Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAn infinite-rank summand from iterated Mazur pattern satellite knots
85
0
0.0
(
0
)
Authors
Wenzhao Chen
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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. For M in the 3-sphere, G is a tree if and only if there is a Fox reimbedding of M which carries horizontal circles to a complete collection of complementary meridian circles. On the other hand, if the connectivity graph of the complement of M is a tree, then there is a level-preserving reimbedding of M so that its complement is a connected sum of handlebodies. Corollary: The width of a satellite knot is no less than the width of its pattern knot. In particular, the width of K_1 # K_2 is no less than the maximum of the widths of K_1 and K_2.
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 non compact analogues of the classical Lissajous knots. We show that there are infinitely many Chebyshev knots with $phi = 0.$ We also show that every knot is a Chebyshev knot.
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 $ a le 4. $ We study the knots $H (2n-1, 2n, 2n+1),$ the knots $H(5,n,n+1),$ and give a table of the simplest harmonic knots.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
