Area of Research
Low dimensional topology, knots links and braids, generalised braids and associated algebra. An elementary introduction to braid theory in postscript form can be downloaded by clicking here
Recent work
  1. The theory of knots and links has given rise to much algebraic theory which has been invented to try and understand the very difficult problems in this area. An example of such an algebraic object is a rack which has been extensively explained by myself and [C.P.Rourke ] of the Mathematics Institute at Warwick. ps paper to download Ref: "Racks and Links in Codimension Two" (with C. Rourke) {Journal of Knot Theory and its Ramifications 4 (1992) 343--406.} In further work we have combined to work with [ B.J.Sanderson, ] also of Warwick, on classifying spaces of racks and general spaces of this type which are built up out of cubes. ps paper to download. Ref: "Trunks and Classifying Spaces" (with B.Sanderson and C.P. Rourke) {Applied Categorical Structures} 3 (1995) pp 321--356 and ps paper to download. Ref: "An introduction to Species and the Rack Space" (with B.Sanderson and C.Rourke). {Topics in Knot Theory: Kluwer Academic Publishers 1993 pp 33--55} or [ B.J.Sanderson, ] For a history of this idea, which is dated 1990, see the following scans of documents more etc etc etc etc etc etc etc etc etc etc etc For 2 recent papers on the arXiv see or here.
  2. Welded knots and braids occupy a position between classical and virtual varieties. Here is a paper
    3. with Colin Rourke and [Richard Rimanyi]. Ref: The Braid--Permutation Group (with R.Rimanyi and C.Rourke). {Topology} 36 (1997) pp 123--135
  3. In work with [D.Rolfsen] and Zhu zhuj@math.ubc.ca from the University of British Columbia, we have determined which braids commute with a standard generator of the braid group and associated questions have been answered in the singular braid monoid.ps paper to download. Ref: "Centralisers in the braid group and singular braid monoid." (with D.Rolfsen and J.Zhu). {l'Enseignment Math\'ematique} 42 (1996) pp 75--96
  4. With [C.P.Rourke ] and [E.Keyman] we have shown that the singular braid monoid embeds in a group. Moreover we have answered various Alexander and Markov type questions for the associated links of this and other generalised braids.
    5. ps paper to download
  5. A paper with Raul Varela
    6. answers a question of Dehornoy concerning braids and the shift operator.
  6. A five author paper "Ordering the braid groups" (with Michael T. Greene, Dale Rolfsen, Colin Rourke and Bert Wiest). {Pacific Journal of Math} 191 (1999) 49--74 shows how the Dehornoy ordering of the braid groups can be interpreted topologically.ps paper to download
  7. With [Louis Kauffman ] and Mercedes Jordan [mmpe3@sussex.ac.uk ]we have written about the birack and biquandle. This is the analogue for virtual links of the rack and quandle of classical links. ps paper to download
  8. A program to calculate various virtual link invariants can be found at Andy Bartholomew's website [click here] . Amongst other things the program shows that the Kishino knot is non-trivial: paper to follow shortly.

A chapter of my book on geometry can be downloaded here.
Click here for a preprint which shows how generalised quaternions can be used to find invariants of virtual knots

Click here for current work

Finally, on a suggestion by Tony Fisher, here are four knots everyone should know how to tie. A sheet bend for tying two ropes together. A bowline for a non-slip loop. A round turn and two half hitches. for tying a rope to something. With modern synthetic ropes, it is better with at least three half-hitches. A figure eight knot for a ``stop'' on a piece of rope.

[ Topology Algebra and Geometry Group ] [ Graduate Research Centre ] [ School of Mathematical Sciences ] [ University of Sussex]
rogerf@Sussex.ac.uk - School of Mathematical Sciences.
Last update 30/9/2002