Average four-genus of two-bridge knots

TL;DR

The paper investigates the asymptotic behavior of the ratio between the smooth 4-genus $g_4(K)$ and the Seifert genus $g(K)$ for large 2-bridge knots presented as 4-plats. It combines combinatorial partition counting to analyze the average Seifert genus with a cobordism-based, random-walk argument to bound the average 4-genus, proving that $\lim_{n\to\infty} \langle g_4/g \rangle_n=0$ (and, equivalently, $\lim_{n\to\infty} \langle g_4 \rangle_n / n = 0$). The approach shows that although $g_4$ can be as large as $g$ for individual knots, its average growth is sublinear in the crossing-number parameter, due to cancellations and cobordism reductions. This result highlights the potential for similar sublinear behavior in other knot families and informs understanding of slice- and concordance-related properties in large classes of knots.

Abstract

We prove that the expected value of the ratio between the smooth four-genus and the Seifert genus of two-bridge knots tends to zero as the crossing number tends to infinity.

Figures (3)

• Figure 1: 2-bridge knot diagrams $C(2,2)$ and $C(-2,-2,4,-2)$
• Figure 2: Reducing the diagram $C(-2,-2,4,-2)$
• Figure 3: Cobordism of genus one (two saddle moves)

Theorems & Definitions (3)

• Theorem 1
• Lemma 1
• Lemma 2