By Dominic Welsh

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

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Extra info for Complexity: Knots, Colourings and Countings

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L II aiaj = OJ Each biparition B of G wil correspond to exactly two spin confgurations (depending on which side is +ve or -ve). However the contribution to Z wil be the same in both cases and given by H(a) = H(BCT) = -J IE(G)I + 2J IE(B)I. CT ijEA A E C =? II aiaj = L. 5) CT Z(G) lEI L (tanh(K))IAI2JVI (cosh(K)) AEC 2 L e+ßJIE(G)I-2ßJIE(B)1 (cosh(K)) I ";IZJVICo(tl1nh(K)) B where the sum is over all biparitions B. Therefore Z(G) = 2eßJ1E(G)1 Le-2ßJIE(B)I. where CG(x) is the generating function of the number of Eulerian subgraphs ofG.

Let D be a link diagram with a portion HS shown D ìO A-' -t(, and let D' be obtained from D by removing the loop. Then + + --- (D) = A( ~ ) + A-I( ) 0 ) (A + A-I( _A2 - A-2))(D') -A-3(D'). A-' ) ~ --- A-' ) ~ again using Reidemeister II equivalence. This shows ì\ obtained by forgetting about the orientation. moves I, II and III. Using Reidemeister II equivalence, V ,A- where weD) is the writhe of D and where ( ) for an oriented diagram is However, no matter which way we oricnt thc string in D the writhe of the crossing is -ve.

1. 'C Hammersley and Mazzarino (1983). J82) or Hintermann, Kunz and 70 Statistical physics Wu (1978). However it does not appear easy to make these arguments rigorous in this context, and the situation seems not dissimilar from that in ordinary percolation when it took 16 years before Kesten (1980) and Wierman (1981) were able to give rigorous justifications of the exact values obtained by Sykes and Essam (1964). 1 is just an extremely brief introduction to what is now a vibrant area of applied probabilty.

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