By Dominic Welsh

In response to lectures on the complex learn Institute of Discrete utilized arithmetic in June 1991, those notes hyperlink algorithmic difficulties coming up in knot conception, statistical physics and classical combinatorics for researchers in discrete arithmetic, laptop technology and statistical physics.

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**Extra info for Complexity: Knots, Colourings and Countings**

**Example text**

Consider the trefoil T ßT(t) = 1 - t + t2. 2) yields \7T = \7 L + z\7 M where M Q Proof of invariance In order to show that a given polynomial is in fact a knot/link invariant, it is necessary and sufcient to show that the invariant in question is unchanged under each of the three Reidemeister moves. 2 We wil ilustrate this in detail in the next section. First note that it is not difcult to find nontrivial knots which have the same Alexander polynomial as the Ilnknot - one such example is the pretzel knot (- 3,5,7).

Consider the generating function 11'1 B(G; À, s) = L sibi(À), i=O Clearly bo(À) is the chromatic polynomial of G and like PG(À) we see that the following relationships hold. 3) If G is connected then provided e is not a loop or coloop, B(G;)", s) = B(G~;)", s) + (s - 1)B(G~;)", s). 4) B(G;)", s) = sB(G~) if e is a loop. L lleA) = 1. 5) B(G;)", s) = (s +). 14) We wil sometimes use w( G) to denote the random confguration produced by ¡i, and P¡i to denote the associated probabilty distribution. Thus, in paricular, lleA) = P¡irw(G) = Al.

2) yields \7T = \7 L + z\7 M where M Q Proof of invariance In order to show that a given polynomial is in fact a knot/link invariant, it is necessary and sufcient to show that the invariant in question is unchanged under each of the three Reidemeister moves. 2 We wil ilustrate this in detail in the next section. First note that it is not difcult to find nontrivial knots which have the same Alexander polynomial as the Ilnknot - one such example is the pretzel knot (- 3,5,7). , Cn) where Ci is the niimber of crossings in the ith tassle, where if Ci is negative the crossings am In the opposite sense.