arXiv subjects

Boudewijn Bosch

Publications and source records attributed to Boudewijn Bosch.

3 recordsLinked to original sources

arXiv2026

Perturbed-Alexander Invariants via Quantum Cluster Algebras

A perturbative expansion of knot invariants is derived using quantum cluster algebras. By interpreting the $R$-matrix of $U_q(\mathfrak{sl}_2)$ as a cluster transformation and introducing an auxiliary parameter $\epsilon$, we derive a perturbed $R$-matrix expressed in terms of Heisenberg algebra generators arising from the representation theory of the quantum cluster algebra. The resulting knot invariant has a zeroth-order term equal to $\Delta_K(T)^{-1}$, the reciprocal of the Alexander polynomial, while higher-order terms in $\epsilon$ produce perturbed-Alexander invariants in line with the construction by Bar-Natan and Van der Veen. Our construction combines the Schr\"odinger representation of the quantum torus algebra with cluster mutation combinatorics and is illustrated with a Mathematica implementation and explicit examples.

math.GT↗

arXiv2025

Tensors, Gaussians and the Alexander Polynomial

In this paper, we formally show that the Alexander polynomial of a knot can be computed through contractions of a Gaussian function. We build on the approach of Bar-Natan and Van der Veen to universal knot invariants using (perturbed) Gaussian functions. We endow the Weyl--Heisenberg algebra with an XC-structure. Given a knot, the universal invariant arising from the XC-algebra induces a Gaussian function whose partition function recovers the Alexander polynomial. This formalism allows for the addition of suitable perturbations to the Gaussian function, from which perturbative expansions of knot invariants can be derived.

math.GT↗

arXiv2024

The Large-Color Expansion Derived from the Universal Invariant

The colored Jones polynomial associated to a knot admits an expansion of knot invariants known as the large-color expansion or Melvin-Morton-Rozansky expansion. We will show how this expansion can be derived from the universal invariant arising from a Hopf algebra $\mathbb{D}$, as introduced by Bar-Natan and Van der Veen. We utilize a Mathematica implementation to compute the universal invariant $\mathbf{Z}_{\mathbb{D}}(\mathcal{K})$ up to a certain order for a given knot $\mathcal{K}$, allowing for experimental verification of our theoretical results.

math.GT↗