From integrals to combinatorial formulas of finite type invariants -- a case study

We obtain a localized version of the configuration space integral for the Casson knot invariant, where the standard symmetric Gauss form is replaced with a locally supported form. An interesting technical difference between the arguments presented here and the classical arguments is that the vanishing of integrals over hidden and anomalous faces does not require the well known "involution tricks". The integral formula easily yields the well-known arrow diagram expression for regular knot diagrams, first presented in the work by Polyak and Viro. Moreover, it yields an arrow diagram count for the multicrossing knot diagrams, such as petal diagrams and gives a new lower bound for the {\em {\"u}bercrossing number}. Previously, the known arrow diagram formulas were applicable only to the regular knot diagrams.