Normal forms for rational 3-tangles

Abstract

A collection of properly embedded three disjoint simple arcs in Sigma 0,6 represents a rational 3-tangle. In this paper, we define a normal form of collections of three disjoint bridge arcs for a given rational 3-tangle. We show that there is a sequence of operations called normal jump moves which makes a path between arbitrary two elements in the set of normal forms of the same rational 3-tangle. We believe that the normal form would give a clue to classify rational 3-tangles. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data