### 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 × I,{x_{1},⋯,x_{n}}× 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 s_{i} 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.

Original language | English |
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Article number | 1750094 |

Journal | Journal of Knot Theory and its Ramifications |

Volume | 26 |

Issue number | 13 |

DOIs | |

Publication status | Published - 2017 Nov 1 |

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### Keywords

- knot
- Stick number
- tangle

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Knot Theory and its Ramifications*,

*26*(13), [1750094]. https://doi.org/10.1142/S0218216517500948