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Andrew Bartholomew

Publications and source records attributed to Andrew Bartholomew.

11 recordsLinked to original sources
arXiv2025

Generalised Biracks and the Birack Polynomial Invariant

We generalise the finite biquandle colouring invariant to a polynomial invariant based on labelling a knot diagram with a finite birack that reduces to the biquandle colouring invariant in that case. The polynomial is an invariant of a class of knot theories amenable to a generalisation of theorem of Trace on regular homotopy. We take the opportunity to reprise the relevant generalised knot theory and the theory of generalised biracks in the light of this work and recent developments.

math.GT↗
arXiv2023

Planar Doodles: Their Properties, Codes and Classification

We present those properties of planar doodles, especially when regarded as 4-valent graphs, that enable us to classify them into {\it prime} and {\it super prime} doodles by analogy to a knot sum. We describe a method for partially characterising a doodle diagram by a {\it doodle code} that describes the complementary regions of the diagram and use that code to enumerate all possible prime and super prime doodle diagrams via their dual graph. In addition we explore the relationship between planar doodles and twin groups, and note that a theorem of Tutte means that super prime doodles have a Hamiltonian circuit. We hope to expand upon this last point in a follow-up paper.

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arXiv2021

Doodles and commutator identities

A doodle is a collection of immersed circles without triple intersections in the $2$-sphere. It was shown by the second author and P.~Tayler that doodles induce commutator identities (identities amongst commutators) in a free group. In this paper we observe this idea more closely by concentrating on doodles with proper noose systems and elementary commutator identities. In particular we show that there is a bijection between cobordism classes of colored doodles and weak equivalence classes of elementary commutator identities.

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arXiv2018

Colorings and doubled colorings of virtual doodles

A virtual doodle is an equivalence class of virtual diagrams under an equivalence relation generated by flat version of classical Reidemesiter moves and virtual Reidemsiter moves such that Reidemeister moves of type 3 are forbidden. In this paper we discuss colorings of virtual diagrams using an algebra, called a doodle switch, and define an invariant of virtual doodles. Besides usual colorings of diagrams, we also introduce doubled colorings.

math.GT↗
arXiv2018

On Gauss codes of virtual doodles

We discuss Gauss codes of virtual diagrams and virtual doodles. The notion of a left canonical Gauss code is introduced and it is shown that oriented virtual doodles are uniquely presented by left canonical Gauss codes.

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arXiv2018

Doodles on surfaces

Doodles were introduced in [R. Fenn and P. Taylor, Introducing doodles, Topology of low-dimensional manifolds, pp. 37--43, Lecture Notes in Math., 722, Springer, Berlin, 1979] but were restricted to embedded circles in the 2-sphere. Khovanov, [M. Khovanov, Doodle groups, Trans. Amer. Math. Soc. 349 (1997), 2297--2315], extended the idea to immersed circles in the 2-sphere. In this paper we further extend the range of doodles to any closed orientable surface. Uniqueness of minimal representatives is proved, and various example of doodles are given with their minimal representatives. We also introduce the notion of virtual doodles, and show that there is a natural one-to-one correspondence between doodles on surfaces and virtual doodles on the plane.

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arXiv2010

Update of "Biquandles of Small Size and some Invariants of Virtual and Welded Knots"

In this paper we give the results of a computer search for biracks of small size and we give various interpretations of these findings. The list includes biquandles, racks and quandles together with new invariants of welded knots and examples of welded knots which are shown to be non-trivial by the new invariants. These can be used to answer various questions concerning virtual and welded knots. As an application we reprove the result that the Burau map from braids to matrices is non injective and give an example of a non-trivial virtual (welded) knot which cannot be distinguished from the unknot by any linear biquandles. (This is a revised version of an earlier paper of the same name)

math.GT↗
arXiv2010

Biquandles of Small Size and some Invariants of Virtual and Welded Knots

In this paper we give the results of a computer search for biracks of small size and we give various interpretations of these findings. The list includes biquandles, racks and quandles together with new invariants of welded knots and examples of welded knots which are shown to be non-trivial by the new invariants. These can be used to answer various questions concerning virtual and welded knots. As an application we reprove the result that the Burau map from braids to matrices is non injective and give an example of a non-trivial virtual (welded) knot which cannot be distinguished from the unknot by any linear biquandles.

math.GT↗
arXiv2007

New Invariants of Long Virtual Knots

This paper extends the construction of invariants for virtual knots to virtual long knots and introduces two new invariant modules of virtual long knots. Several interesting features are described that distinguish virtual long knots from their classical counterparts with respect to their symmetries and the concatenation product.

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arXiv2006

Quaternionic Invariants of Virtual Knots and Links

In this paper we define and give examples of a family of polynomial invariants of virtual knots and links. They arise by considering certain 2$\times$2 matrices with entries in a possibly non-commutative ring, for example the quaternions. These polynomials are sufficiently powerful to distinguish the Kishino knot from any classical knot, including the unknot.

math.GT↗