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What's known about computational complexity of different types of knot invariant polynomials?

For example, Evaluating Jones Polynomial is known to be #P hard. Is there any reference that surveys such complexity results on other knot polynomials?

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2 Answers 2

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The old (1990) paper, "On the computational complexity of the Jones and Tutte polynomials" (Cambridge link), shows that determining the Jones polynomial of alternating links is #P-hard, as the OP notes. Then, much later, the 2012 book Quantum Triangulations (eds.: Carfora, Marzuoli), says this (p.233):
 Quantum Triangulations p.233
See the Wikipedia entry on BPQ for a definition: essentially, solvable in polynomial time on a quantum computer, with bounded error probability. It is conjectured that BPQ $\supset$ P.

In the same book, there follows a section entitled "Efficient Quantum Processing of Colored Jones Polynomials," with several references.

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Prof. O'Rourke, that is a beautiful modern reference; thanks very much. –  Arnab May 27 '12 at 19:45

Complexity: Knots, Colourings and Counting By D. J. A. Welsh

Has pretty extensive information.

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"The aim of these notes is to link algorithmic problems arising in knot theory with statistical physics and classical combinatorics. Apart from the theory of computational complexity needed to deal with enumeration problems, introductions are given to several of the topics, such as combinatorial knot theory, randomized approximation models, percolation, and random cluster models." Note: Written a decade ago, so possibly out of date on some topics. –  Joseph O'Rourke May 26 '12 at 0:01

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