10 problems

Let $K(p,q,p-kq,1,1)$ be the knots of the second type in Theorem, with the parameters satisfying the hypotheses of that theorem. Hyperbolicity conjecture. All of the knots…

Twisted torus knot fiberedness conjecture. A twisted torus knot is fibered if and only if its Alexander polynomial is monic.

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 $K(p,q,r,n)$ be a twisted torus knot, and suppose that the parameters $n$ and $r$ determine whether it is hyper-, end-, or middle-Seifert-fibered with respect to a particular h…

Let $p$ and $q$ be relatively prime positive integers, let $r$ be an integer, and let $K(p,q;2,r)$ denote the twisted torus knot obtained by twisting $r$ full twists on two adjacen…

Essential-torus and compositeness conjecture. A twisted torus knot $T(p,q;r,s)$ has an essential torus in its exterior if and only if it is a knot in Theorem 1.1 or $r=qk$ for some…

A twisted torus knot $T(p,q;r,s)$ is obtained by twisting a torus knot along strands, with the parameters $p,q,r,s$ specifying the construction. Theorem 1 gives a sufficient condit…

Let $K_m=T(7,17,2,10m-4)$, and let $(V,W)$ be the unique minimal genus-two Heegaard splitting of its exterior $E(K_m)=S^3-N(K_m)$. Let $(V',W')$ be the boundary stabilization of…

Let $K=T(p,q,2,r)$ be a twisted torus knot, let $E(K)=S^3-N(K)$ be its exterior, and suppose that $K$ is not $5cmu$-primitive. The unique genus-two splitting conjecture. The knot e…

Let $E(K)=S^3-N(K)$ be an exterior of one of the twisted torus knots considered in the paper. The genus-three splitting conjecture. These manifolds possess irreducible yet weakly r…