TY - JOUR

T1 - Stick number of tangles

AU - Huh, Youngsik

AU - Lee, Jung Hoon

AU - Taniyama, Kouki

N1 - Funding Information:
The first author (corresponding author) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2016R1D1A1B01008044).
Funding Information:
The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education (2015R1D1A1A01056953).
Funding Information:
The third author was partially supported by Grant-in-Aid for Challenging Exploratory Research (No. 15K13439) and Grant-in-Aid for Scientific Research(A) (No. 16H02145), Japan Society for the Promotion of Science.
Publisher Copyright:
© 2017 World Scientific Publishing Company.

PY - 2017/11/1

Y1 - 2017/11/1

N2 - 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,{x1,⋯,xn}× I) as a pair, where D is a 2-disk, I is the unit interval and each xi 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 (s1,s2,⋯,sn) 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.

AB - 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,{x1,⋯,xn}× I) as a pair, where D is a 2-disk, I is the unit interval and each xi 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 (s1,s2,⋯,sn) 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.

KW - Stick number

KW - knot

KW - tangle

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U2 - 10.1142/S0218216517500948

DO - 10.1142/S0218216517500948

M3 - Article

AN - SCOPUS:85034086320

VL - 26

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

SN - 0218-2165

IS - 13

M1 - 1750094

ER -