By Louis H Kauffman

This priceless booklet is an advent to knot and hyperlink invariants as generalized amplitudes for a quasi-physical method. The calls for of knot idea, coupled with a quantum-statistical framework, create a context that certainly and powerfully contains a unprecedented diversity of interrelated themes in topology and mathematical physics. the writer takes a basically combinatorial stance towards knot idea and its kin with those matters. This stance has the benefit of delivering direct entry to the algebra and to the combinatorial topology, in addition to actual principles.

The publication is split into elements: half I is a scientific direction on knots and physics ranging from the floor up, and half II is a collection of lectures on quite a few subject matters relating to half I. half II contains subject matters corresponding to frictional houses of knots, family members with combinatorics, and knots in dynamical platforms.

during this new version, a piece of writing on digital Knot conception and Khovanov Homology has beed additional.

Readership: Physicists and mathematicians.

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**Example text**

In due course. The Trefoil is Knotted. I conclude this section with a description of how to prove that the trefoil is knotted. indd 21 18/10/12 3:13 PM 22 colors (R-red, B-blue, P-purple). p R I claim that, with an appropriate notion of coloring, this property of being three-colored can be preserved under the Reidemeister moves. For example: p p However, note that under a type I move we may be forced to retain only one color at a vertex: Thus I shall say that a knot diagram K is three-colored if each arc in K is assigned one of the three colors ( R, B, P), all three colors occur on the diagram and each crossing carries either three colors or one color.

W (iv) (The Whitehead Link). The Whitehead link has linking number zero however one orients it. indd 36 18/10/12 3:13 PM 37 (W)~A(~ )+A-'~ d(ct:§ ) )+W')(-A-')(~ ~A(-A')(~ )-r•(cp ~ (-A')( ) ) c:f? ) -A -•(E•) (v) Here Kn is a torus link of type (2,n). ) (~~~···~)=A(~) + A-1(€? OCX:: .. indd 37 ) + A- 1( -A- 3 t- 1. 18/10/12 3:13 PM 38 (K1} = -A 3 (K2} =A( -A3) + (-1)1 A-3·2+2 = -A4- A-4 + (-1? A- 10 = -A6 -A- 2 +A- 6 -A- 10 . (K3} = A(-A4 - A-4 ) Use this procedure to show that no torus knot of type (2, n) (n > 1) is ambient isotopic to its mirror image.

Let U be any link shadow. Then there is a choice of over/under structure for the crossings of U forming a diagram K so that K is alternating. ) Proof. Shade the diagram U in two colors and set each crossing so that it has the form that is - so that the A-regions at this crossing are shaded. The picture below This completes the proof. indd 42 II 18/10/12 3:13 PM 43 Example. Now we come to the center of this section. Consider the bracket polynomial, (K), for an alternating link diagram K. If we shade K as in the proof above, so that every pair of A-regions is shaded, then the state S obtained by splicing each shading 1---+ splice will contribute where i(S) is the number of loops in this state.