The lexicographic order on fork-noodle deck monomials

Let $N$ be a noodle and $F$ a fork of Forks, noodles and the LKB intersection pairing, and let $\mathrm{\Phi }$ be the two-variable covering homomorphism of The two-variable covering homomorphism.

Order. The lexicographic order on the deck monomials $q^a{t}^b$ is $q^a{t}^b\le q^{a^{\mathrm{\prime }}}{t}^{b^{\mathrm{\prime }}}⟺a<a^{\mathrm{\prime }},\text{ or }a=a^{\mathrm{\prime }}\text{ and }b\le b^{\mathrm{\prime }}\mathrm{.}$ The $q$-exponent is compared first. A monomial $m_{i,j}$ from a finite list is maximal if $m_{i,j}\ge m_{i^{\mathrm{\prime }},j^{\mathrm{\prime }}}$ for all pairs of the list.

Labelled intersections. Put the tine edge $T(F)$ and the noodle $N$ in transverse position and let $z_1,\dots ,z_l$ be the intersection points of $N$ with $T(F)$, ordered along $N$. Choose a parallel copy $F^{\mathrm{\prime }}$ so that the tine edge $T(F^{\mathrm{\prime }})$ meets $N$ transversely at points $z_1^{\mathrm{\prime }},\dots ,z_l^{\mathrm{\prime }}$, where $z_i$ and $z_i^{\mathrm{\prime }}$ are joined by a short arc of $N$ lying in the narrow strip between $T(F)$ and $T(F^{\mathrm{\prime }})$. For $i,j\in \{1,\dots ,l\}$ define ${\delta }_{i,j}=\{{\alpha }_1,{\alpha }_2\}\{{\beta }_1,{\beta }_2\}\{{\gamma }_1,{\gamma }_2\},$ the arc in $C$ assembled from the following embedded arcs in $D\setminus P$: ${\alpha }_1$ from $d_1$ to $z$ along the handle of $F$; ${\alpha }_2$ from $d_2$ to $z^{\mathrm{\prime }}$ along the handle of $F^{\mathrm{\prime }}$; ${\beta }_1$ from $z$ to $z_i$ along $T(F)$; ${\beta }_2$ from $z^{\mathrm{\prime }}$ to $z_j^{\mathrm{\prime }}$ along $T(F^{\mathrm{\prime }})$; ${\gamma }_1$ from $z_i$ to $d_k$ along $N$, where $k\in \{1,2\}$ is such that ${\gamma }_1$ does not pass through $z_j^{\mathrm{\prime }}$; and ${\gamma }_2$ from $z_j^{\mathrm{\prime }}$ to $d_{k^{\mathrm{\prime }}}$ along $N$, where $k^{\mathrm{\prime }}\in \{1,2\}$ is such that ${\gamma }_2$ does not pass through $z_i$ (necessarily $k\mathrm{\ne }{k}^{\mathrm{\prime }}$: the earlier point along $N$ returns to $d_1$ and the later point to $d_2$).

Monomial and sign labels. The pair $(z_i,z_j^{\mathrm{\prime }})$ carries the deck monomial $m_{i,j}=q^{a_{i,j}}{t}^{b_{i,j}}:=\mathrm{\Phi }({\delta }_{i,j}),$ which is a well-defined element of $\pm q^{\mathbb{Z}}{t}^{\mathbb{Z}}\subset \mathrm{\Lambda }$; the exponent $a_{i,j}$ satisfies $a_{i,j}=(a_{i,i}+a_{j,j})\mathrm{/}2$ by Bigelow 2001 Lemma 2.1. The pair also carries the sign ${ϵ}_{i,j}=-(-1{)}^{b_{i,i}+b_{j,j}+b_{i,j}}\in \{\pm 1\},$ the sign of the transverse intersection of the surfaces $\mathrm{\Sigma }(N)$ and $\mathrm{\Sigma }(F)$ at the point over $\{z_i,z_j^{\mathrm{\prime }}\}$.

The list of labelled pairs $(z_i,z_j^{\mathrm{\prime }},{ϵ}_{i,j},m_{i,j})$ is kept distinct from its sum: the unsummed labelled intersections are the geometric data just described, whereas the Laurent polynomial obtained after collecting equal monomials is the pairing value $⟨N,F⟩={\sum }_{i,j=1}^l{ϵ}_{i,j}{m}_{i,j},$ whose representative-independence and finiteness are proved in The fork-noodle pairing is well defined and equivariant. This definition fixes the order, the labels and the distinction, as used by the extremal-term lemma and the geometric computation of the pairing.