A permanent formula for the Jones polynomial
Quantum Algebra
2012-03-01 v3 Combinatorics
Quantum Physics
Abstract
The permanent of a square matrix is defined in a way similar to the determinant, but without using signs. The exact computation of the permanent is hard, but there are Monte-Carlo algorithms that can estimate general permanents. Given a planar diagram of a link L with crossings, we define a 7n by 7n matrix whose permanent equals to the Jones polynomial of L. This result accompanied with recent work of Freedman, Kitaev, Larson and Wang provides a Monte-Carlo algorithm to any decision problem belonging to the class BQP, i.e. such that it can be computed with bounded error in polynomial time using quantum resources.
Cite
@article{arxiv.0705.4548,
title = {A permanent formula for the Jones polynomial},
author = {Martin Loebl and Iain Moffatt},
journal= {arXiv preprint arXiv:0705.4548},
year = {2012}
}