I am reading Kawamuro's paper on Morton-Franks-Williams inequality (https://arxiv.org/abs/math/0509169). It says that a knot $K$ with braid index $b$ and maximal/minimal degrees of the variable $v$ being $d_+$ ($d_-$) in its HOMFLY-PT polynomial $P(v, z)$ satisfies inequality $$1 + \frac{d_+ - d_-}{2} \le b_K$$
and that Jones noted (in Hecke algebra representations of braid groups and link polynomials) that on all but five prime knots up to 10 crossings: 9-42, 9-49, 10-132, 10-150, 10-156 the inequality is sharp. (The same can be found in https://mathworld.wolfram.com/Morton-Franks-WilliamsInequality.html).
I wanted to verify this and started with 2-braid knot $K = 3_1$. Since $P(3_1; v, z) = (2v^{\color{blue}{2}}-v^{\color{red}{4}})+v^2z^2$ we have $\color{blue}{d_- = 2}$ and $\color{red}{d_+ = 4}$ and the MFW inequality says that $1 + 2/2 = 2 \le 2$. That's something unexpected: inequality isn't strict and $3_1$ does not appear on the list of exceptions.
The same problem arises for 3-braid knot $K = 4_1$. We have $P(4_1; v, z) = (v^{\color{blue}{-2}}-1+v^{\color{red}{2}})-z^2$, so $\color{blue}{d_- = -2}$ and $\color{red}{d_+ = 2}$ which leads to $1 + 4/2 = 3 \le 3$ and again, this is not strict.
Question: what's wrong with my calculations? Should I use different notion of degree since $P$ is a bivariate polynomial?
