We construct an algorithm that lists all closed essential surfaces in the complement of a knot that lies on the fiber of a trefoil or figure eight knot. Such knots are Berge knots and hence admit lens space surgeries. Furthermore they may have arbitrarily large hyperbolic volume. Using this algorithm we concoct large volume Berge knots of two flavors: those whose complement contains arbitrarily...

We give an explicit polynomial of degree 14 in three real variables x, y and z such that the zero set gives the solid trefoil knot. The polynomial depends on two further parameters which enable a deformation from an embedded torus. We use only elementary methods such that the proofs are also accessible to graduate math work groups for pupils in secondary schools. The results can be easily visua...

We study the mechanical response of elastic rods bent into open knots, focusing on the case of trefoil and cinquefoil topologies. The limit of a weak applied tensile force is studied both analytically and experimentally: the Kirchhoff equations with self-contact are solved by means of matched asymptotic expansions; predictions on both the geometrical and mechanical properties of the elastic equ...

A knot diagram is called everywhere n-trivial, if it turns into an unknot diagram by switching any set of n of its crossings. We show several partial cases of the conjecture that the knots with everywhere 1-trivial knot diagrams are exactly the trivial, trefoil and figure eight knots. We discuss a relation to 2-trivadjacent knots and prove some properties of such knots mainly concerning their H...

For any finite link L in R we prove the existence of a high-energy complex-valued eigenfunction of the hydrogen atom such that its nodal set contains a union of connected components diffeomorphic to L. This problem goes back to Berry, who constructed such eigenfunctions in the case where L is the trefoil knot or the Hopf link and asked the question about the general result.