On the Question of Genericity of Hyperbolic Knots

A well-known conjecture in knot theory says that the proportion of hyperbolic knots among all of the prime knots of $n$ or fewer crossings approaches $1$ as $n$ approaches infinity. In this article, it is proved that this conjecture contradicts several other plausible conjectures, including the 120-year-old conjecture on additivity of the crossing number of knots under connected sum and the conjecture that the crossing number of a satellite knot is not less than that of its companion.

NONSURJECTIVE SATELLITE OPERATORS AND PIECEWISE-LINEAR CONCORDANCE

We exhibit a knot $P$ in the solid torus, representing a generator of first homology, such that for any knot $K$ in the 3-sphere, the satellite knot with pattern $P$ and companion $K$ is not smoothly slice in any homology 4-ball. As a consequence, we obtain a knot in a homology 3-sphere that does not bound a piecewise-linear disk in any homology 4-ball.

Knots that admit infinitely many closed essential surfaces

Let K be a knot in S3 that admits two disjoint, separating, essential surfaces with boundary, where one of them has an accidental annulus. Then, K admits infinitely many closed essential surfaces unless the annulus is "centered". This implies for example that any non-satellite knot that admits a 4-puntured essential sphere and any alternating knot that admits a closed essential surface of genus 2 admit infinitely many closed essential surfaces.

TWISTED TORUS KNOTS WITH ESSENTIAL TORI IN THEIR COMPLEMENTS

A twisted torus knot is a torus knot with a number of full-twists on some adjacent strands. In this paper, we show that if a twisted torus knot is a satellite knot then the number of full-twists is generically at most two.

EXTENDING VAN COTT'S BOUNDS FOR THE τ AND s-INVARIANTS OF A SATELLITE KNOT

This paper generalizes Van Cott's bounds on the Heegaard–Floer τ-invariant and Rasmussen's s-invariant of cabled knots to apply to all satellite knots.

THE PROBLEM OF DETECTING THE SATELLITE STRUCTURE OF A LINK BY MONOTONIC SIMPLIFICATION

In a recent work "Arc-presentation of links: Monotonic simplification", Dynnikov shows that each rectangular diagram of the unknot, composite link, or split link can be monotonically simplified into a trivial, composite, or split diagram, respectively. The following natural question arises: Is it always possible to simplify monotonically a rectangular diagram of a satellite knot or link into one where the satellite structure is seen? Here we give a negative answer to that question both for knot and link cases. An example of a torus embedding that cannot be obtained from ordinary "thin" torus by methods of paper [2] is also constructed.

DEHN SURGERIES ON 2-BRIDGE LINKS WHICH YIELD REDUCIBLE 3-MANIFOLDS

We completely determine which Dehn surgeries on 2-bridge links yield reducible 3-manifolds. Further, we consider which surgery on one component of a 2-bridge link yields a torus knot, a cable knot and a satellite knot in this paper.

KNOTS, SATELLITE OPERATIONS AND INVARIANTS OF FINITE ORDER

Let sQ be the satellite operation on knots defined by a pattern (V, Q), where V is a standard solid torus in S3 and Q ⊂ V is a knot that is geometrically essential in V. It is known (Kuperberg [5]) that if v is any knot invariant of order n ≥ 0, then v ◦ sQ is also a knot invariant of order ≤ n. We show that if the knot Q has the winding number zero in V, then the satellite map [Formula: see text] passes n-equivalent knots into (n + 1)-equivalent ones. Kalfagianni [4] has defined for each nonnegative integer n surgery n-trivial knots and studied their properties. It is known that for each n every surgery n-trivial knot is n-trivial. We show that for each n there are n-trivial knots which do not admit a non-unitary n-trivializer that show they to be surgery n-trivial. Przytycki showed [12] that if a knot Q is trivial in S3 and is embedded in V in such a way that it is k-trivial inside V and if a knot [Formula: see text] is m-trivial, then the satellite knot [Formula: see text] is (k + m + 1)-trivial. We establish a version of aforementioned Przytycki's result for surgery n-triviality, refining thus a construction for surgery n-trivial knots suggested by Kalfagianni.