Colin McCulloch | Search arXiv

arXiv2026

Computations of Floer Lasagna Modules for Traces of Negative L-Space Knots

We show that for all negative $L$-space knots and framings $n\geq2g(K)$, if a certain cobordism map between cables of $K$ is non-vanishing, then the Floer lasagna module of the knot trace of $K$ with framing $-n$ is infinite dimensional. Further, for the maps on the hat flavour of link Floer homology induced by band maps, quasi-stabilisations and pair of pants cobordisms we give a description of the map on the $E^{\infty}$-page of Ozsvath and Szabo's spectral sequence for forgetting components.

math.GT↗

arXiv2022

Around Van den Bergh's double brackets for different bimodule structures

A double Poisson bracket, in the sense of M. Van den Bergh, is an operation on an associative algebra $A$ which induces a Poisson bracket on each representation space $\operatorname{Rep}(A,n)$ in an explicit way. In this note, we study the impact of changing the Leibniz rules underlying a double bracket. This change amounts to make a suitable choice of $A$-bimodule structure on $A\otimes A$. In the most important cases, we describe how the choice of $A$-bimodule structure fixes an analogue to Jacobi identity, and we obtain induced Poisson brackets on representation spaces. The present theory also encodes a formalisation of the widespread tensor notation used to write Poisson brackets of matrices in mathematical physics.

math.RT↗