Abstract:
Gordian complex of knots was defined by Hirasawa and
Uchida as the simplicial complex whose vertices are knot isotopy classes
in S
3
. Later Horiuchi and Ohyama defined Gordian complex of virtual
knots using v-move and forbidden moves. In this paper we discuss Gordian complex of knots by region crossing change and Gordian complex
of virtual knots by arc shift move. Arc shift move is a local move in
the virtual knot diagram which results in reversing orientation locally
between two consecutive crossings. We show the existence of an arbitrarily high dimensional simplex in both the Gordian complexes, i.e., by
region crossing change and by the arc shift move. For any given knot
(respectively, virtual knot) diagram we construct an infinite family of
knots (respectively, virtual knots) such that any two distinct members
of the family have distance one by region crossing change (respectively,
arc shift move). We show that that the constructed virtual knots have
the same affine index polynomial.
