M-convexity of the support of the Alexander polynomial refinement

Establish that for every alternating link K, the support in Z^4 of the Laurent polynomial P_K(x,y,z,w) is an M-convex set.

Background

The support of P_K is the set of exponent vectors of its nonzero monomials. M-convexity is a discrete-convexity condition generalizing matroid basis exchange and relating to generalized permutahedra. The conjecture is motivated by the fact that M-convex support is a defining feature of denormalized Lorentzian polynomials, although P_K itself is not always denormalized Lorentzian.

References

We posit:

For any alternating link $K$, the polynomial $P_K$ has $M$-convex support in $Z4$.

— An Alexander Polynomial Refinement for Alternating Links, with Trapezoidal Properties  (2608.28484 - Boninger, 28 Aug 2026) in Introduction, subsection “Fox's trapezoidal conjecture”; Conjecture 1.4 (labelled conjecture:m_convex)