3-Bridgeness under adding crossings to alternating 3-bridge knots in a 3-bridge representation

Abstract

In [B. Kwon and S. Kang, Rectangle conditions and families of 3-bridge prime knots, Topol. Appl. 291 (2021) 107453], using the set [Formula: see text] of all essential alternating rational 3-tangles for positive integer [Formula: see text], the authors showed that all knot diagrams in the numerator closure set [Formula: see text] and the denominator closure set [Formula: see text] with [Formula: see text] are 3-bridge prime knot diagrams. In this paper, for [Formula: see text] we construct a set [Formula: see text] of additions of alternating rational tangles in [Formula: see text]. The set [Formula: see text] generalizes [Formula: see text] and contains it as a subset for some [Formula: see text]. We show that any closure set [Formula: see text] on [Formula: see text] so that the resulting diagrams are reduced and alternating knot diagrams represent alternating 3-bridge prime knot diagrams. Since a tangle diagram in [Formula: see text] is constructed inductively from a tangle diagram in [Formula: see text] by adding only one crossing positively, the result of this paper supports the conjecture that 3-bridge property is preserved under one-crossing alternating addition positively to alternating 3-bridge knots in 3-bridge representations.