The Brylinski beta function of a surface

TL;DR

This paper introduces a Brylinski beta function analogue for surfaces in Euclidean space, analyzes its properties as a meromorphic function, and computes initial residues for a surface in three dimensions.

Contribution

It extends Brylinski's knot beta function concept to surfaces, providing new insights into its meromorphic structure and residue calculations.

Findings

Defined a Brylinski beta function analogue for surfaces

Proved the function is meromorphic on the complex plane

Computed initial residues for a surface in three dimensions

Abstract

An analogue of Brylinski's knot beta function is defined for a submanifold of d-dimensional Euclidean space. This is a meromorphic function on the complex plane. The first few residues are computed for a surface in three dimensional space.