abstract: We shall mainly survey differential geometric approaches to higher order linking and braiding, due to several authors, and in particular those aimed at realizing topological invariants of links and braids via holonomy of suitable flat connections. The necessary geometric, topological, and hydrodynamical background will be developed, and an attempt will be made to place the topics discussed in a wider perspective, by underlining the various relationships with other areas of mathematics and physical applications.
1. Prologue: the Gauss linking number; de Rham cohomology and Poincare'
2. (Co)homology and fundamental group of a link complement.
3. The Chen-Milnor presentation.
The Magnus expansion and Milnor invariants.
4. Connection theory: parallel transport and curvature. Flat connections. Nilpotent connections.
5. "Topological" nilpotent connections associated with a link.
Massey numbers. The Turaev - Porter theorem.
6. Pure braid groups, Arnol'd's relations and Knizhnik-Zamolodchikov connections.
Nilpotent flat connections and representations of the pure braid group.
A. BENVEGNU'and M. SPERA, Low-dimensional representations of the pure
braid group via nilpotent flat connections, Quantum Topology (to appear).
R. BOTT, L.T. TU - Differential forms in algebraic topology,
R. FENN Techniques in Geometric Topology, LMS, 1983
J. HEBDA, C.TSAU An Approach to Higher Order Linking Invariants
Through Holonomy and Curvature, Trans.Amer.Math.Soc. (to appear)
P. PAPI and C. PROCESI, Invarianti di nodi Quaderno U.M.I. 45,
(Pitagora, Bologna, 1998) (in Italian).
V. PENNA and M. SPERA, Higher order linking numbers, curvature
and holonomy, J.Knot Theory Ram. 11 (2002), 701-723.
M. SPERA, A survey on the differential and symplectic geometry
of linking numbers, Milan J.Math. 74 (2006), 139-197.