- #1

lekh2003

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Hello everyone! I'm currently self studying knot theory and I am at the point where I am looking at its relationship with other fields. I am a math and physics student, but my physics understanding is far behind my understanding of math. Hence, I would really like some help interpreting some sections of a paper I'm reading: https://www.sciencedirect.com/science/article/pii/S0960077997000957.

Currently, I am looking at the section on quantum link invariants, and specifically the simple case of a trivial knot in a spacetime diagram.

I understand this section pretty well, until they say that it is "natural" to take a vector space of the form ##V\otimes V##. I don't really think I'm familar with this notation? I assume that it is a tensor product. Could someone give me a TLDR of what exactly this represents mathematically? Then, what exactly is meant by "factor of the tensor product"?

This section also confuses me. However, I think I simply do not understand the structure of ##V\otimes V##, and hence I don't really know how to interpret ##M^{ab}## and ##M_{ab}##. For that matter, what do (1) and ##(e_{ab})## represent. Are these vectors or elements of ##V\otimes V##?

I'd love it if someone could help me out, I'd love to have a better grasp of this content.

Currently, I am looking at the section on quantum link invariants, and specifically the simple case of a trivial knot in a spacetime diagram.

I understand this section pretty well, until they say that it is "natural" to take a vector space of the form ##V\otimes V##. I don't really think I'm familar with this notation? I assume that it is a tensor product. Could someone give me a TLDR of what exactly this represents mathematically? Then, what exactly is meant by "factor of the tensor product"?

This section also confuses me. However, I think I simply do not understand the structure of ##V\otimes V##, and hence I don't really know how to interpret ##M^{ab}## and ##M_{ab}##. For that matter, what do (1) and ##(e_{ab})## represent. Are these vectors or elements of ##V\otimes V##?

I'd love it if someone could help me out, I'd love to have a better grasp of this content.

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