I highly recommend that you check out some of the recent work by Dror Bar-Natan. In particular, he videotapes and posts online all his talks: http://www.math.toronto.edu/drorbn/Talks/

In some of his recent talks, he explains various connections between virtual knots and questions in knot theory but also in representation theory and Lie theory.

I (Theo, who posted this answer originally) have set the answer to CW after it was pointed out that I really should have written more. Rather than me writing more, I'd like to invite others to elaborate if they'd like (I'm not an expert). To start it off, the following paragraph is precisely a comment by Daniel Moskovich.

Dror's programme, in some sense, is to turn quantum topology on its head by making algebra (Lie algebras in particular) the primary object, and topological knotted stuff the tool used to understand them. Specifying a universal finite-type invariant for KTG's is the same thing as specifying a (nice) associator. In the same sense, specifying a universal finite-type invariant for v-KTG's should really explain Etingoff-Kazhdan quantization of Lie bialgebras. This is discussed in his Montpellier talk. It's an ambitious programme, but one with every reason to succeed, IMHO.

I highly recommend that you check out some of the recent work by Dror Bar-Natan. In particular, he videotapes and posts online all his talks: http://www.math.toronto.edu/drorbn/Talks/

In some of his recent talks, he explains various connections between virtual knots and questions in knot theory but also in representation theory and Lie theory.

I (Theo, who posted this answer originally) have set the answer to CW after it was pointed out that I really should have written more. Rather than me writing more, I'd like to invite others to elaborate if they'd like (I'm not an expert). To start it off, the following paragraph is precisely a comment by Daniel Moskovich.

Dror's programme, in some sense, is to turn quantum topology on its head by making algebra (Lie algebras in particular) the primary object, and topological knotted stuff the tool used to understand them. Specifying a universal finite-type invariant for KTG's is the same thing as specifying a (nice) associator. In the same sense, specifying a universal finite-type invariant for v-KTG's should really explain Etingoff-Kazhdan quantization of Lie bialgebras. This is discussed in his Montpellier talk. It's an ambitious programme, but one with every reason to succeed, IMHO.

I highly recommend that you check out some of the recent work by Dror Bar-Natan. In particular, he videotapes and posts online all his talks: http://www.math.toronto.edu/drorbn/Talks/

In some of his recent talks, he explains various connections between virtual knots and questions in knot theory but also in representation theory and Lie theory.

I highly recommend that you check out some of the recent work by Dror Bar-Natan. In particular, he videotapes and posts online all his talks: http://www.math.toronto.edu/drorbn/Talks/

In some of his recent talks, he explains various connections between virtual knots and questions in knot theory but also in representation theory and Lie theory.

I (Theo, who posted this answer originally) have set the answer to CW after it was pointed out that I really should have written more. Rather than me writing more, I'd like to invite others to elaborate if they'd like (I'm not an expert). To start it off, the following paragraph is precisely a comment by Daniel Moskovich.

Dror's programme, in some sense, is to turn quantum topology on its head by making algebra (Lie algebras in particular) the primary object, and topological knotted stuff the tool used to understand them. Specifying a universal finite-type invariant for KTG's is the same thing as specifying a (nice) associator. In the same sense, specifying a universal finite-type invariant for v-KTG's should really explain Etingoff-Kazhdan quantization of Lie bialgebras. This is discussed in his Montpellier talk. It's an ambitious programme, but one with every reason to succeed, IMHO.