A representation of Milnor's triple linking number by chord diagrams and doodle invariants
Ryosuke Hirata
TL;DR
This work expresses Milnor's triple linking number $\bar{\mu}_L(123)$ for a three-component link as a sum of a doodle invariant $-\mu(C_1,C_2,C_3)$ and a collection of chord-diagram–type contributions, computed modulo the gcd $\Delta_L(123)$. Building on Mellor–Melvin's geometric interpretation, the authors couple Seifert-surface data with generalized chord diagrams built from a projection (doodle) of the link, showing that the triple linking number decomposes into a doodle term and explicit interaction terms $\langle j i, G_{L_k}\rangle$ over the alternating group $\mathfrak{A}_3$. The main result formalizes this decomposition: $\bar{\mu}_L(123) \equiv -\mu(C_1,C_2,C_3) - \sum_{(i,j,k)\in \mathfrak{A}_3} \langle j i, G_{L_k}\rangle \pmod{\Delta_L(123)}$, and the proof ties the signed triple points produced by surface stretching to these chord-diagram interactions. Overall, the paper provides a computational framework that parallels Lin–Wang's approach for degree-two knot invariants, translating Milnor's invariant into planar doodle data plus chord-diagram counts with explicit moduli.
Abstract
From the work of X. S. Lin and Z. Wang, it follows that degree two knot invariant admits a decomposition into the sum of a Gauss diagram count and a term involving Arnold invariants. In this paper we establish an analogous description for Milnor's triple linking number - likewise of degree two - showing that it can be represented in terms of counts of certain chord diagrams together with doodle invariants.