Magnus Representation of Genome Sequences

We introduce an alignment-free method, the Magnus Representation, to analyze genome sequences. The Magnus Representation captures higher-order information in genome sequences. We combine our approach with the idea ofk-mers to define an effectively computable Mean Magnus Vector. We perform phylogenetic analysis on three datasets: mosquito-borne viruses, filoviruses, and bacterial genomes. Our results on ebolaviruses are consistent with previous phylogenetic analyses, and confirm the modern viewpoint that the 2014 West African Ebola outbreak likely originated from Central Africa. Our analysis also confirms the close relationship betweenBundibugyo ebolavirusandTaï Forest ebolavirus. For bacterial genomes, our method is able to classify relatively well at the family and genus level, as well as at higher levels such as phylum level. The bacterial genomes are also separated well into Gram-positive and Gram-negative subgroups.

Morita’s trace maps on the group of homology cobordisms

Morita introduced in 2008 a [Formula: see text]-cocycle on the group of homology cobordisms of surfaces with values in an infinite-dimensional vector space. His [Formula: see text]-cocycle contains all the “traces” of Johnson homomorphisms which he introduced 15 years earlier in his study of the mapping class group. In this paper, we propose a new version of Morita’s [Formula: see text]-cocycle based on a simple and explicit construction. Our [Formula: see text]-cocycle is proved to satisfy several fundamental properties, including a connection with the Magnus representation and the LMO homomorphism. As an application, we show that the rational abelianization of the group of homology cobordisms is non-trivial. Besides, we apply some of our algebraic methods to compare two natural filtrations on the automorphism group of a finitely-generated free group.

The kernel of the Magnus representation of the automorphism group of a free group is not finitely generated

We show that the abelianization of the kernel of the Magnus representation of the automorphism group of a free group is not finitely generated.