Understanding Tensors & Knot Theory in Physics

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In summary, the section confuses the author because they do not understand the structure of ##V\otimes V##. However, the author does understand that summing finitely many of these vectors results in a matrix representation of any matrix.
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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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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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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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1. What are tensors and how are they used in physics?

Tensors are mathematical objects that describe the relationships between vectors and other tensors in a specific space. In physics, they are used to represent physical quantities, such as force, momentum, and stress, which have both magnitude and direction. Tensors are also used to describe the curvature of space-time in Einstein's theory of general relativity.

2. What is knot theory and how is it related to physics?

Knot theory is a branch of mathematics that studies the properties of mathematical knots, which are closed loops in three-dimensional space. In physics, knot theory is used to study the topology of physical systems, such as the behavior of particles in a magnetic field or the entanglement of quantum states. Knot theory is also used in the study of polymers and DNA molecules.

3. How do tensors and knot theory contribute to our understanding of the universe?

Tensors and knot theory play a crucial role in modern physics, particularly in theories like general relativity and quantum field theory. Tensors are used to describe the fundamental forces of nature, such as gravity and electromagnetism, and to model the behavior of matter at a microscopic level. Knot theory helps us understand the structure and behavior of complex systems, from the smallest particles to the largest cosmic structures.

4. What are some practical applications of tensors and knot theory?

Tensors and knot theory have numerous practical applications in fields such as engineering, computer graphics, and materials science. In engineering, tensors are used to analyze stress and strain in structures, while knot theory is used to study the properties of knots in ropes and cables. In computer graphics, tensors are used to model the deformation of objects, and knot theory is used to create realistic simulations of hair and fur. In materials science, knot theory is used to study the properties of polymers and other complex materials.

5. Are tensors and knot theory difficult to understand?

Like any branch of mathematics, tensors and knot theory can be challenging to grasp at first. However, with patience and practice, anyone can develop a basic understanding of these concepts. Many online resources, books, and courses are available to help students learn about tensors and knot theory in physics. It is also helpful to have a strong foundation in linear algebra and calculus before diving into these topics.

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