On the Combinatorics of Smoothing

TL;DR

This paper develops a systematic approach using graph spectral theory and a modified theorem to count connected components in smoothed knot diagrams, aiding in the computation of knot invariants.

Contribution

It introduces a novel method combining spectral graph theory and a modified Zulli theorem to analyze smoothing invariants in knot theory.

Findings

Provides a systematic way to count components after smoothing

Applies method to pretzel knots for specific smoothing configurations

Enhances computational tools for knot invariants

Abstract

Many invariants of knots rely upon smoothing the knot at its crossings. To compute them, it is necessary to know how to count the number of connected components the knot diagram is broken into after the smoothing. In this paper, it is shown how to use a modification of a theorem of Zulli together with a modification of the spectral theory of graphs to approach such problems systematically. We give an application to counting subdiagrams of pretzel knots which have one component after oriented and unoriented smoothings.