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

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).