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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., Comment: 28 pages, many figures and tables

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.

In this paper we look at which Alexander and Markov theories can be defined for generalized knot theories

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.

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.

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.

This paper is a concise introduction to virtual knot theory, coupled with a list of research problems in this field., Comment: 65 pages, 24 figures. arXiv admin note: text overlap with arXiv:math/0405428

The classical trefoil is famous for having a three-colouring which distinguishes it from the unknot. The three-colouring is also notorious for not distinguishing the right handed from the left handed trefoil. However with a bit of tweaking the three colours can also be used for this task. What lies behind the method is a new operation on biracks called doubling which converts the 3-colour quandle into a biquandle. Colouring with this biquandle distinguishes the right handed from the left handed trefoil. Equivalently it defines an element of the homology of the quandle or biquandle classifying space.

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)

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., Comment: 14 pages, computer output of small biquandles, new classical, virtual and welded knot invariants

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., Comment: 14 pages 11 figures

Let $A, B$ be invertible, non-commuting elements of a ring $R$. Suppose that $A-1$ is also invertible and that the equation $$[B,(A-1)(A,B)]=0$$ called the fundamental equation is satisfied. Then an invariant $R$-module is defined for any diagram of a (virtual) knot or link. Solutions in the classic quaternion case have been found by Bartholomew, Budden and Fenn. Solutions in the generalised quaternion case have been found by Fenn in an earlier paper. These latter solutions are only partial in the case of $2\times2$ matrices and the aim of this paper is to provide solutions to the missing cases., Comment: 11 pages, accepted by JKTR

In this paper we push forward results on the invariant ${\cal F}$-module of a virtual knot investigated by the first named author where ${\cal F}$ is the algebra with two invertible generators $A,B$ and one relation $A^{-1}B^{-1}AB-B^{-1}AB= BA^{-1}B^{-1}A-A$. For flat knots and links the two sides of the relation equation are put equal to unity and the algebra becomes the Weyl algebra. If this is perturbed and the two sides of the relation equation are put equal to a general element, $q$, of the ground ring, then the resulting module lays claim to be the correct generalization of the Alexander module. Many finite dimensional representations are given together with calculations., Comment: 18 pages, 5 figures, accepted by Journal of Geometry and Physics

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., Comment: 21 pages, 13 figures, accepted by JKTR

In this paper we show how generalized quaternions, including 2X2 matrices, can be used to find solutions of a non-commuting equation intimately connected with braid groups. These solutions can then be used to find polynomial invariants of virtual knots and links., Comment: 25 pages, 13 figures

This paper is an introduction to the theory of virtual knots and links and it gives a list of unsolved problems in this subject., Comment: 33 pages, 7 figures, LaTeX document

The main result of this paper is a new classification theorem for links (smooth embeddings in codimension 2). The classifying space is the rack space (defined in [Trunks and classifying spaces, Applied Categorical Structures, 3 (1995) 321--356]) and the classifying bundle is the first James bundle (defined in "James bundles" math.AT/0301354). We investigate the algebraic topology of this classifying space and report on calculations given elsewhere. Apart from defining many new knot and link invariants (including generalised James--Hopf invariants), the classification theorem has some unexpected applications. We give a combinatorial interpretation for \pi_2 of a complex which can be used for calculations and some new interpretations of the higher homotopy groups of the 3--sphere. We also give a cobordism classification of virtual links., Comment: This paper is largely extracted from our January 1996 preprint `James bundles and applications' available at http://www.maths.warwick.ac.uk/~cpr/ftp/james.ps Version 2: minor corrections

We study cubical sets without degeneracies, which we call square sets. These sets arise naturally in a number of settings and they have a beautiful intrinsic geometry; in particular a square set C has an infinite family of associated square sets J^i(C), i=1,2,..., which we call James complexes. There are mock bundle projections p_i:|J^i(C)|-->|C| (which we call James bundles) defining classes in unstable cohomotopy which generalise the classical James--Hopf invariants of Omega(S^2). The algebra of these classes mimics the algebra of the cohomotopy of Omega(S^2) and the reduction to cohomology defines a sequence of natural characteristic classes for a square set. An associated map to BO leads to a generalised cohomology theory with geometric interpretation similar to that for Mahowald orientation [M Mahowald, Ring Spectra which are Thom complexes, Duke Math. J. 46 (1979) 549--559] and [B Sanderson, The geometry of Mahowald orientations, SLN 763 (1978) 152--174]., Comment: This paper is extracted from our January 1996 preprint `James bundles and applications' available at: http://www.maths.warwick.ac.uk/~cpr/ftp/james.ps

We study equations over torsion-free groups in terms of their `t-shape' (the occurences of the variable t in the equation). A t-shape is good if any equation with that shape has a solution. It is an outstanding conjecture that all t-shapes are good. In [Klyachko's methods and the solution of equations over torsion-free groups, l'Enseign. Maths. 42 (1996) 49--74] we proved the conjecture for a large class of t-shapes called amenable. In [Tesselations of S^2 and equations over torsion-free groups, Proc. Edinburgh Maths. Soc. 38 (1995) 485--493] Clifford and Goldstein characterised a class of good t-shapes using a transformation on t-shapes called the Magnus derivative. In this note we introduce an inverse transformation called blowing up. Amenability can be defined using blowing up; moreover the connection with differentiation gives a useful characterisation and implies that the class of amenable t-shapes is strictly larger than the class considered by Clifford and Goldstein., Comment: 8 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTMon1/paper8.abs.html

We give an explicit geometric argument that Artin's braid group $B_n$ is right-orderable. The construction is elementary, natural, and leads to a new, effectively computable, canonical form for braids which we call left-consistent canonical form. The left-consistent form of a braid which is positive (respectively negative) in our order has consistently positive (respectively negative) exponent in the smallest braid generator which occurs. It follows that our ordering is identical to that of Dehornoy, constructed by very different means, and we recover Dehornoy's main theorem that any braid can be put into such a form using either positive or negative exponent in the smallest generator but not both. Our definition of order is strongly connected with Mosher's normal form and this leads to an algorithm to decide whether a given braid is positive, trivial, or negative which is quadratic in the length of the braid word., Comment: 24 pages, 10 figures