Wooly Graphs : A Mathematical Framework For Knitting

TL;DR

This paper develops a mathematical graph-based framework for modeling and analyzing knitting structures, establishing complexity results and efficient algorithms for certain classes of knitting graphs.

Contribution

It introduces novel graph models for knitting objects and explores their computational complexity, bridging textile arts and graph theory.

Findings

NP-hardness of general knitting graph problems

Polynomial algorithms for specific knitting techniques

Framework for generating knitting objects from graphs

Abstract

This paper aims to develop a mathematical foundation to model knitting with graphs. We provide a precise definition for knit objects with a knot theoretic component and propose a simple undirected graph, a simple directed graph, and a directed multigraph model for any arbitrary knit object. Using these models, we propose natural categories related to the complexity of knitting structures. We use these categories to explore the hardness of determining whether a knit object of each class exists for a given graph. We show that while this problem is NP-hard in general, under specific cases, there are linear and polynomial time algorithms which take advantage of unique properties of common knitting techniques. This work aims to bridge the gap between textile arts and graph theory, offering a useful and rigorous framework for analyzing knitting objects using their corresponding graphs and for generating knitting objects from graphs.