Kauffman bracket state sum

The input is a nonempty unoriented regular link diagram $D$: a finite immersed planar projection of one or more circles with finitely many transverse double points, no triple points, and over/under data at each double point. Let $C(D)$ be its finite crossing set. At each crossing take the local chart in which the over-strand runs southwest to northeast and is drawn continuously over the under-strand, which runs northwest to southeast with a gap at the intersection. The A-smoothing joins the two left endpoints and the two right endpoints, with weight $A$; the B-smoothing joins the two top endpoints and the two bottom endpoints, with weight $A^{-1}$. These labels are transported with orientation-preserving changes of local coordinates. Put $\delta =-A^2-A^{-2}$. The output is the Laurent polynomial $⟨D⟩$.

1. Set an accumulator $p:=0$ in $\mathbb{Z}[A^{\pm 1}]$.
2. For each function $S:C(D)\to \{A,B\}$, smooth every crossing according to $S$, obtaining a crossingless diagram with $\mathrm{\mid }S\mathrm{\mid }$ circles. Count the A- and B-smoothings as $a(S)$ and $b(S)$, and replace $p\text{by}p+A^{a(S)-b(S)}{\delta }^{\mathrm{\mid }S\mathrm{\mid }-1}\mathrm{.}$
3. Return $p$ after all states have been processed.

The input is a mathematical diagram object, not a serialized string. The procedure specifies no parsing or malformed-string rejection behavior. Since the diagram has at least one component, every complete smoothing has $\mathrm{\mid }S\mathrm{\mid }\ge 1$ and every exponent of $\delta$ is nonnegative.

Kauffman bracket smoothing and loop value

Kauffman, An Invariant of Regular Isotopy (1990), §III, printed pp. 430–432

Kauffman, Combinatorial Knot Theory and the Jones Polynomial (2022), §2

Roberts, Knots Knotes (1999), §4.3

Funar, A Crash Course in Quantum Invariants of Knots and 3-Manifolds, Chapter 3, §1.1