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# Knot Group

A knot k is an embedding (ie a continuous 1-1 map) of into and we shall restrict our attention to tame knots which have only a finite number of loops in In some situations it is convenient for the proofs to consider the knots to be embedded in by adding a point at infinity to

Introductory Mathematica packages, and in particular TORUSKNOT.ma for drawing the trefoil and other knots, can be found via the webpage Mathematica for curves and surfaces. The knot group of k is so the trivial knot, has knot group Armstrong [2] Chapter 10 gives a procedure for using Van Kampen's theorem to obtain knot groups in terms of generators (one per overpass) and relations (one per crossing) in suitable projections of the knot onto a plane. For example, the trefoil knot has knot group Abelianizing a knot group G, that is by taking its quotient by the commutator subgroup

always yields the free group on one generator, The first homology group H1(X) of a connected compact subset is generated by classes of loops round the edges of simplices and actually coincides with the quotient of the fundamental group by its commutator subgroup. Click here to go to a website for knot theory The knot group can be represented by the so-called Alexander Polynomial through a combinatorial algorithm applied to a suitably `nice' projection (eg. no crossings project onto one another) of the knot onto a plane [2]. The Alexander polynomial is unaltered by a mirror reflection of the knot. For the knots with 6 or less overcrossings in a nice projection, the inequivalent Alexander polynomials up to a factor tk are given in the table below. It uses the notation of Lickorish[8], where knot mn is the nth knot type having m overcrossings.
 Knot mn Alexander Polynomial 31 (Trefoil) +1-t+t2 41 (Figure eight) -1+3t-t2 51 +1-t+t2-t3+t4 52 +2-3t+2t2 61 (Stevedore's) -2+5t-2t2 62 -1+3t-3t2+3t3-t4 63 +1-3t+5t2-3t3+t4
Find out more about Alexander by clicking about Alexander

Next: Bibliography Up: Introducing Knots Previous: Simplicial Complex
Kit Dodson
2000-01-24