Knot Group

A knot k is an embedding (ie a continuous 1-1 map) of $\mathbb{S} ^1$ into $\mathbb{E} ^3,$ and we shall restrict our attention to tame knots which have only a finite number of loops in $\mathbb{E} ^3.$ In some situations it is convenient for the proofs to consider the knots to be embedded in $\mathbb{S} ^3$ by adding a point at infinity to $\mathbb{E} ^3.$

  

Figure 2: Left: a torus and on it the graph of a map from a circle to itself. Right: a torus knot.

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 $\pi_1(\mathbb{E} ^3\setminus k),$ so the trivial knot, $k=\mathbb{S} ^1,$ has knot group $\mathbb{Z} .$ 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 $\{a,b\vert aba=bab\}.$ Abelianizing a knot group G, that is by taking its quotient by the commutator subgroup

$$C=\{x^{-1}y^{-1}xy\vert x,y\in G\}$$

always yields the free group on one generator, $G/C\cong \mathbb{Z} .$ 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 t^k are given in the table below. It uses the notation of Lickorish[8], where knot m_n is the n^th 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

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