Tensors and Knots

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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.
1643443346786.png

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"?
1643443496777.png

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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Is there a non-paywall version of this paper? It's been years since I was last institutionalized.
 
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lekh2003
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I don't know about the legality of such actions, but there do exist ways to access the paper without institution access.
 
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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. 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.
 
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