Bracket polynomial

The bracket polynomial of any (unoriented) link diagram ${\displaystyle L}$, denoted ${\displaystyle \langle L\rangle }$, is a polynomial in the variable ${\displaystyle A}$, characterized by the three rules:

• ${\displaystyle \langle \bigcirc \rangle =1}$, where ${\displaystyle \bigcirc }$ is the standard diagram of the unknot
• 
• ${\displaystyle \langle \bigcirc \sqcup L\rangle =(-A^{2}-A^{-2})\langle L\rangle }$

The pictures in the second rule represent brackets of the link diagrams which differ inside a disc as shown but are identical outside. The third rule means that adding a circle disjoint from the rest of the diagram multiplies the bracket of the remaining diagram by ${\displaystyle -A^{2}-A^{-2}}$.

• Louis H. Kauffman, State models and the Jones polynomial. Topology 26 (1987), no. 3, 395--407. (introduces the bracket polynomial)

• Weisstein, Eric W. "Bracket Polynomial". MathWorld.