Multi-Virtual Knot Theory
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914393939247104 |
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| author | Kauffman, Louis H |
| author_facet | Kauffman, Louis H |
| contents | This paper discusses a generalization of virtual knot theory that we call multi-virtual knot theory. Multi-virtual knot theory uses a multiplicity of types of virtual crossings. As we will explain, this multiplicity is motivated by the way it arises first in a graph-theoretic setting in relation to generalizing the Penrose evaluation for colorings of planar trivalent graphs to all trivalent graphs, and later by its uses in a virtual knot theory. As a consequence, the paper begins with the graph theory as a basis for our constructions, and then proceeds to the topology of multi-virtual knots and links. The second section of the paper is a review of our previous work (See arXiv:1511.06844). The reader interested in seeing our generalizations of the original Penrose evaluation, can begin this paper at the beginning and see the graph theory. A reader primarily interested in multi-virtual knots and links can begin reading in section 4 with references to the earlier part of the paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_07499 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multi-Virtual Knot Theory Kauffman, Louis H Geometric Topology Combinatorics 05C15, 57K10, 57K12, 57K14 This paper discusses a generalization of virtual knot theory that we call multi-virtual knot theory. Multi-virtual knot theory uses a multiplicity of types of virtual crossings. As we will explain, this multiplicity is motivated by the way it arises first in a graph-theoretic setting in relation to generalizing the Penrose evaluation for colorings of planar trivalent graphs to all trivalent graphs, and later by its uses in a virtual knot theory. As a consequence, the paper begins with the graph theory as a basis for our constructions, and then proceeds to the topology of multi-virtual knots and links. The second section of the paper is a review of our previous work (See arXiv:1511.06844). The reader interested in seeing our generalizations of the original Penrose evaluation, can begin this paper at the beginning and see the graph theory. A reader primarily interested in multi-virtual knots and links can begin reading in section 4 with references to the earlier part of the paper. |
| title | Multi-Virtual Knot Theory |
| topic | Geometric Topology Combinatorics 05C15, 57K10, 57K12, 57K14 |
| url | https://arxiv.org/abs/2409.07499 |