Stick number of tangles

Abstract

An n-string tangle is a pair (B, A) such that A is a disjoint union of properly embedded n arcs in a topological 3-ball B. And an n-string tangle is said to be trivial (or rational) a, if it is homeomorphic to (D x I, {x(1),..., x(n)} x I) as a pair, where D is a 2-disk, I is the unit interval and each x(i) is a point in the interior of D. A stick tangle is a tangle each of whose arcs consists of finitely many line segments, called sticks. For an n-string stick tangle its stick-order is defined to be a nonincreasing sequence (s(1), s(2),..., s(n)) of natural numbers such that, under an ordering of the arcs of the tangle, each si denotes the number of sticks constituting the ith arc of the tangle. And a stick-order S is said to be trivial, if every stick tangle of the order S is trivial. In this paper, restricting the 3-ball B to be the standard 3-ball, we give the complete list of trivial stick-orders.