# A state sum invariant for regular isotopy of links having a polynomial number of states - Mathematics > Algebraic Topology

A state sum invariant for regular isotopy of links having a polynomial number of states - Mathematics > Algebraic Topology - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

Abstract: The state sum regular isotopy invariant of links which I introduce in thiswork is a generalization of the Jones Polynomial. So it distinguishes any pairof links which are distinguishable by Jones-. This new invariant, denoted {\emVSE-invariant} is strictly stronger than Jones-: I detected a pair of linkswhich are not distinguished by Jones- but are distinguished by the newinvariant. The full VSE-invariant has $3^n$ states. However, there are usefulspecializations of it parametrized by an integer k, having$On^k=\sum {\ell=0}^k {n \choose \ell} 2^\ell$ states. The link with morecrossings of the pair which was distinguished by the VSE-invariant has 20crossings. The specialization which is enough to distinguish corresponds to k=2and has only 801 states, as opposed to the $2^{20} = 1,048,576$ states of theJones 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 Jonespolynomial when the number of crossings makes the computation of thispolynomial impossible. For instance, for $k=2$ the specialization of theVSE-invariant of a link with $n=500$ crossings can be computed in a fewminutes, since it has only $2 n^2+1 = 500,001$ states.

Autor: Sostenes Lins

Fuente: https://arxiv.org/