Narrow Results

We show that for each positive integer n>1 a simple C_n-move on knots, introduced by Habiro, is equivalent to the (non-oriented) insertion in a knot via a tangle map the geometric pure braid n-commutator of the form p_n=[p_n-1,n,[p_n-2,n-1,…, [p_1,2,p_0,1]…]∈P_n+1, where P_n+1 is the subgroup of pure braids of the braid group B_n+1 on n+1 strands. We relate the insertions of the pure braid commutators in knots to the operations of inverting and mirroring the knots.

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.