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.

翻译：本文旨在建立一种基于图论的编织建模数学基础。我们为具有纽结理论成分的编织对象提供了精确定义，并为任意编织对象提出了简单无向图、简单有向图及有向多重图三种模型。利用这些模型，我们提出了与编织结构复杂度相关的自然分类。通过该分类体系，我们探究了判断给定图是否存在对应类别编织对象的计算复杂度问题。研究表明，虽然该问题在一般情况下属于NP难问题，但在特定情形下，利用常见编织技术的独特性质，可设计出线性与多项式时间复杂度的算法。本工作致力于弥合纺织艺术与图论之间的鸿沟，为通过对应图结构分析编织对象以及从图生成编织对象提供了严谨有效的理论框架。