Concerning a Bound Problem in Knot Theory

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.

Locations

• Transactions of the American Mathematical Society - View - PDF