A Reciprocity Formula on Multicurves

TL;DR

This paper establishes a reciprocity formula for a power series associated with multicurves on punctured surfaces, generalizing previous results and providing new insights into intersection counting.

Contribution

It introduces a reciprocity formula for intersection-based power series on multicurves, extending prior work to more general multicurve configurations.

Findings

Derived a new reciprocity formula for multicurve intersection series

Generalized previous reciprocity results to multicurves with no contractible components

Provides a framework for counting intersections on punctured surfaces

Abstract

Given a specific collection of curves on an oriented surface with punctures, we associate a power series by counting its intersections with multicurves. This paper presents a reciprocity formula on the power series when multicurves with no component contractible to a puncture are concerned, as a generalization of the reciprocity presented in arXiv:1612.02518.