One-two-way pass-move for knots and links

Abstract

We define a local move for knots and links called the {\em one-two-way pass-move}, abbreviated briefly as the {\em $1$-$2$-move}. The $1$-$2$-move is motivated from the pass-move and the $\#$-move, and it is a hybrid of them. We show that the equivalence under the $1$-$2$-move for knots is the same as that of the pass-move: a knot $K$ is $1$-$2$-move equivalent to an unknot (a trefoil respectively) if and only if the Arf invariant of $K$ is $0$ ($1$ respectively). On the other hand, we show that the number of $1$-$2$-moves behaves differently from the number of pass-moves.