On Krebes' tangle

TL;DR

This paper investigates conditions under which a genus-1 tangle can embed in knots, using homological obstructions from branched covers, and applies this to Krebes' example to derive divisibility constraints.

Contribution

It introduces a homological obstruction method for embedding genus-1 tangles in knots and applies it to Krebes' tangle example to establish new divisibility conditions.

Findings

Any closure passing through the hole an odd number of times has determinant divisible by three.

If Krebes' tangle embeds in the unknot, the arc must pass through the hole an even number of times.

Homological obstructions can determine embedding possibilities of tangles in knots.

Abstract

A genus-1 tangle G is an arc properly embedded in a standardly embedded solid torus S in the 3-sphere. We say that a genus-1 tangle embeds in a knot K in S^3 if the tangle can be completed by adding an arc exterior to the solid torus to form the knot K. We call K a closure of G. An obstruction to embedding a genus-1 tangle G in a knot is given by torsion in the homology of branched covers of S branched over G. We examine a particular example A of a genus-1 tangle, given by Krebes, and consider its two double-branched covers. Using this homological obstruction, we show that any closure of A obtained via an arc which passes through the hole of S an odd number of times must have determinant divisible by three. A resulting corollary is that if A embeds in the unknot, then the arc which completes A to the unknot must pass through the hole of S an even number of times.