## A state sum invariant for regular isotopy of links having a polynomial number of states

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Lins, Sostenes

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The state sum regular isotopy invariant of links which I introduce in this
work is a generalization of the Jones Polynomial. So it distinguishes any pair
of links which are distinguishable by Jones'. This new invariant, denoted {\em
VSE-invariant} is strictly stronger than Jones': I detected a pair of links
which are not distinguished by Jones' but are distinguished by the new
invariant. The full VSE-invariant has $3^n$ states. However, there are useful
specializations of it parametrized by an integer k, having
$O(n^k)=\sum_{\ell=0}^k {n \choose \ell} 2^\ell $ states. The link with more
crossings of the pair which was distinguished by the VSE-invariant has 20
crossings. The specialization which is enough to distinguish corresponds to k=2
and has only 801 states, as opposed to the $2^{20} = 1,048,576$ states of the
Jones polynomial of the same link. The full VSE-invariant of it has $3^{20} =
3,486,784,401$ states. The VSE-invariant is a good alternative for the Jones
polynomial when the number of crossings makes the computation of this
polynomial impossible. For instance, for $k=2$ the specialization of the
VSE-invariant of a link with $n=500$ crossings can be computed in a few
minutes, since it has only $2 n^2+1 = 500,001$ states.

Comment: 17 pages, 12 figures, some colored

Comment: 17 pages, 12 figures, some colored

##### Keywords

Mathematics - Algebraic Topology, Mathematics - Combinatorics