14 problems

A polygonal walk model is a random model producing knots. Frisch–Wasserman conjecture. The probability that the random knot is an unknot tends to $0$ as the size of the walk tends…

Even–Hass–Linial–Nowik conjecture. The normalized invariant $\frac{v_m}{n^m}$ of any finite-type knot invariant $v_m$ of order $m$ converges weakly to a limit distribution in the p…

A grid walk model is a random model producing knots; write $K_n$ for the random knot at size $n$. Delbruck's conjecture. The probability that $K_n$ is knotted tends to $1$ as $n$ t…

Alexander-invariant limit conjecture. For $t=-1$ (respectively, $t=i$), there exist numbers $c_t$, $\gamma_t$, and $\alpha_t$ such that the sequence of random variables

Let $\Gamma$ be the interface curve joining opposite corners of a three-dimensional cube of size $N$, and let $L_N$ be its length, defined as the number of tetrahedra crossed by th…

In dimension $3$, let $K^\text{Arm}(n,w,k,r)$ be the probability that a length-$k$ arc of a random open arm is an $r$-local knot, and let $K^\text{Pol}(n,w,k,r)$ be the correspondi…

Let $K_n$ be a random knot sampled from the Random Jump, Petaluma, or Grid model. Random-knot universality conjectures. The following properties are expected: with high probability…

Suppose the normalized pair of invariants for a random Petaluma knot is … where $c_2$ is the Casson invariant and $v_3$ is the third-order finite type invariant. Petaluma full-supp…

Let $K_{2n+1}$ be a random knot in the Petaluma model, and let $v_m$ be a finite type invariant of order $m$. Petaluma finite-type scaling conjecture. The normalized invariant … we…

Let $K_{2n+1}$ be the random knot generated by a uniformly random permutation in the Petaluma model, and fix a knot $K$. Petaluma individual-knot conjecture. The probability … deca…

Let $L_{mn}$ be the linking number of a random two-component link whose components have $n$ and $m$ segments, respectively, with vertices sampled independently from the same three-…

Let $K_n$ be the knot obtained from a uniformly random closed self-avoiding $n$-step walk on the lattice $\mathbb{Z}^3$. Delbruck's conjecture. The knot $K_n$ is knotted with high…

Delbrück–Frisch–Wasserman conjecture. The resulting knot is non-trivial with high probability; equivalently, the probability that it is the unknot decays to zero as the number of s…

Let $T(3,n+1)$ be the random knot model obtained from Chebyshev billiard table diagrams, and let $P(T^{(n)}_K)$ denote the probability that the resulting diagram represents a fixed…