Bar-Natan [@BN]. Meanwhile, the swaddling construction (the first method mentioned above) is preferred by Dylan Thurston [@Th] and appears more recently in papers by Poirier ([@Poi] and by Lescop [@LeS]. The Poirier paper also gives a brief explanation for the equivalence between the two constructions ([@Poi], remark 6.17).

Acknowledgements
----------------

I would like to thank Prof. Dror Bar-Natan, who had the idea for this paper, and who helped me enormously at every stage of getting it written. I also wish to thank Tomotada Ohtsuki for his many helpful suggestions, Emanuel Farjoun for his generous help with the topological side of things, and Raymond Lickorish for his helpful comments.

The swaddling map construction
==============================

Our problem when attempting to transport the Gauss self-linking integral to knots is what to do about points of the form $(x,x)$, for which the Gauss map is not defined. We cannot simply ignore them, since this would force us to integrate over the space $C_{2}(S^{1})$ which is not compact, so that the Gauss integral would not be guaranteed to converge. If we want to find an invariant based on the concept of self-linking, we have no choice but to extend the Gauss function to points of 