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Is there a good survey paper which describes the general ideas of Vassiliev's invariant? I am not an expert on knot theory, many references are too technical for me.

Could Vassiliev's invariants be defined for general embeddings (rather than knot theory)? What are the useful references?

Thanks.

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    $\begingroup$ You may enjoy Chapter 7 of "Knots, Mathematics with a Twist" by Alexei Sossinsky. $\endgroup$ – skupers Oct 14 at 15:21
  • $\begingroup$ There is also Vassiliev's original exposition. It's not the most concise, but it gives a fairly uniform exposition: bookstore.ams.org/mmono-98 $\endgroup$ – Ryan Budney Oct 15 at 15:50
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A short introduction to some ideas of this theory is Kontsevich, M. (1993). Vassiliev's Knot Invariants. Advances in Soviet Mathematics. Vol 16, Part 2. I think that also the following two papers give a general idea of Vassiliev's invariants: Bar-Natan, D. Finite Type Invariants (this is an overview written for the Encyclopedia of Mathematical Physics), Bar-Natan, D. (1995). The fundamental Theorem of Vasilliev's Invariants. Lecture Notes.

If you are interested in a good introduction to these invariants which is written for readers with little background in knot theory, I'd suggest you Chmutov, S., Duzhin, S., & Mostovoy, J. (2012). Introduction to Vassiliev Knot Invariants. Cambridge: Cambridge University Press. Here, some technical results are not entirely proven (they instead refer to other papers for the details); moreover, the book also deals with more advanced topics, so I think that this is a really good reference.

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    $\begingroup$ Chmutov, Duzhin, and Mostovoy also covers more "general embeddings": they discuss briefly how to extend Vassiliev invariants to links and tangles. $\endgroup$ – Calvin McPhail-Snyder Oct 15 at 13:54
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Along with the references provided already, I think many people like New points of view in knot theory by Joan Birman. She gives a good explanation of the connections between these finite type invariants and knot polynomials.

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