Concerning a bound problem in knot theory

Treybig, L. B.

doi:10.1090/S0002-9947-1971-0278289-8

Concerning a bound problem in knot theory
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Trans. Amer. Math. Soc. 158 (1971), 423-436 Request permission

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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