🪢Knot Theory Unit 7 – The Jones Polynomial and Kauffman Bracket

What's the Big Idea?

• The Jones polynomial is a knot invariant that distinguishes different knots and links
• Assigns a polynomial to each knot, which remains unchanged under ambient isotopy
• Provides a powerful tool for studying and classifying knots
• Kauffman bracket is a closely related polynomial that simplifies the computation of the Jones polynomial
• Together, these concepts form the foundation for modern knot theory and its applications
• Connects knot theory with other areas of mathematics, such as statistical mechanics and quantum field theory
• Offers insights into the fundamental properties and behavior of knots and links

Key Concepts and Definitions

• Knot: A closed, non-self-intersecting curve embedded in three-dimensional space
  • Can be thought of as a tangled piece of string with its ends joined together
• Link: A collection of knots that may be intertwined or linked together
• Ambient isotopy: A continuous deformation of the space surrounding a knot, without allowing the knot to pass through itself
  • Knots are considered equivalent if they can be transformed into each other through ambient isotopy
• Knot invariant: A quantity or property that remains unchanged under ambient isotopy
  • Helps distinguish different knots and links
• Polynomial: A mathematical expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents
• Skein relation: A recursive formula that relates the polynomials of knots or links that differ only in a small region
  • Enables the computation of the Jones polynomial and Kauffman bracket

Historical Background

• Knot theory has its roots in the 19th century, with early work by mathematicians such as Carl Friedrich Gauss and Peter Guthrie Tait
• In the 1920s, Kurt Reidemeister introduced the Reidemeister moves, which laid the foundation for the study of knot equivalence
• The Jones polynomial was discovered by Vaughan Jones in 1984, revolutionizing the field of knot theory
  • Jones was awarded the Fields Medal in 1990 for this groundbreaking work
• Louis Kauffman introduced the Kauffman bracket in 1987 as a simplified approach to computing the Jones polynomial
• Since then, numerous other knot invariants and polynomial invariants have been discovered, expanding our understanding of knots and their properties
• The study of knots has found applications in various fields, including physics, chemistry, and biology

The Jones Polynomial Explained

• The Jones polynomial, denoted as $V_K(t)$, is a Laurent polynomial associated with a knot $K$
• It is computed using a skein relation, which relates the polynomials of three links that differ only in a small region
  • The skein relation for the Jones polynomial is: $t^{-1} V_{L_+}(t) - t V_{L_-}(t) = (t^{1/2} - t^{-1/2}) V_{L_0}(t)$
• The Jones polynomial is invariant under the Reidemeister moves, making it a true knot invariant
• To compute the Jones polynomial, start with a knot diagram and apply the skein relation recursively until the diagram is reduced to a collection of unknots
  • The polynomial of an unknot is defined as 1
• The Jones polynomial can detect chirality, as mirror images of a chiral knot will have different polynomials
• However, the Jones polynomial is not a complete invariant, as there exist distinct knots with the same Jones polynomial

Kauffman Bracket: Breaking It Down

• The Kauffman bracket, denoted as $\langle K \rangle$, is a polynomial invariant closely related to the Jones polynomial
• It simplifies the computation of the Jones polynomial by eliminating the need for orientation in the knot diagram
• The Kauffman bracket is defined using the following rules:
  • $\langle \emptyset \rangle = 1$ (empty diagram)
  • $\langle K \sqcup \bigcirc \rangle = (-A^2 - A^{-2}) \langle K \rangle$ (disjoint union with a circle)
  • \langle \raisebox{-2pt}{\includegraphics[height=0.4cm]{L+.png}} \rangle = A \langle \raisebox{-2pt}{\includegraphics[height=0.4cm]{L0.png}} \rangle + A^{-1} \langle \raisebox{-2pt}{\includegraphics[height=0.4cm]{Linf.png}} \rangle (skein relation)
• To compute the Kauffman bracket, apply the skein relation recursively until the diagram is reduced to a collection of disjoint circles
• The Jones polynomial can be obtained from the Kauffman bracket by a simple substitution and normalization: $V_K(t) = (-A)^{-3w(K)} \langle K \rangle|_{A=t^{-1/4}}$ where $w(K)$ is the writhe of the knot $K$
• The Kauffman bracket is not a true knot invariant, as it is not invariant under the first Reidemeister move
  • However, this issue is resolved when converting to the Jones polynomial

