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A survey on the differential and symplectic geometry of linking numbers. (English) Zbl 1165.58004

From the text: The aim of the present survey mainly consists in illustrating some recently emerged differential and symplectic geometric aspects of the ordinary and higher order linking numbers of knot theory, within the modern geometrical and topological framework, constantly referring to their multifaceted physical origins and interpretations.
The paper is organized as follows. Section 2 is somewhat preparatory and it is centered around the concept of helicity, which allows for a natural introduction of the Gauss linking number in a manner tailored to our future purposes. We both discuss the classical vector analytical and the modern differential form theoretic formulation (abelian Chern-Simons action), concluding with a short digression on topological quantum field theory which will lend motivation to our subsequent constructions.
In Section 3, after recalling equivariant moment maps, we discuss the manifold of (mildly – in a sense to be specified) smooth singular knots (“closed vortex filaments”) in a three-fold (sticking to the \(\mathbb R^3\) case) introduced by J.-L. Brylinski [Loop spaces, characteristic classes and geometric quantization. Progress in Mathematics. 107. Boston, MA: Birkhäuser (1993; Zbl 0823.55002)], which possesses natural symplectic and Riemannian structures (formally) combining into a Kähler one, proceeding subsequently to elucidate its hydrodynamical content, in the framework of the geometrical interpretation of Euler’s equation for a perfect (i.e. incompressible, inviscid) fluid in terms of coadjoint orbits of volume preserving diffeomorphisms.
In the following section (4), after a detour on Lagrangian submanifold theory, we discuss the Morse family interpretation of the abelian Chern-Simons action (with knot insertion) set forth in [A. Besana, Framed knots, Lagrangian submanifolds and geometric quantization. Ph.D. Thesis, Università degli Studi di Milano (2004) and A. Besana and M. Spera, J. Knot Theory Ramifications 15, No. 7, 883–912 (2006; Zbl 1114.53065)], leading to a Maslov theoretical interpretation of the writhe of a knot (after a choice of a plane projection thereof).
We then pass (in Section 5) to geometric quantization issues, reviewing the Bohr-Sommerfeld interpretation of the Feynman-Onsager condition arising in quantum vortex theory, again developed in [Besana and Spera (loc. cit.)], in which the Gauss linking number plays a pivotal role. Next, we discuss higher order linking phenomena via the differential geometric apparatus of [V. Penna and M. Spera, J. Knot Theory Ramifications 11, No. 5, 701–723 (2002; Zbl 1027.57007)], in terms of Chen-Hain-Tavares nilpotent “topological” connections, focusing on the basic steps of the construction (which is strongly reminiscent of Chern-Weil theory). This leads to a holonomy interpretation of Massey higher order linking numbers, and to a short proof of a weak version of the Turaev-Porter theorem, stating equality with the so-called Milnor higher order linking numbers, defined group combinatorially. Then we address magnetic relaxation and its topological bounds, a field which has recently witnessed a massive flurry of activity, starting from the seminal work of Arnold, Moffatt and Freedman. We prove a possibly new result in this direction, in the context of higher order linking, for almost trivial (i.e. Brunnian) links which involves Arnold’s “Helicity Bounds Energy Theorem” together with the intepretation, originating in [Penna and Spera (loc. cit.)], of higher order linking numbers in terms of suitable ordinary linking numbers.
The following section is devoted to pointing out some possible further fruitful connections of the above theory with the work of M. A. Berger on higher order braiding and the Kontsevich integral [J. Phys. A, Math. Gen. 34, No. 7, 1363–1374 (2001; Zbl 0984.70016), Lett. Math. Phys. 55, No. 3, 181–192 (2001; Zbl 0986.57010)] and with the theory developed in the final section of [Besana and Spera (loc. cit.)] aiming at a geometric quantization interpretation of Laughlin’s wave functions employed in the theory of the Fractional Quantum Hall Effect. As a new application of the previous Chen integral theoretic techniques, we recover Berger’s 3-braid invariant via parallel transport of a nilpotent flat connection manufactured from the Arnold identity, defined on the configuration space \(X_3\) of distinct points on the complex plane, thereby yielding a (Heisenberg group) representation of its fundamental group, that is, the pure braid group \(P_3\). The nontrivial entries of the parallel transport matrix will give second and third order (pure) braid invariants.
A short final section is devoted to some concluding remarks and open problems. The paper is accompanied by a certain amount of hand-drawn pictures, mostly taken from [Penna and Spera (loc. cit.] and [Besana and Spera (loc. cit.)], but depicted anew, again in view of clarity enhancement.

MSC:

58D10 Spaces of embeddings and immersions
53D50 Geometric quantization
53D12 Lagrangian submanifolds; Maslov index
58J28 Eta-invariants, Chern-Simons invariants
76B47 Vortex flows for incompressible inviscid fluids
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