Compute the Kauffman polynomial from a BMW braid closure

Input and output. The input is one finite ASCII string. Its prescribed form is n;ell;word, where n and ell are canonical decimal natural numbers (0, or a nonzero decimal digit followed by zero or more decimal digits), and word is empty if $\mathrm{\ell }=0$ or is exactly $\mathrm{\ell }$ comma-separated tokens of the form i+ or i-, with i a canonical decimal natural number. There is no whitespace or trailing character. The output is either the rejection symbol REJECT or an element of the converted coefficient ring $T_K=\mathbb{Z}[i,A^{\pm 1},Z,d^{\pm 1},q^{\pm 1}]\mathrm{/}(i^2+1,Z(d-1)+A+A^{-1},q-q^{-1}-iZ)\mathrm{.}$ Steps 2–5 use the Dubrovnik coefficient extension $T$, with $a=\rho$ and $d=\delta$; Step 6 uses the ring homomorphism $\phi :T⟶T_K,(a,z,d,q)⟼(-iA,iZ,d,q)\mathrm{.}$

1. Parse and validate. If the input is not in the stated syntax, if $\mathrm{\ell }$ is not the number of letter tokens, if $n=0$, or if any token index fails $1\le i<n$, return REJECT and halt. Otherwise read the tokens as the finite signed Artin word $\beta ={\sigma }_{i_1}^{{\epsilon }_1}\cdots {\sigma }_{i_{\mathrm{\ell }}}^{{\epsilon }_{\mathrm{\ell }}},{\epsilon }_j\in \{-1,1\}\mathrm{.}$

2. Compute the closure data. Form the BMW product $b_{\beta }=g_{i_1}^{{\epsilon }_1}\cdots g_{i_{\mathrm{\ell }}}^{{\epsilon }_{\mathrm{\ell }}}$. Set $w={\sum }_{j=1}^{\mathrm{\ell }}{\epsilon }_j$ and let $c$ be the number of cycles, including fixed points, of the permutation $s_{i_1}\cdots s_{i_{\mathrm{\ell }}}$ of $\{1,\dots ,n\}$.

3. Enumerate a matching basis. Work in the unit rectangle $R=[0,1{]}^2$ with top and bottom endpoints at $(j\mathrm{/}(n+1),1)$ and $(j\mathrm{/}(n+1),0)$, respectively, for $1\le j\le n$. Fix a prefix-free binary encoding of finite rational polygonal diagrams with over/under data, with rational coordinates written as reduced numerator/positive-denominator pairs. Number the tangle endpoints in cyclic order $1,2,\dots ,n,\bar{n},\dots ,\bar{2},\bar{1}$. Enumerate perfect matchings recursively: pair the least unmatched endpoint with each greater unmatched endpoint in increasing order, then continue on the remaining endpoints. For each matching $\mu$, scan finite diagram codes in increasing code length and then lexicographically. Choose the first code of a rational polygonal tangle diagram with that connector, no closed component and no strand self-crossing, in which the open strands are oriented from their lower-numbered to their higher-numbered endpoint, ordered by initial endpoint, and every crossing is over on the first traversed strand. This is the standard descending representative $T_{\mu }$. Give each polygonal segment its distinct code label and its orientation inherited from its strand. The spanning lemma identifies the standard descending class for each connector.

