Wooly Graphs : A Mathematical Framework For Knitting

TL;DR

The paper establishes Wooly Graphs as a graph-theoretic framework to model knitting by defining knit objects with knot-theoretic aspects and three graph representations: $G=(V,E)$, $Y=(V,E)$, and $B=(V,E,C)$. It introduces a taxonomy of knitting complexity (classes 0–3) tied to graph properties, and studies the knittability problem, showing NP-hardness in general while providing linear-time solutions for $1$-knittable DAGs and polynomial-time flow-based methods for broader DAG cases. Core contributions include formal definitions of knit graphs, and complexity results that connect Hamiltonian paths, Eulerian paths, and $k$-path covers to knittability, enabling both analysis and potential generation of knitting patterns from graphs. The framework lays groundwork for graph-driven knitting design and points to future work on elasticity, cables, brioche, and higher-dimensional or reversible knitting structures, broadening the intersection of textile arts and graph theory.

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.

Figures (7)

• Figure 1: A common "Knit Front and Back" (kfb) stitch operation which increases the number of stitches in a row by 1 is shown in knitting (left), following the path of yarn (middle) and the resulting graph (right).
• Figure 2: An example of a nontrivial knot formed by looping the loose ends of yarn again through their respective stitches and then connecting them. We color each stitch to show 5-colorability.
• Figure 3: An example of how a single node connects to the rest of the knitting graph. Here $S_n$ notes a single stitch with its connections. $S_l$ and $S_r$ connect along the Hamiltonian path, which can also be seen in the red strand. $S_a$ and $S_b$ connect through a loop connection.
• Figure 4: An example yarn graph for simple stockinette knitting. Here, the edges that are on the Hamiltonian are shown in black and the ones only on the Eulerian (yarn) path are shown in blue
• Figure 5: Example graph modeling from Counts_directed_2018 (colors adjusted) The blue edges represent the yarn, the red edges represent interlocking (which is actually creating the 'real' loops). Note purple edges are not visible in this section since they only appear on the exterior where the rows increment.

Theorems & Definitions (30)

• Definition 1: $1$-Knit Object
• Definition 2: $k$-Knit Object
• Definition 3: $1$-Knit Finished Object
• Definition 4: $k$-Knit Finished Object
• Definition 5: Knitting Vertices
• Definition 6: Knitting Graph
• Definition 7: Yarn Graph
• Definition 9: Directed Knitting Graph
• Theorem 10
• proof