Alexander–Beck modules detect the unknot

Twisted Alexander Polynomials Detect the Unknot

The group of a nontrivial knot admits a finite permutation representation such that the corresponding twisted Alexander polynomial is not a unit.

Does the Jones Polynomial Detect the Unknot?

We address the question: Does there exist a non-trivial knot with a trivial Jones polynomial? To find such a knot, it is almost certainly sufficient to find a non-trivial braid on four strands in the kernel of the Burau representation. I will describe a computer algorithm to search for such a braid.

Does the Jones Polynomial Detect the Unknot ? Stephen

A probabilistic method to detect regulatory modules

MOTIVATION The discovery of cis-regulatory modules in metazoan genomes is crucial for understanding the connection between genes and organism diversity. RESULTS We develop a computational method that uses Hidden Markov Models and an Expectation Maximization algorithm to detect such modules, given the weight matrices of a set of transcription factors known to work together. Two novel features ...

Homological Characterization of the Unknot

Given a knot K in the 3-sphere, let QK be its fundamental quandle as introduced by D. Joyce. Its first homology group is easily seen to be H1(QK) ∼= Z. We prove that H2(QK) = 0 if and only if K is trivial, and H2(QK) ∼= Z whenever K is non-trivial. An analogous result holds for links, thus characterizing trivial components. More detailed information can be derived from the conjugation quandle: ...