arXiv subjects

William Rushworth

Publications and source records attributed to William Rushworth.

12 recordsLinked to original sources

arXiv2026

Some link homologies in $ \mathbb{RP}^3 $

We introduce extensions of Khovanov homology and the Lee and Bar-Natan spectral sequences for links in $ \mathbb{RP}^3 $. These extensions are distinct to those previously defined by Asaeda-Przytycki-Sikora (and Gabrov\v{s}ek's generalization), Chen, and Manolescu-Willis. The new Lee and Bar-Natan theories each yield Rasmussen invariants (that are distinct to one another). The invariant extracted from the new Lee homology is distinct to that defined by Manolescu-Willis; it is unclear if the same is true for the new Bar-Natan homology and that defined by Chen.

math.GT↗

arXiv2022

On knots that divide ribbon knotted surfaces

We define a knot to be half ribbon if it is the cross-section of a ribbon 2-knot, and observe that ribbon implies half ribbon implies slice. We introduce the half ribbon genus of a knot K, the minimum genus of a ribbon knotted surface of which K is a cross-section. We compute this genus for all prime knots up to 12 crossings, and many 13-crossing knots. The same approach yields new computations of the doubly slice genus. We also introduce the half fusion number of a knot K, that measures the complexity of ribbon 2-knots of which K is a cross-section. We show that it is bounded from below by the Levine-Tristram signatures, and differs from the standard fusion number by an arbitrarily large amount.

math.GT↗

arXiv2022

On ribbon graphs and virtual links

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 invariant of ribbon graphs, and vice versa.

math.GT↗

arXiv2021

Minimal crossing number implies minimal supporting genus

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 stable equivalence class. This is achieved by constructing a new parity theory for virtual links. As corollaries, we prove that the crossing, bridge, and ascending numbers of a classical link do not decrease when it is regarded as a virtual link. This extends corresponding results in the case of virtual knots due to Manturov and Chernov.

math.GT↗

arXiv2020

A parity for 2-colourable links

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 Gaussian parity, to which it reduces on virtual knots. We show that the 2-colour parity descends to a parity on free links. We compare the 2-colour parity to other parity theories of virtual links, focusing on a theory due to Im and Park. The 2-colour parity yields a strictly stronger invariant than the Im-Park parity. We introduce an invariant, the 2-colour writhe, that takes the form of a string of integers. The 2-colour writhe is a concordance invariant, and so obstructs sliceness. It is also an obstruction to amphichirality and chequerboard colourability within a concordance class.

math.GT↗

arXiv2020

Ascent concordance

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 them in which $ S $ is simple, then $ M $ must be complex. That is, there are cases in which low complexity of the surface does not imply low complexity of the $3$-manifold. Specifically, we show that there exist concordant links in thickened surfaces between which a concordance can only be realised by passing through thickenings of higher genus surfaces. We exhibit an infinite family of such links that are detected by an elementary method and other families of links that are not detectable in this way. We investigate an augmented version of Khovanov homology, and use it to detect these families. Such links provide counterexamples to an analogue of the Slice-Ribbon conjecture.

math.GT↗

arXiv2020

Generalized Fishburn numbers and torus knots

Andrews and Sellers recently initiated the study of arithmetic properties of Fishburn numbers. In this paper, we prove prime power congruences for generalized Fishburn numbers. These numbers are the coefficients in the $1-q$ expansion of the Kontsevich-Zagier series $\mathscr{F}_{t}(q)$ for the torus knots $T(3,2^t)$, $t \geq 2$. The proof uses a strong divisibility result of Ahlgren, Kim and Lovejoy and a new "strange identity" for $\mathscr{F}_{t}(q)$.

