Bulletin of the
Korean Mathematical Society BKMS

$n$-writhes denoted by $J_n(K)$ are virtual knot invariants for $n\neq 0$ and are closely associated with coefficients of some polynomial invariants of virtual knots. In this work, we investigate the variations of $J_n(K)$ under arc shift move and conclude that $n$-writhes $J_n(K)$ vary randomly in the sense that it may change by any random integer value under one arc shift move. Also, for each $n\neq 0$ we provide an infinite family of virtual knots which can be distinguished by $n$-writhes $J_n(K)$, whereas odd writhe $J(K)$ fails to do so.

Keywords: Writhes, virtual knot, local moves

MSC numbers: 57K10, 57K12

Supported by: This research work as a part of the SERB project MTR/2021/000394.