8.2 The Kauffman polynomial: definition and properties

The Kauffman polynomial is a powerful tool in knot theory, offering a two-variable approach to distinguishing knots and links. It builds on the Kauffman bracket, incorporating writhe normalization to create a more robust invariant than its predecessors.

This polynomial's strength lies in its ability to differentiate between certain knots that simpler invariants can't. It's also closely related to the Jones polynomial, showcasing the interconnected nature of knot invariants in this fascinating field of mathematics.

The Kauffman Polynomial

Definition of Kauffman polynomial

• Two-variable polynomial invariant for knots and links denoted as $F(L,a,x)$
  • $L$ represents the link or knot being studied
  • $a$ and $x$ are variables used in the polynomial
• Defined using the Kauffman bracket $\langle L \rangle$, a Laurent polynomial in the variable $A$ that is a state sum invariant of unoriented link diagrams
• Obtained from the Kauffman bracket by substituting $A = a^{-1/4}$ and introducing a writhe normalization factor $a^{-w(L)/4}$
  • Writhe $w(L)$ measures the self-linking of a knot or link diagram, with each positive crossing contributing +1 and each negative crossing contributing -1
• The Kauffman polynomial is defined as: $F(L,a,x) = a^{-w(L)/4} \langle L \rangle |_{A=a^{-1/4}}$

Calculation for simple knots

• Compute the Kauffman bracket of the link diagram using the following rules:
  1. $\langle \bigcirc \rangle = 1$ for the unknot $\bigcirc$
  2. $\langle L \sqcup \bigcirc \rangle = (-A^2 - A^{-2}) \langle L \rangle$ for disjoint union $\sqcup$
  3. \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 for link diagrams differing only at one crossing
• Calculate the writhe of the link diagram by summing the contributions of each crossing
• Substitute $A = a^{-1/4}$ in the Kauffman bracket and multiply by the writhe normalization factor $a^{-w(L)/4}$ to obtain the Kauffman polynomial

Kauffman vs Jones polynomials

• The Kauffman polynomial is a generalization of the Jones polynomial $V(L,t)$, a one-variable polynomial invariant for knots and links
• The Kauffman polynomial can be specialized to the Jones polynomial by setting $x = -A^3$ and $a = A^4$: $V(L,t) = F(L,t^{-1/4},-t^{-3/4})|_{t=A^{-4}}$
• Both polynomials are invariant under Reidemeister moves and can distinguish certain knots and links

Invariance under Reidemeister moves

• The Kauffman polynomial is invariant under Reidemeister moves, local modifications of link diagrams that do not change the underlying knot or link
  • Reidemeister move I: A twist can be added or removed from a strand
  • Reidemeister move II: Two strands can be moved over or under each other
  • Reidemeister move III: A strand can be moved over or under a crossing

• The Kauffman polynomial is invariant under Reidemeister moves, local modifications of link diagrams that do not change the underlying knot or link
  • Reidemeister move I: A twist can be added or removed from a strand
  • Reidemeister move II: Two strands can be moved over or under each other
  • Reidemeister move III: A strand can be moved over or under a crossing

Invariance under Reidemeister move I:

Consider a link diagram with a single twist [positive twist] and its Kauffman bracket:

$\langle \text{[positive twist]} \rangle = A \langle \text{[straight strand]} \rangle + A^{-1} \langle \text{[straight strand]} \rangle = (-A^3 - A^{-3}) \langle \text{[straight strand]} \rangle$

1. The writhe of [positive twist] is 1, so the Kauffman polynomial is:

   $F(\text{[positive twist]},a,x) = a^{-1/4} (-a^{-3/4} - a^{3/4}) \langle \text{[straight strand]} \rangle = (-a^{-1} - a) F(\text{[straight strand]},a,x)$

2. Similarly, for a negative twist [negative twist], the Kauffman polynomial is:

   $F(\text{[negative twist]},a,x) = (-a^{-1} - a) F(\text{[straight strand]},a,x)$

3. This shows that the Kauffman polynomial is invariant under Reidemeister move I, up to multiplication by $(-a^{-1}-a)$

• Similar arguments demonstrate invariance under Reidemeister moves II and III