Knitting Knots & the Framing Anomaly

TL;DR

This paper explores the mathematical and physical aspects of twisting faults in circular knitting, linking them to framing curves, the Gauss linking integral, and Chern-Simons gauge theory, through experimental and computational methods.

Contribution

It introduces a novel connection between knitting faults and advanced mathematical concepts like framing and the Gauss linking integral, with experimental validation.

Findings

Knitted bands with different framing values were successfully created.

Three techniques for computing framing using the Gauss linking integral were demonstrated.

A brief connection to Chern-Simons gauge theory was established.

Abstract

We study the twisting fault emerging in circular knitting and its relation to the mathematical concepts of framing curves and the Gauss linking integral. We create three knitted bands with framing zero, one, and negative two, and use three different techniques to compute the framing using the Gauss linking integral. We also briefly mention the connection to in Chern-Simons gauge theory.