math.NT↗

arXiv2019

Ascent sliceness

We introduce the notion of ascent sliceness of virtual knots. A representative of a virtual knot is an embedding $ S^1 \hookrightarrow Σ_{g} \times I $, for $ Σ_g $ a closed connected oriented surface of genus $ g $; the virtual knot represented is slice if there exists a pair consisting of a disc $ D $ and an oriented $ 3 $-manifold $ M $, such that $ D \hookrightarrow M \times I $, $ \partial M = Σ_{g} $, and $ \partial D = S^1 $ (the image of the embedding). This definition of sliceness exemplifies that a cobordism of virtual links is a pair consisting of a surface and a $ 3 $-manifold; in addition to analysing the surfaces, as is done in classical knot theory, we may analyse the $ 3 $-manifolds appearing in cobordisms between virtual knots. In particular, consider a Morse function on the $ 3 $-manifold $ M $: away from critical points the level sets are surfaces, and we may ask how the genus of these surfaces changes as we move through the cobordism. Roughly, a slice virtual knot $ K $ with genus-minimal representative $ S^1 \hookrightarrow Σ_{g} \times I $ is ascent slice if, given any disc and $ 3 $-manifold pair $ ( D, M ) $ as above, and any Morse function $ f : M \rightarrow I $, the surface $ Σ_{g+1} $ appears as a level set of $ f $. We use an augmented version of doubled Khovanov homology to define a property which implies ascent sliceness for slice virtual knots of minimal supporting genus $ 1 $.

math.GT↗

arXiv2017

Additional gradings on generalisations of Khovanov homology and invariants of embedded surfaces

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 surfaces and of surfaces embedded in thickened $3$-manifolds. In particular, the invariants of embedded surfaces are expressed in terms of certain diagrams related to the thickened $3$-manifold, so that we refer to them as picture-valued invariants. This paper contains the first instance of such invariants for $2$-dimensional objects. The additional gradings are defined using cohomological and homotopic information of surfaces: using this information we decorate the smoothings of the standard Khovanov cube, before transferring the decorations into algebra.

math.GT↗

arXiv2017

Computations of the slice genus of virtual knots

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 direct analogy to that of a classical knot. However, it may be defined, equivalently, as follows: a representative of a virtual knot is an embedding of $ S^1 $ into a thickened surface $ Σ_g \times I $; what is the minimal genus of oriented surfaces $ S \hookrightarrow M \times I $ with the embedded $ S^1 $ as boundary, where $ M $ is an oriented $ 3 $-manifold with $ \partial M = Σ_g $? We compute and estimate the slice genus of all virtual knots of $4$ classical crossings or less. We also compute or estimate the slice genus of $46$ virtual knots of $5$ and $6$ classical crossings whose slice status is not determined in the work of Boden, Chrisman, and Gaudreau. The computations are made using two distinct virtual extensions of the Rasmussen invariant, one due to Dye, Kaestner, and Kauffman, the other due to the author. Specifically, the computations are made using bounds on the two extensions of the Ramussen invariant which we construct and investigate. The bounds are themselves generalisations of those on the classical Rasmussen invariant due, independently, to Kawamura and Lobb. The bounds allow for the computation of the extensions of the Rasmussen invariant in particular cases. As asides we identify a class of virtual knots for which the two extensions of the Rasmussen invariant agree, and show that the extension due to Dye, Kaestner, and Kauffman is additive with respect to the connect sum.

math.GT↗

arXiv2017

On the virtual Rasmussen invariant

We produce chain-level generators of the virtual Lee complex $ Kh ' (V ) $ and use them to convert the computable bounds on the Rasmussen invariant of classical knots due to Kawamura and Lobb into bounds on the virtual Rasmussen invariant as defined by Dye, Kaestner, and Kauffman. We also exhibit a class of diagrams for which the bounds are tight. In addition, we use the chain-level generators to show that the virtual Rasmussen invariant is additive with respect to connect sum.

math.GT↗

arXiv2017

Doubled Khovanov Homology

We define a homology theory of virtual links built out of the direct sum of the standard Khovanov complex with itself, motivating the name doubled Khovanov homology. We demonstrate that it can be used to show that some virtual links are non-classical, and that it yields a condition on a virtual knot being the connect sum of two unknots. Further, we show that doubled Khovanov homology possesses a perturbation analogous to that defined by Lee in the classical case and define a doubled Rasmussen invariant. This invariant is used to obtain various cobordism obstructions; in particular it is an obstruction to sliceness. Finally, we show that the doubled Rasmussen invariant contains the odd writhe of a virtual knot, and use this to show that knots with non-zero odd writhe are not slice.

math.GT↗