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Transactions of the Moscow Mathematical Society

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Combinatorial computation of combinatorial formulas for knot invariants

Author: V. A. Vassiliev
Translated by: the author
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 66 (2005).
Journal: Trans. Moscow Math. Soc. 2005, 1-83
MSC (2000): Primary 57M27, 57M25
Published electronically: October 28, 2005
MathSciNet review: 2193429
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Abstract: We construct a homology algebraic algorithm for computing combinatorial formulas of all finite degree knot invariants. Its input is an arbitrary weight system, i.e., a virtual principal part of a finite degree invariant, and the output is either a proof of the fact that this weight system actually does not correspond to any knot invariant or an effective description of some invariant with this principal part, i.e., a finite collection of easily described singular chains of full dimension in the space of spatial curves such that the value of this invariant on any generic knot is equal to the sum of multiplicities of these chains in a neighborhood of the knot. (In examples calculated by now, the former possibility never occurred.) This algorithm is formally realized over $ \mathbb{Z}_2$, but its generalization to the case of arbitrary coefficients is just a technical task. The algorithm is based on the study of a complex of chains in the space of smooth curves in the three-dimensional space with a fixed flag of directions, and also in the discriminant variety of this space of curves.

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Additional Information

V. A. Vassiliev
Affiliation: Steklov Mathematical Institute and Poncelet Laboratory (UMI 2615 of CNRS and Independent University of Moscow)

Keywords: Knot, invariant, combinatorial formula, discriminant, combinatorial algorithm, spectral sequence, subalgebraic chain, simplicial resolution.
Published electronically: October 28, 2005
Additional Notes: Supported in part by grants RFBR-01-01-00660, INTAS–00-0259, grant 1972.2003.01 of President of Russia, and Program “Contemporary Mathematics" of the Mathematical division of Russian Ac. Sci.
Article copyright: © Copyright 2005 American Mathematical Society

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