On the self-linking of knots

Abstract

This note describes a subcomplex F of the de Rham complex of parametrized knot space, which is combinatorial over a number of universal ‘‘Anomaly Integrals.’’ The self-linking integrals of Guadaguini, Martellini, and Mintchev [‘‘Perturbative aspects of Chern–Simons field theory,’’ Phys. Lett. B 227, 111 (1989)] and Bar-Natan [‘‘Perturbative aspects of the Chern–Simons topological quantum field theory,’’ Ph.D. thesis, Princeton University, 1991; also ‘‘On the Vassiliev Knot Invariants’’ (to appear in Topology)] are seen to represent the first nontrivial element in H0(F)—occurring at level 4, and are anomaly free. However, already at the next level an anomalous term is possible.

A homotopy-theoretic view of Bott–Taubes integrals and knot spaces

“…In [8], Bott and Taubes constructed knot invariants by considering a bundle over Emb.S 1 ; R 3 /. The fiber of this bundle is a compactification of a configuration space of q C t points in R 3 , q of which lie on the knot.…”

“…The bundle we consider is essentially the one considered by Bott and Taubes [8], who integrated differential forms along the fiber to get knot invariants. By doing this "integration" homotopy-theoretically, we are able to produce integral cohomology classes.…”

Vassiliev invariants for links from Chern-Simons perturbation theory

“…This seems to be the case as shown in [40] and the resulting invariants might be related to string link invariants [41]. Finally, the invariants presented in this paper should be also regarded from the approach proposed in [24]. We plan to report on these and some other issues related to Vassiliev invariants in future work.…”

“…This procedure has been applied to obtain Vassiliev knot invariants up to order six for all prime knots up to six crossings [19] and for all torus knots [23]. These geometrical invariants have also been studied by Bott and Taubes [24] using a different approach. The relation of this approach to the one in [19] has been studied recently in [25].…”

Higher-Dimensional BF Theories¶in the Batalin-Vilkovisky Formalism:¶The BV Action and Generalized Wilson Loops

“…There are a couple of interesting cases when M does not satisfy Assumption 1, but one can define the superpropagator anyway. First, when M = R m (see subsection 9.1) all boils down to looking for (the higher-dimensional generalization of) Bott and Taubes's [9] tautological forms, as described in [16]. Second, when M is a rational homology sphere, one can generalize the construction of [7] (which does not yield a closed η, so that extra diagrams must be introduced to correct for it) or alternatively remove one point, as suggested in [29], and essentially reduce to the previous case.…”