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We show there exists a topologically slice knot $K$ such that the knots ${M^n(K)}_{n=0}^infty$ obtained by iterated satellite operations by the Mazur pattern span an infinite-rank summand of the smooth knot concordance group. This answers a question raised by Feller-Park-Ray.
We show that the crossing number of a satellite knot is at least 10^{-13} times the crossing number of its companion knot.
We consider compact 3-manifolds M having a submersion h to R in which each generic point inverse is a planar surface. The standard height function on a submanifold of the 3-sphere is a motivating example. To (M, h) we associate a connectivity graph G
We define Lie algebras from a hyperbolic knot in the 3-sphere. Since the definitions in terms of group homology are analogous to Goldman Lie algebra, we discuss relations among these Lie algebras.
A Chebyshev knot is a knot which admits a parametrization of the form $ x(t)=T_a(t); y(t)=T_b(t) ; z(t)= T_c(t + phi), $ where $a,b,c$ are pairwise coprime, $T_n(t)$ is the Chebyshev polynomial of degree $n,$ and $phi in RR .$ Chebyshev knots are n
The harmonic knot $H(a,b,c)$ is parametrized as $K(t)= (T_a(t) ,T_b (t), T_c (t))$ where $a$, $b$ and $c$ are pairwise coprime integers and $T_n$ is the degree $n$ Chebyshev polynomial of the first kind. We classify the harmonic knots $H(a,b,c)$ for