17 problems

Let $L$ be a positive integer, let $a_1,\ldots,a_{2L}$ satisfy $a_i\geq 2$, and set $D=a_1+\cdots+a_{2L}$. For the braid word $x_1^{a_1}x_2^{a_2}\dots x_1^{a_{2L-1}}x_2^{a_{2L}}$,…

Asymptotic Vassiliev-invariant conjecture. The numbers $\log_{v_2(K_i)}v_3(K_i)$, which are well-defined for $K_i\ne !3_1$, converge to $2$ as $i\to\infty$. This predicts a precise…

Let $\beta\in B_n$ be an $n$-braid, let $\widehat{\beta}$ be its closure, and let $V(t)$ denote the Jones polynomial of $\widehat{\beta}$. For a complex number $t$, choose a square…

Let $B_n$ be the braid group on $n$ strands, with standard generators $\sigma_1,\sigma_2,\dots,\sigma_{n-1}$. For a braid, its closure is the link obtained by joining corresponding end…

Let $B_n$ be a braid group, and let its Temperley–Lieb representation be the representation into the Temperley–Lieb algebra. A knot is non-trivial if it is not isotopic to the unkn…

Let $b$ be an integer, and consider quasi-alternating links whose Jones polynomial has breadth $b$, where the breadth is the difference between the highest and lowest exponents occ…

Odd-degree Jones coefficient nonvanishing conjecture. For every $k\geq 1$,

The Alexander polynomial can be defined using a method known as colouring. The Jones-polynomial colouring problem. Can the Jones polynomial be defined via colouring? The source pre…

Let $K$ be a knot, and let $V(K)$ denote its Jones polynomial. A knot is non-trivial if it is not isotopic to the unknot. Jones's conjecture. There exists a non-trivial knot $K$ su…

A rational knot is represented by an integer sequence presentation, and a Jones rational coincidence is a pair of rational knots with the same Jones polynomial. The pivoting-pairs…

Chbili–Qazaqzeh's conjecture. The coefficients satisfy

Let $L$ be a link whose Jones polynomial is written as … where $m\geq 0$, $a_0\neq 0$, and $a_m\neq 0$. Chbili–Qazaqzeh conjecture. If $L$ is a prime quasi-alternating link, other…

Breadth–determinant conjecture. If $L$ is a quasi-alternating link, then

Condensed three-braid conjecture. One has

Let $\kappa=\kappa_\beta$ be the closure of a braid $\beta\in Br_m$ with writhe $w$, and let $P_{\mathrm{coker}}(t)$ denote the associated cokernel polynomial. Cokernel polynomial…

Jp-speciality conjecture. Every adequate non-alternating link is Jp-special.

Jp-speciality conjecture. All alternating knots or links, except those belonging to the family given by Conway symbol $n$—the torus links $(2,n)$, $n\ge2$—are not Jp-special.