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Geometric Topology Seminar
August 3, 2017 14:00, Moscow, Steklov Mathematical Institute, room 534
 


A combinatorial formula for M-invariant

P. M. Akhmet'ev

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Abstract: M-invariant of isotopy classes for classical 3-component links are well-defined using an integral formula over a finite-dimensional configuration space of the links. Therefore this invariant is a finite-type invariant (of the order 7 in the sense of Vassiliev). This invariant is used for physical applications to investigate properties of magneto-static configuration in the ideal magnetohydrodynamics.
This invariant is a polynomial function of the first two coefficients of the normalized Conway polynomial for proper sub-links:(C_0(1,2) С_0(2,3), C_0(3,1), C_1(1), C_1(2), C_1(3), C_1(1,2), C_1(2,3), C_1(3,1), C_1(1,2,3); the collection of 10 variables). An explicit expression is known up to an odd polynomial of the order not great then 7 of pairwise linking numbers of the components С_0(1,2), C_0(2,3), C_0(3,1). Nevertheless, one may calculate M-invariant for an arbitrary 3-component links using the following additional normalization property (this is a new result): For the standard Hopf 3-component links the value of M equals to +1.

Language: English

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