4. Build generator matrices by skein reduction. For every signed generator $g_i^{\pm 1}$, form its left-multiplication matrix in the ordered basis $\{T_{\mu }\}$. The column at $T_{\mu }$ is computed by reducing $g_i^{\pm 1}{T}_{\mu }$. For each initial reducer input (a matrix column here or a closed trace diagram in Step 5), first fix pairwise disjoint rational crossing boxes, each meeting only its two crossing branches, in four boundary ports, and containing no polygonal vertex. Choose the first admissible finite tuple in the length-then-lexicographic rational coding order of Step 3. Make the boxes small enough that every component contains a segment portion outside the boxes. Subdivide all polygonal segments at their box-boundary ports. Give the portions outside the boxes fresh distinct labels in their finite diagram-code order; designate the lexicographically smaller coordinate endpoint of each portion as its initial endpoint. Retain these portions, labels, and endpoints throughout that reduction tree. Inside each box record either the original crossing pairing or one of its two smoothing pairings; switches change only over/under data, and smoothings change only this pairing. In each current diagram, orient open strands from the smaller to the larger boundary endpoint label and order them by their initial endpoints. Order closed components by their least portion labels. Start each closed component along its least-labelled portion from its designated initial endpoint to its other endpoint, then follow its unique continuation. Inside a crossing box follow its current pairing. No portion is split or relabelled by a switch or smoothing, and every resulting component contains a labelled portion. This determines the traversal even when the designated directions of successive portions disagree. Traverse the open strands followed by the closed components in this order, and select the first crossing whose first encountered branch is under. Let $R(D)$ denote the coordinate vector of the current tangle expression in the matching basis. If the selected crossing is $D_{+}$ in the fixed chart, put $\eta =1$; if it is $D_{-}$, put $\eta =-1$. Replace $R(D)=R(D^{\mathrm{s}\mathrm{w}})+\eta z_D(R(D_0)-R(D_{\mathrm{\infty }})),$ using the fixed through smoothing $D_0$ and cup-cap smoothing $D_{\mathrm{\infty }}$. Recurse on all three resulting terms. The switched branch has one fewer non-descending crossing and each smoothing has one fewer crossing. At a descending tangle leaf, use the traversal to lift each already traversed segment over the remaining diagram, removing self-crossings while fixing the boundary; retain crossings between distinct open strands in their ordered layers. Remove the resulting framing curls with factors $a$ or $a^{-1}$ and each closed component with factor $d$. Record the remaining standard matching tangle and add its coefficient to that matching coordinate.

5. Apply the matrices and evaluate the trace row. Start with the coordinate vector of the identity tangle and apply the generator matrices from the rightmost braid letter to the leftmost, obtaining $v_{\beta }$, the coordinate vector of ${\mathrm{\Phi }}_n(b_{\beta })$. For each matching $\mu$, close $T_{\mu }$ by the following fixed untwisted planar closure. For $1\le j\le n$ put $x_j=j\mathrm{/}(n+1)$ and ${ϵ}_j=1\mathrm{/}(j+1)$, and add the rational polygonal arc

   $C_j:(x_j,1)\to (x_j,1+{ϵ}_j)\to (1+{ϵ}_j,1+{ϵ}_j)\to (1+{ϵ}_j,-{ϵ}_j)\to (x_j,-{ϵ}_j)\to (x_j,0)\mathrm{.}$

   These arcs lie outside $R$ except at their specified endpoints and are pairwise disjoint: if $j<k$, then $x_j<x_k$ and ${ϵ}_j>{ϵ}_k$, so their exterior lanes are nested. Orient $C_j$ from top endpoint $j$ to bottom endpoint $\bar{j}$, give its segments fresh distinct labels, and use this same arc system for every matching. It represents the untwisted closure used in the definition of ${\tau }_n$. Evaluate the closed diagram by the deterministic traversal and skein recursion of Step 4: at a descending leaf, straighten the traversal-ordered components, remove their framing curls with factors $a,a^{-1}$, remove each free circle with factor $d$, and use $P(\mathrm{\varnothing })=1$ for the empty diagram.

   Set $t_{\mu }=d^{-n}P(\widehat{T_{\mu }}),{\tau }_n(b_{\beta })=t^{\mathsf{T}}{v}_{\beta }\mathrm{.}$

6. Convert parameters and return. Substitute $a=-iA$ and $z=iZ$ in the expression $d^{n-1}{\tau }_n(b_{\beta })$ computed in Step 5, leaving $d,q$ fixed and interpreting all coefficients in $T_K$. Return $-i^w(-1{)}^c{A}^{-w}\phi \text{\hspace{0.17em}⁣}(d^{n-1}{\tau }_n(b_{\beta }))\mathrm{.}$

BMW product and coefficient ring

Tangle skein algebra

BMW generator map

Descending matching spanning

Matching independence

Termination and output formula

Finite tangle skein algebra

BMW generator map

Standard matching spanning set

Matching independence

Normalized closure trace

Definitions of the fixed-diagram values D, L, and F

Braid closure and its components

Exponent sum

Artin generators as geometric half twists

Goodman and Hauschild, Affine Birman–Wenzl–Murakami Algebras and Tangles in the Solid Torus

Kauffman, An Invariant of Regular Isotopy

Poulain d’Andecy, Fusion for the Yang–Baxter Equation and the Braid Group (2020)