On Some Symplectic Aspects of Knot Framings

Abstract

The present article delves into some symplectic features arising in basic knot theory. An interpretation of the writhing number of a knot (with reference to a plane projection thereof) is provided in terms of a phase function analogous to those encountered in geometrical optics, its variation upon switching a crossing being akin to the passage through a caustic, yielding a knot theoretical analogue of Maslov's theory, via classical fluidodynamical helicity. The Maslov cycle is given by knots having exactly one double point, among those having a fixed plane shadow and lying on a semi-cone issued therefrom, which turn out to build up a Lagrangian submanifold of Brylinski's symplectic manifold of (mildly) singular knots. A Morse family (generating function) for this submanifold is determined and can be taken to be the Abelian Chern–Simons action plus a source term (knot insertion) appearing in the Jones–Witten theory. The relevance of the Bohr–Sommerfeld conditions arising in geometric quantization are investigated and a relationship with the Gauss linking number integral formula is also established, together with a novel derivation of the so-called Feynman–Onsager quantization condition. Furthermore, an additional Chern–Simons interpretation of the writhe of a braid is discussed and interpreted symplectically, also making contact with the Goldin–Menikoff–Sharp approach to vortices and anyons. Finally, a geometrical setting for the ground state wave functions arising in the theory of the Fractional Quantum Hall Effect is established.

A Survey on the Differential and Symplectic Geometry of Linking Numbers

“…74 (2006) Geometry of Linking Numbers 147 as in the preceding subsection. This simple observation plays an important role in our Lagrangian submanifold theory interpretation ( [20,19]). …”

“…The (highly differential geometrically flavoured) theory of Kontsevich ([57, 12, 77]) can be viewed as the farthest reaching generalization of Gauß' ideas and at the same time perhaps the closest in spirit thereto. However we are going to comment only briefly thereupon in the sequel, since the scope of this note is, by contrast, rather modest: we are going to review, keeping technical details to a minimum and concentrating on the basic ideas, some newly discovered symplectic and differential geometric interpretations of ordinary and higher order linking numbers ( [20,84]), within the general geometric framework -manufactured, among others, by Arnol'd, Marsden and Weinstein, and Brylinski -trying and properly place them within the existing research lines in topological fluid mechanics, quantum field theory and knot theory per se. Nevertheless, we shall often intermingle the flow of the exposition with short digressions (which can be skipped by expert readers) providing some background material on the various topics involved, in order to improve readability for a larger audience.…”

“…In the following section (4), after a detour on Lagrangian submanifold theory, we discuss the Morse family interpretation of the abelian ChernSimons action (with knot insertion) set forth in [20,19], 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 [20,19], in which the Gauß linking number plays a pivotal role.…”

“…The following section is devoted to pointing out some possible further fruitful connections of the above theory with the work of Berger on higher order braiding and the Kontsevich integral ( [17,18]) and with the theory developed in the final section of [20] aiming at a geometric quantization interpretation of Laughlin's wave functions employed in the theory of…”