128 problems

Reduced 2-loop identification conjecture.

First-polynomial identification conjecture.

Universal-invariant equivalence conjecture. The datum of the universal invariant $Z_\mathbb{D}(K)$ is equivalent to any of the three pieces of data in the preceding theorem.

Diagonal determination conjecture. The family of polynomials $\rho_K^{i,i}$, $i>0$, is completely determined by the Alexander polynomial of $K$.

Higher-order determination conjecture. The value of $Z_\mathbb{D}(K)$ is completely determined by the Alexander polynomial $\Delta_K$ and the family of polynomials $\rho_K^{i,0}$,…

For every knot $K\subset S^3$, let $F_K^N(x,q)=F_K^{\mathfrak{sl}_N,\mathrm{sym}}(x,q)$ denote the series associated with symmetric representations, let $\Delta_K(x)$ be the Alexan…

Let $\mathcal{C}$ denote the smooth knot concordance group. The knot invariant $r(K)$ is defined for knots $K$ and takes values in $\mathbb{Q}$. Concordance homomorphism conjecture…

Resolution conjecture. The generalized Kauffman bracket is given by

Let $W_n$ be the weight system defined in the paper, let $Z$ denote the Kontsevich integral, and let $\nabla_n$ denote the degree-$n$ coefficient of the multivariable Conway potent…

Full-twist odd-superbridge conjecture. For $n\ge3$ and $k\ge0$, the closure of this braid is a $(2n-1)$-superbridge knot.

Trefoil connected-sum conjecture. Every nontrivial knot $K$ satisfies

Odd-superbridge knots conjecture. There are infinitely many $(2n-1)$-superbridge knots for any positive integer $n\ge3$.

Let $K_m'(t)$ be the modified family of virtual knot diagrams illustrated in the paper. Let $\langle K\rangle_N$ denote the $N$-strand bracket polynomial. The modified Family B con…

Let $K_D(t)$ denote the virtual knot diagrams in Family A depicted in the paper's schematic, and let $\langle K\rangle_N$ denote the $N$-strand bracket polynomial. The Family A det…

Let $K$ be a virtual knot diagram. The 3-strand bracket polynomial conjecture. The 3-strand bracket polynomial detects all non-trivial virtual knot diagrams. This would establish t…

Let $K$ be a link. Let $A_p(K)$ be its Alexander module, let $A'_p(K)$ be the submodule of elements annihilated by $h$, and let $\operatorname{Rep}'(K,\mathbb{E}_p)$ be the annihil…

A rational $(m,q)$-move is the local move corresponding to the indicated rational tangle, and its inverse and mirror-image moves are also allowed. The Fox $11$-coloring space of a…

Let $\mathcal{C}=(\mathbb{Z}/2)\langle a_1,\ldots,a_n\rangle/I$ be the characteristic algebra, and let $X$ be the affine scheme in $\mathbb{A}^n$ over $\mathbb{Z}/2$ obtained from…

Let $(A,\partial)$ be the differential graded algebra of a Legendrian knot, let $\varepsilon$ be an augmentation of $A$, and let the first-order Poincaré–Chekanov polynomial be the…

Let $K$ be a Legendrian knot with maximal Thurston–Bennequin number. Let its characteristic algebra be abelianized by imposing $vw=wv$ for all elements $v,w$, and regard the result…

Let $Q(x)=x^3-\alpha x^2-\beta x-1$, and let $H(Q,n)=\mathbb{C}[B_n]/(Q(b_j);\,j=1,\dots,n-1)$ be the cubic Hecke algebra. The invariant $I_{(\alpha,\beta)}$ is defined by a Markov…

Let $K$ be a knot, let $\lambda$ be the coupling parameter, and let $\bigcirc$ denote the unknot. Let $\Delta(K;z)$ be the Alexander polynomial evaluated at $z$, and set … Let…

A Vassiliev invariant of order $k$ is a knot invariant that vanishes on higher-order differences of knots. Let $v_k$ be any Vassiliev invariant of order $k$, let $K$ be a knot with…

Links–Gould polynomial conjecture. The invariant $\mathrm{LG}$ completely captures causality in such spacetimes.

Let $K$ be a fibered knot, let $\genusK$ be its genus, let $\hopfK$ be its Hopf invariant, and let $F_K$ have leading term determined by the exponent $\el(K)$. Fibered-knot leading…