Complexity: Knots, Colourings and Countings by Dominic Welsh

By Dominic Welsh

In line with lectures on the complex learn Institute of Discrete utilized arithmetic in June 1991, those notes hyperlink algorithmic difficulties bobbing up in knot conception, statistical physics and classical combinatorics for researchers in discrete arithmetic, machine technological know-how and statistical physics.

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Polynomial 'i L(Z) belonging to the polynomial ring Z¡zj. Equivalent knots and links receive the same polynomial which satisfies the following recursive rules. 1) If L rv U, the unknot, then 'i L(Z) = 1. 1 but are otherwise the same, then 'i L ( X ) = 'i L ( X ) + z'i L ( ) ( ). We call 'i dz) the Conway polynomial of L. The relationship shown in Figure 5,1 is an example of what is known as a skein relation. he link diagram. The Alexander polynomial of L written D. ¡). The Alexander polynomial is well understood, in contrast to later polynomials we wil consider.

21) If L is a knot then VL(e21ri/:I) = 1. 22) If K is a knot then dVK(t)ldti=i = O. It is useful to give an alternative 8tate8 'fwild representation of (L) in terms of concepts more familar in combinaLorlcH. This is contained in the following. 16) Proposition. nÒ L:. is given by L and suppose that the associated signeò graph G = G(D) is connected. V¡)t) = V¡, ilt). VLi#L2(t) = VLi (t)VL2(t). For any subset A ç E(G), let A+ aiid A- denoLe the positive (negative) signed part respectively. 17) (D) AIE-I-IE+I(_A2 _ A-2rCG) x x :E A4CrCX)-lx-I)(_A4 _lrCGJ+IXI-2r(X) XÇE AIE-I-IE+I+2r(G) X X :E A-4Cr(x)-Ix-ll)(_A-4 _lr(GJ+IXI-2r(X).

2. 3. 13) Along Hq, when q is a prime power, for a matroid M of vectors over GF(q), T specialises to the weight enumerator of the linear code over GF(q), determined by M. 6) relating T(M) to T(M*) gives the MacWiliams identity of coding theory. 1 Percolation processes . . . . . 1 Percolation processes media. Its origin, in 1957 in the work of Broadbent & Hammersley, was as . . . a model for molecules penetrating a porous solid, electrons migrating over . . As its name suggests, percolation theory is concerned with flow in random .

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