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All Outputs (7)

On ribbon graphs and virtual links (2022)
Journal Article
Baldridge, S., H. Kauffman, L., & Rushworth, W. (2022). On ribbon graphs and virtual links. European Journal of Combinatorics, 103, Article 103520. https://doi.org/10.1016/j.ejc.2022.103520

We introduce a new equivalence relation on decorated ribbon graphs, and show that its equivalence classes directly correspond to virtual links. We demonstrate how this correspondence can be used to convert any invariant of virtual links into an invar... Read More about On ribbon graphs and virtual links.

Ascent concordance (2021)
Journal Article
Rushworth, W. (2021). Ascent concordance. Algebraic and Geometric Topology, 21(6), 3073–3106. https://doi.org/10.2140/agt.2021.21.3073

A cobordism between links in thickened surfaces consists of a surface $ S $ and a $3$-manifold $M $, with $ S $ properly embedded in $ M \times I $. We show that there exist links in thickened surfaces such that if $(S,M) $ is a cobordism between the... Read More about Ascent concordance.

A parity for 2-colourable links (2021)
Journal Article
Rushworth, W. (2021). A parity for 2-colourable links. Osaka Journal of Mathematics, 58(4), 767–801

We introduce the 2-colour parity. It is a theory of parity for a large class of virtual links, defined using the interaction between orientations of the link components and a certain type of colouring. The 2-colour parity is an extension of the Gauss... Read More about A parity for 2-colourable links.

Minimal crossing number implies minimal supporting genus (2021)
Journal Article
Boden, H., & Rushworth, W. (2021). Minimal crossing number implies minimal supporting genus. Bulletin of the London Mathematical Society, 53(4), 1174-1184. https://doi.org/10.1112/blms.12491

A virtual link may be defined as an equivalence class of diagrams, or alternatively as a stable equivalence class of links in thickened surfaces. We prove that a minimal crossing virtual link diagram has minimal genus across representatives of the st... Read More about Minimal crossing number implies minimal supporting genus.

Computations of the slice genus of virtual knots (2018)
Journal Article
Rushworth, W. (2019). Computations of the slice genus of virtual knots. Topology and its Applications, 253, 57-84. https://doi.org/10.1016/j.topol.2018.11.028

A virtual knot is an equivalence class of embeddings of $ S^1 $ into thickened (closed oriented) surfaces, up to self-diffeomorphism of the surface and certain handle stabilisations. The slice genus of a virtual knot is defined diagrammatically, in d... Read More about Computations of the slice genus of virtual knots.

Additional gradings on generalisations of Khovanov homology and invariants of embedded surfaces (2018)
Journal Article
Olegovich Manturov, V., & Rushworth, W. (2018). Additional gradings on generalisations of Khovanov homology and invariants of embedded surfaces. Journal of Knot Theory and Its Ramifications, 27(9), https://doi.org/10.1142/S0218216518420014

We define additional gradings on two generalisations of Khovanov homology (one due to the first author, the other due to the second), and use them to define invariants of various kinds of embeddings. These include invariants of links in thickened sur... Read More about Additional gradings on generalisations of Khovanov homology and invariants of embedded surfaces.