# Tensors and Knots

• I
Gold Member
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.

Last edited:

Gold Member
Is there a non-paywall version of this paper? It's been years since I was last institutionalized.

MathematicalPhysicist
Gold Member
I don't know about the legality of such actions, but there do exist ways to access the paper without institution access.

Mentor
2022 Award
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. View attachment 296214
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"?
View attachment 296215
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.