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Operator formalism for topology-conserving crossing dynamics in planar knot diagrams. (English) Zbl 1319.82028

Summary: We address here the topological equivalence of knots through the so-called Reidemeister moves. These topology-conserving manipulations are recast into dynamical rules on the crossings of knot diagrams. This is presented in terms of a simple graphical representation related to the Gauss code of knots. Drawing on techniques for reaction-diffusion systems, we then develop didactically an operator formalism wherein these rules for crossing dynamics are encoded. The aim is to develop new tools for studying dynamical behaviour and regimes in the presence of topology conservation. This necessitates the introduction of composite paulionic operators. The formalism is applied to calculate some differential equations for the time evolution of densities and correlators of crossings, subject to topology-conserving stochastic dynamics. We consider here the simplified situation of two-dimensional knot projections. However, we hope that this is a first valuable step towards addressing a number of important questions regarding the role of topological constraints and specifically of topology conservation in dynamics through a variety of solution and approximation schemes. Further applicability arises in the context of the simulated annealing of knots. The methods presented here depart significantly from the invariant-based path integral descriptions often applied in polymer systems, and, in our view, offer a fresh perspective on the conservation of topological states and topological equivalence in knots.

MSC:

82D60 Statistical mechanics of polymers
35K57 Reaction-diffusion equations

Software:

KnotTheory; K2K
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Full Text: DOI arXiv

References:

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