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KNOTS, SATELLITE OPERATIONS AND INVARIANTS OF FINITE ORDER

L. PLACHTA

Institute of Applied Problems of Mechanics and Mathematics of NAS of Ukraine, Naukova 3b, 79 060, Lviv, Ukraine

Institute of Mathematics, University of Gdańsk, 80 952 Wita Stwosza, 57, Gdańsk, Poland

https://doi.org/10.1142/S0218216506004944Cited by:2 (Source: Crossref)

Abstract

Let s_Q be the satellite operation on knots defined by a pattern (V, Q), where V is a standard solid torus in S³ and Q ⊂ V is a knot that is geometrically essential in V. It is known (Kuperberg [5]) that if v is any knot invariant of order n ≥ 0, then v ◦ s_Q is also a knot invariant of order ≤ n. We show that if the knot Q has the winding number zero in V, then the satellite map passes n-equivalent knots into (n + 1)-equivalent ones. Kalfagianni [4] has defined for each nonnegative integer n surgery n-trivial knots and studied their properties. It is known that for each n every surgery n-trivial knot is n-trivial. We show that for each n there are n-trivial knots which do not admit a non-unitary n-trivializer that show they to be surgery n-trivial. Przytycki showed [12] that if a knot Q is trivial in S³ and is embedded in V in such a way that it is k-trivial inside V and if a knot is m-trivial, then the satellite knot is (k + m + 1)-trivial. We establish a version of aforementioned Przytycki's result for surgery n-triviality, refining thus a construction for surgery n-trivial knots suggested by Kalfagianni.

To Professor Louis H. Kauffman who directed my attention to virtual braids

Keywords:

AMSC: 57M25

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