50 problems

Let $K$ be an alternating knot, and write its Alexander polynomial as $\Delta_K(t)=\sum_i a_i t^i$. A sequence is trapezoidal if the absolute values of its terms increase, possibly…

Hoste's conjecture. Then $\Re(t) > -1$. This conjecture concerns the location of zeros of Alexander polynomials for alternating knots. The source recalls it while analyzing zeros i…

Let $K$ be a knot, let $J_{K,n}(q)\in\mathbb{Z}[q^{\pm1}]$ be its $n$-th colored Jones polynomial, and define … Let $\triangle_K$ denote the Alexander polynomial of $K$. Melvin–Mor…

Let $C$ be a line arrangement that is not a union of concurrent lines, and let ${\alpha}$ be a root of its Alexander polynomial $\Delta^1_C(t)$. Suppose that the order of…

Let $\mathcal{K}$ be a knot, let $A(\mathcal{K};q)$ denote its classical Alexander polynomial, and let $A_{\lambda}(\mathcal{K};q)$ denote the colored Alexander polynomial associat…

Li–Ni's conjecture. Then $K$ is an iterated torus knot.

Degtyarev's conjecture. Up to equisingular deformation, $C$ is determined by its set of singularities and its Alexander polynomial.

Alexander-polynomial characterization. The polynomial $P(t)$ is the Alexander polynomial of some Hurwitz $C$-group $G$ if and only if: (i) the roots of $P(t)$ are roots of unity; a…

Factorization conjecture. Alexander polynomials of equivariant slice knots satisfy this factorization condition with $b(g,t)=1$.

Let $K$ be a knot and let $\theta$ be a parameter such that the only $\operatorname{SU}(2)$ holonomy is the trivial one. Let $I_K$ denote the partition-function integral associated…

Let $L$ be a link and let $\mathbf{p}$ be a set of basepoints containing exactly one basepoint in each component of $L$. Write $\mathsf{\Sigma}_2(S^3,L)$ for the double branched co…

Let $L_1\subset T^2\times I$ and $L_2\subset T^2\times I$ be an $m_1\times n$ swatch and an $m_2\times n$ swatch, respectively, where $m_1,m_2,n\in\{1,2,\ldots\}$. Let $H$ be the H…

Let $K$ be a hyperbolic L-space knot, let $b(K)$ denote its bridge number, and suppose that $b(K)=b$. Let $\Delta_K(t)$ denote the Alexander polynomial of $K$. KriMor's conjecture.…

Neuwirth–Burde finiteness conjecture. For any fixed Alexander polynomial, the number of distinct fibered knots having that Alexander polynomial is finite.

Vanishing Alexander polynomial conjecture. There exists some primitive cohomology class $\psi\in H^1(M;{\mathbb Z})$ such that the Alexander polynomial $\Delta_M^\psi(t)$ vanishes.

Let $K\notin \mathcal{D}$ be a knot, and suppose that there is a knot $K'\in \mathcal{D}$ with … Here $\Delta_K(x)$ denotes the Alexander polynomial of $K$, and…

Let $K\subset S^3$ be an oriented knot and let $D\subset S^3$ be an oriented disk intersecting $K$ in two points of opposite sign. A crossing change on $K$ is the operation of perf…

Positive-coefficient conjecture. Every coefficient of $h$ is positive:

Let $K$ be an alternating knot and write its Alexander polynomial as … where $a_i>0$. The refined trapezoidal conjecture. There exists an integer $l\leq n$ such that … and … This c…

Let $K$ be an alternating knot and write its Alexander polynomial as … where $a_i>0$. The stable part is the central consecutive block of equal coefficients, and its length is deno…

Let $K$ be an alternating knot, and write its Alexander polynomial, up to multiplication by a power of $t$ and an overall sign, as … where $a_i>0$. Stoimenow's log-concavity conjec…

Let $K$ be a periodic or freely periodic L-space knot, and let $\Delta_K(t)$ denote its Alexander polynomial. Cyclotomic Alexander polynomial conjecture. $\Delta_K(t)$ is a product…

Let $K$ be an alternating knot with signature $\sigma(K)$, and let $r$ denote the half-length parameter of the central constant part in Fox's trapezoidal coefficient pattern. Hiras…

Let $\hat{Q}_3(m,-m,n,\ell)$ denote the twisted hybrid weaving knot, where $m$ is a natural number and $\ell\in\{-1,0,1\}$. Let $\operatorname{Det}(L)=|\Delta^L(-1)|$ denote the de…

Let $K$ be a knot in $S^3$, let $X_K$ be its exterior, and let $\phi\in H^1(X_K;\mathbb{Z})$ be a generator. A pair $(\rho,\phi)$ is assumed to be admissible, and…