19 problems

A ring polymer is a closed polymer chain, and a ring polymer is nontrivially knotted when its embedding in space has nontrivial knot type. Frisch–Wasserman–Delbrück conjecture. Non…

Airy scaling conjecture. The pressure-induced phase transition is characterized by a tri-critical point, and the associated single-variable scaling function is given by the logarit…

Let $r\ge1$ and $s\ge2$ be given. For staircase polygons and rooted self-avoiding polygons, let $F^{(r,s)}(z)$ and $F^{(r)}(z)$ be the corresponding area amplitude series. Suppose…

Universal amplitude-combination conjecture. All products

A polygon is a closed self-avoiding polymer chain, and its prime factors are the prime knot summands in its knot decomposition. Prime factors are tight and isolated from each other…

Square-polygon asymptotic conjecture. The fraction satisfies

Maximal-fraction conjecture. Among all self-avoiding polygons of length $4L$, the polygon with the largest value of $F_p$ is $\operatorname{Sq}_{L\times L}$.

Convergence conjecture. The cumulative sum satisfies

Self-avoiding polygon Airy conjecture. The derivative of the scaling function is

Let $G_L(q)=\sum_{k=1}^{L^2-1}g_{L,k}q^k$ be the gerrymander polynomial, so that $G_L(1)/2$ counts partitions of an $L\times L$ square into two connected regions. Let $\lambda$ be…

Let $P_L(1)$ denote the number of self-avoiding polygons crossing an $L\times L$ square, and let $\lambda_S$ be their growth constant. The subdominant asymptotic form has a power-l…

Let $P_L(1)$ denote the number of self-avoiding polygons crossing an $L\times L$ square, and let $\lambda_S$ be the common growth constant for polygons and self-avoiding walks cros…

Self-avoiding-polygon exponent conjecture. The exponent in this asymptotic form is exactly

A ring polymer is modeled by a self-avoiding lattice polygon, and its knot type is the topological knot type represented by that polygon. Frisch–Wasserman–Delbrück conjecture. Suff…

Richard–Guttmann–Jensen conjecture. Based on exact enumeration data, $P_r(t,q)$ satisfies a $q$-functional equation of finite degree with polynomial coefficients.

Let $T^\mathrm{F}(x)$ be the weighted transfer matrix for full $s$-blocks, and let $T^\mathrm{H}(x)$ be its restriction to the strongly connected component generated by Hamiltonian…

Hamiltonian-polygon dominance conjecture. Hamiltonian polygons and full $s$-blocks, counted by length rather than span, have the same growth rate:

For self-avoiding polygons, let $F(s)$ be the scaling function in the tricritical scaling form near $(x_c,1)$, and let $x_c$ and $A_0$ denote the constants appearing in that scalin…

Gaussian perimeter limit-law conjecture. The mean and standard deviation satisfy