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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Concerning a bound problem in knot theory

Author: L. B. Treybig
Journal: Trans. Amer. Math. Soc. 158 (1971), 423-436
MSC: Primary 55.20
MathSciNet review: 0278289
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Abstract: In a recent paper Treybig shows that if two knot functions $f,g$ determine equivalent knots, then $f,g$ are the ends of a simple sequence $x$ of knot functions. In an effort to bound the length of $x$ in terms of $f$ and $g$ (1) a bound is found for the moves necessary in moving one polyhedral disk onto another in the interior of a tetrahedron and (2) it is shown that two polygonal knots $K,L$ in regular position can “essentially” be embedded as part of the $1$-skeleton of a triangulation $T$ of a tetrahedron, where (1) all 3 cells which are unions of elements of $T$ can be shelled and (2) the number of elements in $T$ is determined by $K,L$. A number of “counting” lemmas are proved.

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Keywords: Polyhedron, polygonal knot, piecewise linear homeomorphism, simplicial isotopy, free cell, shelling order
Article copyright: © Copyright 1971 American Mathematical Society