Embedding tangles in links

TL;DR

This paper provides a homological method to determine when a tangle can be embedded into a link by analyzing the determinants of associated numerator and denominator links, extending previous results.

Contribution

It extends Krebes' obstruction to embedding tangles into links using homological arguments on branched covers, offering a new criterion based on determinants.

Findings

The gcd of determinants of numerator and denominator links divides the determinant of the embedding link.

The method applies homological techniques to 2-fold branched covers.

Provides a new obstruction criterion for tangle embedding.

Abstract

We reprove and extend a result of David Krebes (J. Knot Theory Ramif. 8 (1999), 321-352) giving an obstruction to embedding a tangle T into a link L. Closing the tangle up in the two obvious ways gives rise to two links, the numerator and denominator links n(T) and d(T). Applying a homological argument to the 2-fold branched covers of these links, we show that the gcd of the determinants of the numerator and denominator divides the determinant of L.