7.2 The Kauffman bracket and its relation to the Jones polynomial

The Kauffman bracket is a powerful tool in knot theory. It assigns a polynomial to unoriented link diagrams, providing a way to distinguish between different knots and links. This invariant forms the foundation for the Jones polynomial.

Calculating the Kauffman bracket involves applying skein relations to simplify link diagrams. Its invariance under Reidemeister moves ensures it remains consistent regardless of how a knot is drawn, making it a reliable method for knot classification.

The Kauffman Bracket

Kauffman bracket properties

• Polynomial invariant of unoriented link diagrams assigns Laurent polynomial in variable $A$ to each unoriented link diagram
• Unknot $\bigcirc$ has Kauffman bracket $\langle \bigcirc \rangle = 1$
• Disjoint union of link $L$ and unknot $L \sqcup \bigcirc$ satisfies $\langle L \sqcup \bigcirc \rangle = (-A^2 - A^{-2}) \langle L \rangle$
• Skein relation for overcrossing \raisebox{-2pt}{\includegraphics[height=0.4cm]{images/overcrossing.png}} expressed as \langle \raisebox{-2pt}{\includegraphics[height=0.4cm]{images/overcrossing.png}} \rangle = A \langle \raisebox{-2pt}{\includegraphics[height=0.4cm]{images/smoothing_A.png}} \rangle + A^{-1} \langle \raisebox{-2pt}{\includegraphics[height=0.4cm]{images/smoothing_B.png}} \rangle

Kauffman bracket vs Jones polynomial

• Jones polynomial $V_L(t)$ obtained from Kauffman bracket by variable substitution $t = A^{-4}$ and normalization factor $(-A)^{-3w(L)}$
  • $w(L)$ represents writhe of oriented link $L$
• Relationship between Kauffman bracket and Jones polynomial: $V_L(t) = (-A)^{-3w(L)} \langle L \rangle |_{A = t^{-1/4}}$

Calculating Jones polynomials

1. Obtain unoriented diagram of knot or link
2. Apply Kauffman bracket skein relation recursively until diagram reduced to linear combination of unknots
3. Evaluate resulting polynomial in variable $A$
4. Substitute $t = A^{-4}$ and multiply by normalization factor $(-A)^{-3w(L)}$ to obtain Jones polynomial $V_L(t)$

Invariance of the Kauffman Bracket

Invariance under Reidemeister moves

• Kauffman bracket invariant under three Reidemeister moves, local modifications of link diagrams not changing underlying link
  • Reidemeister move I:
    • \langle \raisebox{-2pt}{\includegraphics[height=0.4cm]{images/R1_before.png}} \rangle = (-A^3) \langle \raisebox{-2pt}{\includegraphics[height=0.4cm]{images/R1_after.png}} \rangle (positive twist)
    • \langle \raisebox{-2pt}{\includegraphics[height=0.4cm]{images/R1_mirror_before.png}} \rangle = (-A^{-3}) \langle \raisebox{-2pt}{\includegraphics[height=0.4cm]{images/R1_mirror_after.png}} \rangle (negative twist)
  • Reidemeister move II: \langle \raisebox{-2pt}{\includegraphics[height=0.4cm]{images/R2_before.png}} \rangle = \langle \raisebox{-2pt}{\includegraphics[height=0.4cm]{images/R2_after.png}} \rangle
  • Reidemeister move III: \langle \raisebox{-2pt}{\includegraphics[height=0.4cm]{images/R3_before.png}} \rangle = \langle \raisebox{-2pt}{\includegraphics[height=0.4cm]{images/R3_after.png}} \rangle
• Invariance under Reidemeister moves ensures Kauffman bracket well-defined link invariant independent of chosen diagram representation