We will survey various connections, starting with the theory of Vassiliev invariants of classical knots and its relation to Goodwillie-Weiss embedding calculus.
This extends to more recent work by Watanabe on Diff($S^4$) and Fresse-Turchin-Willwacher on higher dimensional embedding theory.
Links:
[1] http://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/de/node/3444
[3] http://www.mpim-bonn.mpg.de/de/node/13203