The HOMFLYPT coefficient ring

Let $A=\mathbb{Z}[v^{\pm 1}]$ and $\mathrm{\Lambda }=A[z]=\mathbb{Z}[v^{\pm 1},z]$ (The Laurent polynomial ring as the principal localisation of Z[t] at t, The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution), and put $z_{-}:=v^{-1}(z+1-v)\in \mathrm{\Lambda }$. Let $R$ be the commutative ring presented by $\boxed{{\displaystyle R:=\mathrm{\Lambda }[z^{-1},u^{\pm 1},s]/(s^2-v,vz{u}^2-(z+1-v)),}}$ i.e. the quotient of the localisation of $\mathrm{\Lambda }[s,u]$ at the multiplicative subset generated by $z$ and $u$ by the two displayed relations (Multiplicative subsets and the localisation $S^{-1}R$ as equivalence classes of fractions, The quotient ring $R\mathrm{/}I$ with $(r+I)(s+I)=rs+I$, Universal property of localisation: maps that invert $S$ factor uniquely through $S^{-1}R$). Write again $v,z,s,u$ for the images in $R$.

Elementary properties. $R$ is a commutative ring with unit; $v$, $z$ and $u$ are units by construction, and $s\cdot (s{v}^{-1})=s^2{v}^{-1}=1$, so $s$ is a unit too, with $s^{-1}=s{v}^{-1}$. The relations read $s^2=v$ and $u^2=z_{-}\mathrm{/}z$, equivalently $vz{u}^2=z+1-v$. The elements $l:=us,m:=s-s^{-1}=s^{-1}(v-1)$ belong to $R$, and $l$ is a unit. No invertibility of $m$ is imposed. The identity $l^{-1}-l=m(uz{)}^{-1}$ holds in $R$.

Universal property. Let $T$ be a commutative ring with unit. Giving a unital ring homomorphism $R\to T$ is exactly the same as giving units $v_0,z_0,u_0,s_0\in T^{\times }$ satisfying $s_0^2=v_0,v_0{z}_0{u}_0^2=z_0+1-v_0;$ the homomorphism then sends $v\mapsto v_0$, $z\mapsto z_0$, $u\mapsto u_0$, $s\mapsto s_0$. The existence direction is the universal property of polynomial rings, localisations and quotients; uniqueness holds because these images determine the images of their inverses, and these generators together with the prescribed inverses generate $R$.

Caveats. $R$ is introduced only to hold the normalisation of The HOMFLYPT polynomial from the Hecke Markov trace; no claim is made that $R$ is a domain or a UFD, and the two square roots $s$ of $v$ and $u$ of $z_{-}\mathrm{/}z$ are formal. The elements $v,z$ are inverted because $v$ is a unit of $\mathrm{\Lambda }$ and the trace identity $\alpha =(uz{)}^{-1}$ inverts $z$; $u$ is inverted because the exponent $e(\beta )$ of a braid may be negative and because $\alpha =(uz{)}^{-1}$ inverts $u$. Inverting $u$ is part of the definition: the relation $u^2=z_{-}\mathrm{/}z$ alone does not force $u$ to be a unit. The element $m=s-s^{-1}$ need not be a unit: the universal specialization $v=z=u=s=1$ sends it to $0\in \mathbb{Z}$. Division by $m$ is therefore only licensed after passing to the localization $R[m^{-1}]$; the quotient ring $R\mathrm{/}(m)$ is defined without inverting $m$.