Hirasawa–Murasugi conjecture on stable coefficients and signature of alternating knots

Let $K$ be an alternating knot with signature satisfying $|\sigma(K)|=2k$, and let its Alexander polynomial be

$$\Delta_K(t)=\Sigma_{j=0}^{2n}(-1)^j a_j t^{2n-j},$$

where $a_j>0$. Hirasawa–Murasugi conjecture. The coefficients satisfy

$$a_0<a_1<\cdots<a_{n-m-1}<a_{n-m}=\cdots=a_{n+m}>a_{n+m+1}>\cdots>a_{2n},$$

and moreover $m\leq k$. This strengthens Fox's trapezoidal conjecture by relating the number of stable coefficients to the knot signature; the paper proves it for two-bridge knots, while the general alternating-knot case remains open.