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- W4318225423 abstract "Virtual knots are defined diagrammatically as a collection of figures, called virtual knot diagrams, that are considered equivalent up to finite sequences of extended Reidemeister moves. By contrast, knots in $mathbb{R}^3$ can be defined geometrically. They are the points of a space $mathbb{K}$ of knots. The knot space has a topology so that equivalent knots lie in the same path component. The aim of this paper is to use sheaf theory to obtain a fully geometric model for virtual knots. The geometric model formalizes the intuitive notion that a virtual knot is an actual knot residing in a variable ambient space; the usual diagrammatic theory follows as in the classical case. To do this, it is shown that there exists a site $(textbf{VK}, J_{textbf{VK}})$ so that its category $text{Sh}(textbf{VK},J_{textbf{VK}})$ of sheaves can be naturally interpreted as the ``space of virtual knots''. A point of this Grothendieck topos, that is a geometric morphism $textbf{Sets} to text{Sh}(textbf{VK})$, is a virtual knot. The virtual isotopy relation is generated by paths in this space, or more precisely, geometric morphisms $text{Sh}([0,1]) to text{Sh}(textbf{VK},J_{textbf{VK}})$. Virtual knot invariants valued in a discrete topological space $mathbb{G}$ are geometric morphisms $text{Sh}(textbf{VK},J_{textbf{VK}}) to text{Sh}(mathbb{G})$, just as classical knot invariants valued in $mathbb{G}$ are continuous functions $mathbb{K} to mathbb{G}$. The embedding of classical knots into virtual knots is also realized as a geometric morphism." @default.
- W4318225423 created "2023-01-27" @default.
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- W4318225423 date "2023-01-24" @default.
- W4318225423 modified "2023-09-29" @default.
- W4318225423 title "A geometric foundation of virtual knot theory" @default.
- W4318225423 doi "https://doi.org/10.48550/arxiv.2301.10318" @default.
- W4318225423 hasPublicationYear "2023" @default.
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