# Transition polynomial as a weight system for binary delta-matroids

## TL;DR

This paper extends the transition polynomial invariant from chord diagrams to ribbon graphs and binary delta-matroids, establishing a new finite type invariant for links based on these generalized structures.

## Contribution

It generalizes the transition polynomial as a weight system for binary delta-matroids, linking it to finite type invariants of links.

## Key findings

- Transition polynomial is a finite type invariant for links.
- Extension of the polynomial to ribbon graphs and delta-matroids.
- Provides a new algebraic framework for link invariants.

## Abstract

To a singular knot K with n double points, one can associate a chord diagram with n chords. A chord diagram can also be understood as a 4-regular graph endowed with an oriented Euler circuit. L. Traldi introduced a polynomial invariant for such graphs, called a transition polynomial. We specialize this polynomial to a multiplicative weight system, that is, a function on chord diagrams satisfying 4-term relations and determining thus a finite type knot invariant. We prove a similar statement for the transition polynomial of general ribbon graphs and binary delta-matroids defined by R. Brijder and H. J. Hoogeboom, which defines, as a consequence, a finite type invariant of links.