On Fox's congruence classes of knots
The authors refine the following idea of \textit{R. H. Fox} in ``Congruence classes of knots'' [Osaka Math. J. 10, 37-41 (1958; Zbl 0084.192)]. Two knot types K, L are said to be congruent modulo (n,q) if and only if L may be obtained from K by a sequence of 1/n\({}_ i\) surgeries along trivial knots with linking numbers \(q_ i\) with the intermediate knots, where the \(n_ i\) are all congruent to 0 mod n, and the \(q_ i\) are all congruent to 0 mod q. The refinement consists in replacing congruence mod q with equality of \(q_ i\) and q (the resulting notion is called q congruence modulo n). Several necessary properties of q congruent mod n and (n,q) congruent knot pairs are given: For example, (n,q) congruent knot pairs have Alexander Matrices congruent modulo \((1-t)\sigma_ n(t\quad q)\) where \(\sigma_ n(x)=(1-x\quad n)/(1-x).\) The authors show that this implies there are infinitely many knot types incongruent modulo (n,q) if \(n>1\) and \(q>2\). On the other hand the authors conjecture: all knots are congruent modulo (2,1); all knots are congruent modulo (2,2).
\(1/n_ i\) surgeries along trivial knots
scientific article; zbMATH DE number 4067874
scientific article