22 problems

Let $K$ be a hyperbolic L-space knot, let $g$ be its Seifert genus, and let $n$ be its braid index. Let $Delta_K(t)$ denote its Alexander polynomial. The hyperbolic L-space knot Al…

Let $K$ be an alternating knot whose Alexander polynomial has a stable part with parameter $m$, as in the trapezoidal coefficient pattern … Let $\sigma(K)$ denote the signature of…

Melvin–Morton–Rozansky conjecture. The semi-classical expansion is

Let $L$ be an alternating link, and write its Conway-normalized Alexander polynomial as … where the coefficients have alternating signs and no internal zeros. Fox's conjecture. The…

Let $K$ be a knot in an integral homology $3$-sphere $M$, and let $Q_n(M,K)$ denote the $n$-loop part of the logarithm of the LMO invariant, for $n\geq 1$. Rozansky's rationality c…

Let $v$ be an arbitrary Vassiliev invariant, and let the coefficients of the Alexander–Conway polynomial generate an algebra. Bar-Natan–Garoufalidis' highest-order conjecture. Ther…

Let … be the Dean knot represented by the closed $p$-braid … where $p>r>1$ and $q,s$ are nonzero integers with $\gcd(q,p)=1$. Let $\Delta_K$ be the Alexander polynomial of $K$. Hir…

Let $L\subset S^3$ be a link with $N$ components, let $\Delta_L(t_1,\ldots,t_N)$ be its Alexander polynomial, and let $\rho\colon \pi L\to \operatorname{SO}(3)$ be an abelian repre…

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…

Perturbed Conway invariant positivity conjecture. If $K\subset S^3$ is a positive knot, then all coefficients of $\delta_1(K)$ are non-positive.

Let a knot have defect defined by the zero structure of coefficients in the differential expansion of its colored knot polynomial, and let its fundamental Alexander polynomial be t…

Universal-invariant conjecture. The universal invariant associated to $B$ has the form

Let $K$ be a knot with trivial Alexander polynomial, and let Theorem denote the paper's main theorem concerning the computational complexity of its coloring invariant. Trivial-Alex…

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 $K$ be an amphicheiral knot, and let $\Delta_K$ be its Alexander polynomial. Stoimenow's sign conjecture. The sign of the leading coefficient of $\Delta_K$ is … The conjecture…

Let $K$ be an amphicheiral knot, and let $\Delta_K$ denote its Alexander polynomial; equivalently, consider the leading coefficient of its Conway polynomial. Stoimenow's square lea…

Let $K$ be a virtual knot. Let $Delta_0(K)(u,v)$ denote the zeroth Alexander polynomial, and let $OW(K)$ denote the odd writhe of $K$. The odd writhe conjecture. … The relationship…

Let $K$ be a torus knot and let $\Delta_K$ be its Alexander polynomial. The torus-knot instanton rank conjecture asserts that the reduced singular instanton homology has rank … whe…

Wheels-part Alexander conjecture. What remains of the wheels part of $Z$ is precisely the multivariable Alexander polynomial.

Let $K$ be a knot with exterior $E_K$, let $E=H_1(\Sigma,\mathbb R)$ for the peripheral torus $\Sigma$, and let $\lambda$ be the longitude direction. Let a regular point of…

Kronheimer–Mrowka's vanishing conjecture. If the symmetrized Alexander polynomial of $K$ is trivial, equivalently if $K$ is an unknot, then

Let $K$ be a knot, and let $\Delta(K;t)$ be its Alexander polynomial. Define … Choose $c\in\Lambda(K)$ such that $|c|=\min\{|z|\mid z\in\Lambda(K)\}$. Polynomial-growth conjecture.…