Amphicheirality of ribbon 2-knots

Abstract

For any classical knot [Formula: see text], we can construct a ribbon [Formula: see text]-knot [Formula: see text] by spinning an arc removed a small segment from [Formula: see text] about [Formula: see text] in [Formula: see text]. A ribbon [Formula: see text]-knot is an embedded [Formula: see text]-sphere in [Formula: see text]. If [Formula: see text] has an [Formula: see text]-crossing presentation, by spinning this, we can naturally construct a ribbon presentation with [Formula: see text] ribbon crossings for [Formula: see text]. Thus, we can define naturally a notion on ribbon [Formula: see text]-knots corresponding to the crossing number on classical knots. It is called the ribbon crossing number. On classical knots, it was a long-standing conjecture that any odd crossing classical knot is not amphicheiral. In this paper, we show that for any odd integer [Formula: see text] there exists an amphicheiral ribbon [Formula: see text]-knot with the ribbon crossing number [Formula: see text].