I Understanding Tensors & Knot Theory in Physics

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The discussion focuses on the relationship between knot theory and tensor products in physics, particularly in the context of quantum link invariants. The notation ##V\otimes V## represents a tensor product of vector spaces, where elements can be viewed as rank one matrices formed by the outer product of vectors. The tensor product is bilinear and allows for operations on individual factors without affecting others, distinguishing it from direct sums. The user seeks clarification on specific mathematical representations, such as ##M^{ab}## and ##M_{ab}##, and expresses a desire for a non-paywall version of the referenced paper. This inquiry highlights the need for a deeper understanding of the mathematical structures involved in knot theory and their applications in physics.
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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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I don't know about the legality of such actions, but there do exist ways to access the paper without institution access.
 
lekh2003 said:
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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