18 problems

Let $K$ be a knot in $bS^3$, let $bbSigmad_2K$ be its 2-fold branched covering, and let $X(bbSigmad_2K)$ be the $bSL_2(bC)$ character variety of $bpi_1(bbSigmad_2K)$. Write…

Let $L$ be a link, let $\mathsf{H}_q(L)$ be its HOMFLYPT difference module, let $[L]$ be its distinguished link class, and let $\mathsf{ev}\colon \mathsf{H}(L)\to \mathcal{F}_r$ be…

Let $L$ be a link, let $\mathsf{H}_q(L)$ be its HOMFLYPT difference module, let $\mathsf{ev}\colon \mathsf{H}_q(L)\to \mathcal{F}_r$ be the evaluation map to colored HOMFLYPT polyn…

Let $L$ be a link, let its three-variable augmentation variety denote the algebraic variety associated to the degree-zero abelianized knot contact homology of $L$, and let…

Let $T$ be a framed oriented $r$-component tangle in ${\mathbb{R}}^3_{x\geq 0}$, let $N(\varLambda_T)$ be its specified neighborhood, and let $H$ denote the reference subspace used…

For a knot $\mathcal{K}$, let $\mathcal{H}_n(\mathcal{K};q,a)$ be the colored HOMFLY-PT invariant in the $n$th symmetric representation. Let $M$ and $L$ act by … with $LM=qML$. A n…

Let $L$ be a link in ${\mathbb R}^3$, let ${\mathcal A}_L$ be its combinatorial knot differential graded algebra, and let $\Lambda_L\subseteq ST^*{\mathbb R}^3$ be the unit conorma…

Let $K$ be a knot, and let $A_K(M,L,x,q)$ denote the minimal recurrence polynomial for the sequence $W_K(h_a)$. Physicists have conjectured the existence of a four-variable polynom…

Ng's conjecture. The cord algebra invariant of $K$ is the $0$-th relative knot contact homology.

A-polynomial augmentation conjecture. The $a$-deformed $A$-polynomial of $SU$-type should be identifiable with the augmentation polynomial of knot contact homology:

Let $\Phi$ be the chain map from the degree-zero knot contact-homology complex at $Q=1$ to the cord algebra of a knot. Cord-algebra isomorphism conjecture. The map $\Phi$ induces a…

Let $K$ be a knot, let $L_K$ be its associated Lagrangian, and let the augmentation variety be the curve parametrizing one-dimensional representations of the knot's differential gr…

Let $K\subset\mathbb{R}^3$ be a knot, let $HC_*(K)$ be its contact homology algebra, and let … Let $\overline{V_K}$ denote the closure of $V_K$. Polynomial-component conjecture. Th…

Quantization conjecture for super-A-polynomials. The $Q$-deformed $A$-polynomial is identified with the augmentation polynomial of knot contact homology. This proposed identificati…

Let $A^{\text{Q-def}}(x,y;a)$ denote the $Q$-deformation of the $A$-polynomial, obtained from the super-$A$-polynomial by setting $t=-1$. Let the augmentation polynomial be the inv…

Let $K$ be a knot in $S^3$, and let $f(U)$ be the polynomial determined by the branch of the zero locus of $\operatorname{Aug}_K$ near $(\lambda,\mu,U)=(0,U,U)$ through … Here…

Let $K$ be a knot, and let $\operatorname{Aug}_K(\lambda,\mu,U)$ be its three-variable augmentation polynomial. When the $U=1$ slice of the augmentation variety has a maximal-dimen…

Let $K$ be a knot. The augmentation variety $V_K$ is the set of augmentation values in $(\mathbb{C}^*)^3$, and the augmentation polynomial is defined when the maximal-dimensional p…