30 citations · 57 across the 16 of their papers we have counts for
18 papers · 1 filter
Planar Doodles: Their Properties, Codes and Classification
Andrew Bartholomew, Roger Fenn
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 analog…
Doodles and commutator identities
Andrew Bartholomew, Roger Fenn, Naoko Kamada +1
A doodle is a collection of immersed circles without triple intersections in the -sphere. It was shown by the second author and P.~Tayler that doodles induce commutator identiti…
Alexander and Markov Theorems for Generalized Knots, I
Andrew Bartholomew, Roger Fenn
In this paper we look at which Alexander and Markov theories can be defined for generalized knot theories
Colorings and doubled colorings of virtual doodles
Andrew Bartholomew, Roger Fenn, Naoko Kamada +1
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 s…
On Gauss codes of virtual doodles
Andrew Bartholomew, Roger Fenn, Naoko Kamada +1
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 p…
Doodles on surfaces
Andrew Bartholomew, Roger Fenn, Naoko Kamada +1
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…
Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory
Roger Fenn, Denis P. Ilyutko, Louis H. Kauffman +1
This paper is a concise introduction to virtual knot theory, coupled with a list of research problems in this field.
Tackling the Trefoils
Roger Fenn
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 f…
Update of "Biquandles of Small Size and some Invariants of Virtual and Welded Knots"
Andrew Bartholomew, Roger Fenn
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 quan…
Biquandles of Small Size and some Invariants of Virtual and Welded Knots
Andrew Bartholomew, Roger Fenn
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 quan…
New Invariants of Long Virtual Knots
Andrew Bartholomew, Roger Fenn, Naoko Kamada +1
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 featur…
Quaternionic Invariants of Virtual Knots and Links
Andrew Bartholomew, Roger Fenn
In this paper we define and give examples of a family of polynomial invariants of virtual knots and links. They arise by considering certain 22 matrices with entries in a p…
Quaternion Algebras and Invariants of Virtual Knots and Links II: The Hyperbolic Case
Steven Budden, Roger Fenn
Let be invertible, non-commuting elements of a ring . Suppose that is also invertible and that the equation called the fundamental equation is…
Weyl Algebras and Knots
Roger Fenn, Vladimir Turaev
In this paper we push forward results on the invariant -module of a virtual knot investigated by the first named author where is the algebra with two invertibl…
Quaternion Algebras and Invariants of Virtual Knots and Links I: The Elliptic Case
Roger Fenn
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 s…
Problems on invariants of knots and 3-manifolds
J. E. Andersen, N. Askitas, D. Bar-Natan +57
This is a list of open problems on invariants of knots and 3-manifolds with expositions of their history, background, significance, or importance. This list was made by editing ope…
The rack space
Roger Fenn, Colin Rourke, Brian Sanderson
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 classi…
Ordering the braid groups
Roger Fenn, Michael T Greene, Dale Rolfsen +2
We give an explicit geometric argument that Artin's braid group is right-orderable. The construction is elementary, natural, and leads to a new, effectively computable, canon…