A polynomial invariant for plane curve complements: Krammer polynomials

TL;DR

This paper introduces the Krammer polynomial, a new multivariate invariant for plane curve complements based on the Krammer representation of the braid group, with computations for specific n-gonal curves.

Contribution

The paper constructs a novel polynomial invariant for curve complements using the Krammer representation, extending Libgober's invariant.

Findings

Krammer polynomial of an essential braid equals zero

Computed Krammer polynomials for certain n-gonal curves

Established properties of the new invariant

Abstract

We use the Krammer representation of the braid group in Libgober's invariant and construct a new multivariate polynomial invariant for curve complements: Krammer polynomial. We show that the Krammer polynomial of an essential braid is equal to zero. We also compute the Krammer polynomials of some certain n-gonal curves.