Combinatorial cusp count and clover invariants

TL;DR

The paper addresses the problem of quantifying locally flat topological cobordisms between torus links $T(m,n)$ and large connected sums of trefoils $3_1^N$ via the cobordism distance $d_\chi$ and the invariant $\sigma_6$. It develops a two-pronged approach: explicit minimal cobordisms for 6-strand torus links and a cabling/twisting (McCoy) construction to extend results to general torus links, enabling affine-error bounds related to $m$ and $n$ and establishing the leading quadratic term $\frac{5}{18} mn$ for $\sigma_6(T(m,n))$. The paper shows that any clover invariant $\rho$ is controlled by these bounds, yielding $\rho(T(m,n)) \ge \frac{5}{18} mn - A m - B n - C$, and demonstrates that $\sigma_6$ essentially dominates the class of clover invariants on torus links. By connecting cusp-count-like questions to topological 4-genus bounds, it provides quantitative control over concordance invariants in this knot-theoretic setting.

Abstract

We construct efficient topological cobordisms between torus links and large connected sums of trefoil knots. As an application, we show that the signature invariant $σ_ω$ at $ω=ζ_6$ takes essentially minimal values on torus links among all concordance homomorphisms with the same normalisation on the trefoil knot.

Figures (2)

• Figure 1: $(a^2cba^3cb)^4=(abc)^{12}$
• Figure 2: Five saddle moves

Theorems & Definitions (9)

• Theorem 1
• Corollary 1
• Proposition 1
• Lemma 1
• proof : Proof of Lemma \ref{['fourstrand']}
• proof : Proof of Proposition \ref{['sixstrand']}
• Lemma 2
• proof : Proof of Lemma \ref{['twisting']}
• proof : Proof of Theorem \ref{['trefoil']}