30 problems

Let $K \subset S^3$ be an alternating knot. Its Alexander polynomial is … Here a polynomial is log-concave when its coefficient sequence satisfies $a_{i-1}a_{i+1}\leq a_i^2$ for ev…

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 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…

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…

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…

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 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 $C$ 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 $K$ 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…

Let $K$ be a knot. Write $c_3(K)$ for its triple-crossing number and let $\Delta(K)$ denote its Alexander polynomial; write $\operatorname{breadth}(\Delta(K))$ for the breadth of t…

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…

Let $K$ be an alternating knot with normalized Alexander polynomial … where $g$ is the genus of $K$. Fox Trapezoidal Conjecture. The coefficients satisfy … Moreover, if…

Let $p$ and $k$ be relatively prime positive integers. Let $K_{p,k}$ be the lens space knot in the homology sphere $Y_{p,k}$ dual to a simple $(1,1)$-knot in a lens space, with par…

Cha's primary decomposition conjecture. The canonical homomorphisms $\mathcal{T}^p_\Delta\to\mathcal{T}_\Delta$ induce an isomorphism

Primary decomposition conjecture. Let $K$ and $J$ be nonslice knots. If $K$ and $J$ have coprime Alexander polynomials, then $K$ and $J$ are not concordant.

Let $K$ be an alternating knot with signature satisfying $|\sigma(K)|=2k$, and let its Alexander polynomial be … where $a_j>0$. Hirasawa–Murasugi conjecture. The coefficients satis…

Let $C$ be a rational cuspidal curve of degree $d$ with critical points $z_1,\ldots,z_n$. Let $K_1,\ldots,K_n$ be the corresponding links of singular points, with Alexander polynom…

Let … be a real zero of $\Delta_{K[1]}(t)=t^2-3t+1$, and let $C$ be the circle with centre $(\frac{\alpha-1}{2},0)$ and radius $\frac{\alpha+1}{2}$. Lunar-domain conjecture. For ev…

For $n\geq1$, let $\Gamma_{2n}$ be the set of Alexander polynomials of alternating knots of genus $n$, and let $\delta_{2n}$ be the maximum of the maximal real part of a zero among…

Large-parameter stability conjecture. There exists a positive integer $N_m$ such that: if $m$ is even, then $K(r_m)$ is $c$-stable for $k\geq N_m$; and if $m$ is odd, then $K(r_m)$…

Let … be the Alexander polynomial of an alternating knot, and let $\sigma(K)$ be its signature. Signature-controlled coefficient conjecture. If $\sigma(K)=0$, then … More generally…