A study of 2-periodic weft-knitted textiles using the theory of knots and links
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| author | Kuzbary, Miriam Markande, Shashank G. Matsumoto, Elisabetta A. Pritchard, Stanley |
| author_facet | Kuzbary, Miriam Markande, Shashank G. Matsumoto, Elisabetta A. Pritchard, Stanley |
| contents | In this study, we use a correspondence between two-periodic weft-knitted textiles and links in the thickened torus to study the former using link invariants. We establish a criterion to identify the set of links whose elements are realized through techniques of weft-knitting leading to new, unconventional types of weft-knitting stitch patterns. A crucial topological underpinning of these links is shown to be their correspondence with ribbon knots and links in Euclidean three-space and equivalently in the three-sphere. Using the mechanics of weft-knitting, we propose a protocol for constructing and enumerating links in the thickened torus that can be knitted as a motif of a weft-knitted textile, and we call such links \emph{swatches}. Based on our analysis of link invariants of swatches, we propose conjectures on hyperbolic structure of the link complements of swatches and their multivariable Alexander polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_08384 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A study of 2-periodic weft-knitted textiles using the theory of knots and links Kuzbary, Miriam Markande, Shashank G. Matsumoto, Elisabetta A. Pritchard, Stanley Geometric Topology General Topology 57K10 In this study, we use a correspondence between two-periodic weft-knitted textiles and links in the thickened torus to study the former using link invariants. We establish a criterion to identify the set of links whose elements are realized through techniques of weft-knitting leading to new, unconventional types of weft-knitting stitch patterns. A crucial topological underpinning of these links is shown to be their correspondence with ribbon knots and links in Euclidean three-space and equivalently in the three-sphere. Using the mechanics of weft-knitting, we propose a protocol for constructing and enumerating links in the thickened torus that can be knitted as a motif of a weft-knitted textile, and we call such links \emph{swatches}. Based on our analysis of link invariants of swatches, we propose conjectures on hyperbolic structure of the link complements of swatches and their multivariable Alexander polynomials. |
| title | A study of 2-periodic weft-knitted textiles using the theory of knots and links |
| topic | Geometric Topology General Topology 57K10 |
| url | https://arxiv.org/abs/2510.08384 |