Polyak-Viro type formula for the Milnor triple linking number of link diagrams with multiple-crossings

TL;DR

This work develops a Polyak-Viro type formula for the Milnor triple linking number $_{123}$ that applies to diagrams with triple or higher crossings, extending the Brooks–Komendarczyk localization program. By explicitly computing the configuration-space integrals associated to the Y-graph and by using arrow/Gauss diagrams, the authors derive a concrete combinatorial formula for $$ in terms of diagram pairings $igraket{oldsymbol{}_k,G_D}$ and their primed counterparts, paralleling the Casson invariant's formula. The paper demonstrates the generalized Polyak-type formulas for both the Casson invariant and the Milnor triple linking number in the presence of multiple crossings, showing how triple-point configurations contribute via explicit half-sum terms while certain contributions vanish by symmetry. The approach provides a direct, computable link between diagrammatic data and topological invariants, with potential utility for Vassiliev theory and higher-dimensional embedding invariants.

Abstract

We obtain Polyak-Viro type formula for the Milnor triple linking number that can be applied to diagrams with triple or more multiple-crossings. The proof is based on the idea of Brooks and Komendarczyk, but is different from theirs in that we explicitly compute the value of configuration space integral associated to the "Y-graph," and is applicable to the original Brooks-Komendarczyk formula for the Casson knot invariant. The results for the Milnor triple linking number were firstly obtained in the first author's master thesis, in which the proof follows the same way as that in the paper of Brooks-Komendarczyk (2024).

Figures (18)

• Figure 2.1: Long knot and 3-component long link
• Figure 2.2: Examples of arrow / Gauss diagrams for double crossings
• Figure 2.3: Examples of arrow / Gauss diagrams for triple crossings; the arc (2) is front of (3)
• Figure 2.4: Examples of arrow / Gauss diagrams for triple crossings; the three arcs are contained in mutually distinct components of a 3-component link, and the arc of $f_1$ is front of that of $f_3$
• Figure 2.5: Arrow diagrams $X$ and $X'_k$ ($k=1,2,3$)

Theorems & Definitions (18)

• Definition 2.1
• Remark 2.2
• Definition 2.3
• Theorem 2.4: BrooksKomendarczyk24, see also PolyakViro01
• Definition 2.5
• Theorem 2.6
• Remark 2.7
• Example 3.1
• Proposition 3.2
• proof