By Dominic Welsh
The purpose of those notes is to hyperlink algorithmic difficulties coming up in knot conception with statistical physics and classical combinatorics. except the speculation of computational complexity had to care for enumeration difficulties, introductions are given to numerous of the themes, equivalent to combinatorial knot idea, randomized approximation types, percolation, and random cluster versions.
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Extra info for Complexity: Knots, Colourings and Counting
1) shows both that sup α∈m(X 1/3 ) |h(α)| X 5/6 (log X)4 , October 6, 2009 13:49 WSPC - Proceedings Trim Size: 9in x 6in ws-procs9x6 ADDITIVE REPRESENTATION IN THIN SEQUENCES 45 and, whenever Q ≤ X 1/3 , that |h(α)| sup XQ−1/2 (log X)4 . 18) λ1 |p1 −p2 |<τ p1 ,p2 ≤X the ﬁrst of these estimates yields |h(λ1 α)h(λ2 α)|2 dτ α X 8/3 (log X)9 , m2 and the same is true for the contribution from m1 , by symmetry. 3. 3, we ﬁnd that c∩K(X 1/3 ,X 1/3 ) |h(λ1 α)h(λ2 α)|2 dτ α X 8/3+ε , an estimate that may also be found on p.
10. The notation used in this memoir is standard, or otherwise explained at the appropriate stage of the proceedings. We write e(α) = exp(2πiα). The distance of a real number α to the nearest integer is α . The integer part of α is [α], and α is the smallest integer n with n ≥ α. We apply the following convention concerning the letter ε. Whenever ε occurs in a statement, it is asserted that this statement is true for all real ε > 0, but constants implicit in Landau or Vinogradov symbols may depend on the actual value of ε.
WOOLEY For comparison, Parsell  works under the weaker hypothesis that λ1 /λ2 is irrational, and obtains a result that is essentially equivalent to N σ(τ, ν) − 0 2τ ν 2 dν = o(N 3 ). 7), as well as an improvement when λ1 /λ2 is algebraic, but not by a power of N . Limitations arise from our current knowledge concerning the zeros of the Riemann zeta function. 7) is O(N 3 exp(−c log N )) for some c > 0, and with only moderate extra eﬀort one obtains a saving that corresponds to the sharpest one currently known in the error term for the prime number theorem.
Complexity: Knots, Colourings and Counting by Dominic Welsh