9 problems

An algebraic knot is a knot arising from an algebraic plane curve singularity; these knots are, in particular, iterated cables of torus knots. The smooth knot concordance group is…

Cable stabilization conjecture. For a given normalized Alexander degree $\widetilde{\mathbf{s}}$, the dimension of…

Let $L_m$ be the $m$-fold cable link of a link $L$, and let $\overline{\mathbf{s}}$ be a normalized Alexander degree. The maps … are the connecting maps in the directed system defi…

Let $K$ and $J$ be distinct iterated cables of tight fibered knots, and let $p>1$ and $q_1,q_2$ be distinct integers relatively prime to $p$. Define … where $K_{p,q}$ denotes the…

Cable expansion conjecture. The $V_n$-polynomial of a knot is a linear combination of the $V_1$-polynomial of the knot and its $(n',1)$-cables for $n' \leq n$, with coefficients in…

Let $K_r=C_{(r,2)}(\bm{4_1})\subseteq S^3$ be the family of cable knots from the preceding conjecture, with $r>8$ odd and monic Alexander polynomial. Let $\{f_m(q)\}$ be the coeffi…

Let $\bm{4_1}$ denote the figure-eight knot, and consider the cable knots … where $r>8$ is odd and the Alexander polynomial is monic. Write the invariant as … Let $\{f_m(q)\}$ be p…

Let $T(p,q;r,s)$ denote a twisted torus knot, with $r\neq 1$ and $r$ not a multiple of $q$. A twisted torus knot is satellite if its exterior contains an essential torus, and a cab…

Let $4ell4$ be an integer, let $K$ be a knot, and let $P_\ell(K)$ denote the $(2,2\ell)$ cable of $K$. Let $P_\ell(\mathcal O)$ be the corresponding cable of the unknot, possibly w…