17 problems

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 $\rho_K^{i,0}$,…

Let $n$ be an odd integer with $n>1$. For each $p$ with $1\leq p\leq \lfloor n/2\rfloor$, let $\Delta_p(t)$ be the suitably normalized generalized Alexander invariant of a long $n$…

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 $K$ 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…

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

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…

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…

Let $L=K_1\cup\cdots\cup K_r$ be an $r$-component link, and let … where $t_i$ corresponds to a meridian of $K_i$. Suppose that $L$ is algebraically split and component-preservingly…

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.…