nLab space of knots

Idea

Given a manifold $\Sigma$, such as the 3-sphere, its space of knots is a topological space parametrizing knots in $\Sigma$, hence something like the topological subspace

$$Emb(S^1, \Sigma) \;\subset\; Maps(S^1,\Sigma)$$

of the space of maps $S^1 \to \Sigma$ on those maps which are embeddings. This space is often denoted just $\mathcal{K}$.

A knot invariant is then equivalently a locally constant function on the space of knots.

References

General

• Allen Hatcher, Spaces of Knots (arXiv:math/9909095)

• Ryan Budney, Topology of spaces of knots in dimension 3, Proceedings of the London Mathematical Society Vol 101 pt. 2 (Sept. 2010) (arXiv:math/0506524)

Using graph complexes

Discussion of the rational homotopy type of knot spaces in terms of graph complexes:

• Dev Sinha, The Topology of Spaces of Knots: Cosimplicial Models, American Journal of Mathematics, Vol. 131, No. 4 (Aug., 2009), pp. 945-980 (36 pages) (arXiv:math/0202287, jstor:40263760)

• Alberto Cattaneo, Paolo Cotta-Ramusino, Riccardo Longoni, Configuration spaces and Vassiliev classes in any dimension, Algebr. Geom. Topol. 2 (2002) 949-1000 (arXiv:math/9910139)

• Alberto Cattaneo, Paolo Cotta-Ramusino, Riccardo Longoni, Algebraic structures on graph cohomology, Journal of Knot Theory and Its Ramifications, Vol. 14, No. 5 (2005) 627-640 (arXiv:math/0307218, doi:10.1142/S0218216505004019, math.GT/0307218, MR2006g:58021)

• Pascal Lambrechts, Victor Tourtchine, Homotopy graph-complex for configuration and knot spaces, Transactions of the AMS, Volume 361, Number 1, January 2009, Pages 207–222 (arxiv:math/0611766)

• Ismar Volić, Section 4 of: Configuration space integrals and the topology of knot and link spaces, Morfismos, Vol 17, no 2, 2013 (arxiv:1310.7224)