Understanding Tensors & Knot Theory in Physics

[lekh2003]

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.

 

[Paul Colby]

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

 

[lekh2003]

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

 

[fresh_42]

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.

Have a read:
https://www.physicsforums.com/insights/what-is-a-tensor/

The elements of $V\otimes V$ are $\sum_\rho u_\rho \otimes v_\rho$ where $u_\rho$ is the first factor and $v_\rho$ the second, both vectors in $V$. $u_\rho \otimes v_\rho$ can be considered as matrix multiplication, but column times row so that they represent a rank one matrix. Summing finitely many of them allows a representation of any $n \times n$ matrix. Hence if you like, you can consider $V\otimes V$ as the set of square matrices. To see it as a matrix, we need basis vectors, e.g. spin.

The crucial points of a tensor product are, that it is bilinear as an ordinary product (distributive law in both arguments) and that $\alpha u\otimes v= u\otimes \alpha v$ for all scalars $\alpha.$

Whether $V\otimes V$ in contrast to $V\oplus V$ is natural, is another question. The main difference is that we have $(x+y)\otimes z=x\otimes z +y\otimes z$ in a tensor product, however, $(x+y,z)\neq (x,z)+(y,z)=(x+y,2z)$ in a direct sum. So the tensor product is natural because we can have operations on the first particle that do not affect the second particle, whereas a sum is always a pair and the single factors cannot be dealt with without the other.

 

[robphy]

Possibly useful reading material at:
http://homepages.math.uic.edu/~kauffman/
https://arxiv.org/a/kauffman_l_1.html (short list?)