy between wrapping and stable unknotting can be made arbitrarily large by constructing twist families with large wrapping number relative to a fixed winding number, using the set $\\mathcal{L}(\\ell, m)$ of links with wind $\\ell$ and wrap $m$. Overall, the work connects stable unknotting numbers to stable slice genus and provides explicit mechanisms to realize large wrapping numbers without increasing $u_s$.

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.

Figures (4)

• Figure 1.1: Twisting $L$$n$--times about $c$; $(n = 1)$
• Figure 1.2: Unknotting numbers of the knots $K_n$ is bounded when $K \cup c$ is a Whitehead link (above) or a Mazur link (below).
• Figure 2.1: $n$--twist and $k$ crossing changes; $\eta = 9, \omega = 3$
• Figure 2.2:

Theorems & Definitions (15)

• Remark 1.1
• Definition 1.2
• Example 1.3: Torus knots
• Example 1.4: Whitehead link and Mazur link
• Theorem 1.5
• Remark 1.7
• Remark 1.8
• Corollary 1.9
• Proposition 1.10
• proof : Proof of Theorem \ref{['stable_unknotting']}.