Connections and Applications

• The Jones polynomial and Kauffman bracket have deep connections to various areas of mathematics and physics
• In statistical mechanics, the Jones polynomial arises in the study of the Potts model and the Yang-Baxter equation
  • The Potts model is a generalization of the Ising model, which describes the behavior of magnetic spins on a lattice
• In quantum field theory, the Jones polynomial is related to the Chern-Simons theory and the Witten invariant
  • Chern-Simons theory is a topological quantum field theory that associates invariants to 3-manifolds and knots
• The Kauffman bracket has applications in DNA topology and the study of enzyme action on DNA
  • Knot theory provides a framework for understanding the entanglement and recombination of DNA molecules
• Knot invariants, including the Jones polynomial, have been used in the study of protein folding and the classification of protein structures
• The Jones polynomial has also found applications in cryptography and quantum computing
  • It has been proposed as a basis for constructing quantum algorithms and error-correcting codes

Worked Examples

1. Compute the Jones polynomial of the trefoil knot:

   • Start with the knot diagram of the trefoil knot
   • Apply the skein relation recursively until the diagram is reduced to unknots
   • Substitute the polynomial of the unknot (1) and simplify the resulting expression
   • The Jones polynomial of the trefoil knot is: $V(t) = t + t^3 - t^4$

2. Compute the Kauffman bracket of the figure-eight knot:

   • Begin with the knot diagram of the figure-eight knot
   • Apply the Kauffman bracket skein relation recursively until the diagram is reduced to disjoint circles
   • Substitute the bracket of the empty diagram (1) and the bracket of a disjoint union with a circle ($(-A^2 - A^{-2}) \langle K \rangle$)
   • Simplify the resulting expression
   • The Kauffman bracket of the figure-eight knot is: $\langle K \rangle = -A^8 - A^4 - A^{-4} - A^{-8}$

3. Determine the Jones polynomial of the figure-eight knot using the Kauffman bracket:

   • Compute the Kauffman bracket of the figure-eight knot (as shown in the previous example)
   • Calculate the writhe of the knot diagram (in this case, $w(K) = 0$)
   • Apply the formula: $V_K(t) = (-A)^{-3w(K)} \langle K \rangle|_{A=t^{-1/4}}$
   • Substitute $A=t^{-1/4}$ and simplify the expression
   • The Jones polynomial of the figure-eight knot is: $V(t) = t^{-2} - t^{-1} + 1 - t + t^2$

Common Pitfalls and Tips

• When computing the Jones polynomial or Kauffman bracket, be careful to apply the skein relations consistently and recursively
  • Double-check your work to ensure that you haven't missed any terms or made algebraic errors
• Remember that the Jones polynomial is invariant under ambient isotopy, while the Kauffman bracket is not invariant under the first Reidemeister move
  • When using the Kauffman bracket to compute the Jones polynomial, don't forget to account for the writhe of the knot diagram
• Keep in mind that the Jones polynomial is not a complete invariant
  • If two knots have the same Jones polynomial, they may still be distinct knots
  • To definitively distinguish knots, you may need to use additional invariants or techniques
• When working with knot diagrams, be mindful of the orientation of the crossings
  • The sign of a crossing (positive or negative) affects the computation of the writhe and, consequently, the Jones polynomial
• Practice computing the Jones polynomial and Kauffman bracket for a variety of knots and links to develop your skills and intuition
  • Start with simple examples and gradually work your way up to more complex knots
• Explore the connections between knot theory and other areas of mathematics and science to deepen your understanding and appreciation of the subject
  • Look for opportunities to apply your knowledge of knot invariants to real-world problems and research questions