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Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)



Parity, free knots, groups, and invariants of finite type

Author: V. O. Manturov
Translated by: G. G. Gould
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 72 (2011), vypusk 2.
Journal: Trans. Moscow Math. Soc. 2011, 157-169
MSC (2010): Primary 57M25, 57M27
Published electronically: January 12, 2012
MathSciNet review: 3184816
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In this paper, on the basis of the notion of parity introduced recently by the author, for each positive integer $m$ we construct invariants of long virtual knots with values in some simply defined group $\mathcal {G}_m$; conjugacy classes of this group play a role as invariants of compact virtual knots. By construction, each of the invariants is unaltered by the move of virtualization. Factorization of the group algebra of the corresponding group leads to invariants of finite order of (long) virtual knots that do not change under virtualization.

The central notion used in the construction of the invariants is parity: the crossings of diagrams of free knots is endowed with an additional structure — each crossing is declared to be even or odd, where even crossings behave regularly under Reidemeister moves.

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Additional Information

V. O. Manturov
Affiliation: People’s Friendship University, Moscow, Russia

Keywords: Knot, virtual knot, free knot, invariant, parity, group, invariant of finite order
Published electronically: January 12, 2012
Article copyright: © Copyright 2012 American Mathematical Society