Reshetikhin-Turaev Invariant of Manifolds

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• nateHI
In summary, the Reshetikhin-Turaev Invariant is a topological invariant that measures the properties of a space, often used in knot theory and three-dimensional topology. It is calculated using the Reshetikhin-Turaev construction and can provide information about the symmetries and structures present in the space. It is obtained from surgery on a link in #S^3# and is related to the colored Jones polynomials of the link.
nateHI
The Reshetikhin-Turavev construction comes with an invariant that is sometimes called the Reshetikhin-Turaev Invariant. I'm currently attempting to wrap my head around this construction but was hoping for a sneak peak to help motivate me. My question is, what does the Reshetikhin-Turaev Invariant measure? I mean, if it's an invariant of a space it must give you some information about the space itself right?

atyy
I stumbled onto the answer to my own question. I'm sufficiently motivated now. Anyway, the Reshetikhin-Turaev Invariant of a 3-manifold obtained from surgery on a link in #S^3# are colored jones polynomials of the link. Roughly (very roughly), calculate the Reshetikhin-Turaev Invariant of a knot compliment and you get the Jones Polynomial. Colored Jones polynomials sometimes have a geometric meaning.

Hello! The Reshetikhin-Turaev Invariant is a very interesting concept in mathematics and it can be a bit difficult to wrap your head around at first. Essentially, the Reshetikhin-Turaev Invariant measures the topological properties of a space. It is often used in the study of knot theory and three-dimensional topology. This invariant is calculated using the Reshetikhin-Turaev construction, which is a method for associating a topological invariant to a three-dimensional manifold. This invariant can be used to distinguish between different manifolds, and it can also provide information about the symmetries and structures present in the space. I hope this helps to motivate you in your understanding of the Reshetikhin-Turaev Invariant. Let me know if you have any further questions!

1. What is the Reshetikhin-Turaev Invariant of Manifolds?

The Reshetikhin-Turaev Invariant of Manifolds is a mathematical concept used in the study of topology and quantum field theory. It is a numerical value assigned to a manifold, which is a mathematical object that represents the shape and structure of a space. This invariant is used to classify manifolds and distinguish between different manifolds that may have similar properties.

2. How is the Reshetikhin-Turaev Invariant calculated?

The Reshetikhin-Turaev Invariant is calculated using a mathematical formula that involves the representation theory of certain quantum groups. This involves a complicated process of manipulating mathematical symbols and equations, and may require advanced knowledge of mathematics and physics to fully understand.

3. What is the significance of the Reshetikhin-Turaev Invariant?

The Reshetikhin-Turaev Invariant is significant because it allows for the classification and study of different manifolds, which are important in many areas of mathematics and physics. It also has applications in quantum computing, as it can be used to detect and correct errors in quantum algorithms.

4. Can the Reshetikhin-Turaev Invariant be calculated for any manifold?

The Reshetikhin-Turaev Invariant can be calculated for many manifolds, but not all. It is most commonly used for closed, orientable 3-manifolds, but can also be extended to certain 4-manifolds. However, there are some manifolds for which the Reshetikhin-Turaev Invariant cannot be calculated, or may not be well-defined.

5. What are some applications of the Reshetikhin-Turaev Invariant?

Aside from its significance in topology and quantum field theory, the Reshetikhin-Turaev Invariant has also been applied in other areas of mathematics and physics, such as knot theory, statistical mechanics, and conformal field theory. It also has potential applications in quantum gravity and the study of topological phases of matter.

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