Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn Fibonacci Knots
139
0
0.0
(
0
)
Authors
Pierre-Vincent Koseleff
Ask ChatGPT about the research
No Arabic abstract
We show that the Conway polynomials of Fibonacci links are Fibonacci polynomials modulo 2. We deduce that, when $ n otequiv 0 Mod 4$ and $(n,j) eq (3,3),$ the Fibonacci knot $ cF_j^{(n)} $ is not a Lissajous knot.
rate research
Read More
A surgery on a knot in 3-sphere is called SU(2)-cyclic if it gives a manifold whose fundamental group has no non-cyclic SU(2) representations. Using holonomy perturbations on the Chern-Simons functional, we prove that the distance of two SU(2)-cyclic surgery coefficients is bounded by the sum of the absolute values of their numerators. This is an analog of Culler-Gordon-Luecke-Shalens cyclic surgery theorem.
In this article, we introduce rack invariants of oriented Legendrian knots in the 3-dimensional Euclidean space endowed with the standard contact structure, which we call Legendrian racks. These invariants form a generalization of the quandle invariants of knots. These rack invariants do not result in a complete invariant, but detect some of the geometric properties such as cusps in a Legendrian knot. In the case of topologically trivial Legendrian knots, we test this family of invariants for its strengths and limitations. We further prove that these invariants form a natural generalization of the quandle invariant, by which we mean that any rack invariant under certain restrictions is equivalent to a Legendrian rack. The axioms of these racks are expressible in first order logic, and were discovered through a series of experiments using an automated theorem prover for first order logic. We also present the results from the experiments on Legendrian unknots involving auto-mated theorem provers, and describe how they led to our current formulation.
For a virtual knot $K$ and an integer $rgeq 0$, the $r$-covering $K^{(r)}$ is defined by using the indices of chords on a Gauss diagram of $K$. In this paper, we prove that for any finite set of virtual knots $J_0,J_2,J_3,dots,J_m$, there is a virtual knot $K$ such that $K^{(r)}=J_r$ $(r=0mbox{ and }2leq rleq m)$, $K^{(1)}=K$, and otherwise $K^{(r)}=J_0$.
Suppose $(M, gamma)$ is a balanced sutured manifold and $K$ is a rationally null-homologous knot in $M$. It is known that the rank of the sutured Floer homology of $Mbackslash N(K)$ is at least twice the rank of the sutured Floer homology of $M$. This paper studies the properties of $K$ when the equality is achieved for instanton homology. As an application, we show that if $Lsubset S^3$ is a fixed link and $K$ is a knot in the complement of $L$, then the instanton link Floer homology of $Lcup K$ achieves the minimum rank if and only if $K$ is the unknot in $S^3backslash L$.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
