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 EAT(k) of all essential alternating rational 3-tangles for positive integer k, the authors showed that all knot diagrams in the numerator closure set C-N(EAT(2l+1)) and the denominator closure set C-D(EAT(2l+2)) with l > 0 are 3-bridge prime knot diagrams. In this paper, for n > 4 we construct a set AAT(4)(n) of additions of alternating rational tangles in EAT(4). The set AAT(4)(n) generalizes EAT(k )and contains it as a subset for some k. We show that any closure set C(AAT(4)(n)) on AAT(4)(n) so that the resulting diagrams are reduced and alternating knot diagrams represent alternating 3-bridge prime knot diagrams. Since a tangle diagram in AAT(4)(n+1) is constructed inductively from a tangle diagram in AAT(4)(n) 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.