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

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dc.creatorLins, Sostenes
dc.date2008-04-26
dc.date.accessioned2026-07-07T09:35:30Z
dc.date.available2026-07-07T09:35:30Z
dc.descriptionThe 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.
dc.description17 pages, 12 figures, some colored
dc.identifierhttps://arxiv.org/abs/0804.4233
dc.identifierhttp://arxiv.org/abs/0804.4233
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/159853
dc.subjectAlgebraic Topology
dc.subjectCombinatorics
dc.titleA state sum invariant for regular isotopy of links having a polynomial number of states
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