Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userEmbedding spheres in knot traces
93
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The trace of $n$-framed surgery on a knot in $S^3$ is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded 2-sphere whose complement has abelian fundamental group. Our characterisation is in terms of classical and computable 3-dimensional knot invariants. For each $n$, this provides conditions that imply a knot is topologically $n$-shake slice, directly analogous to the result of Freedman and Quinn that a knot with trivial Alexander polynomial is topologically slice.
rate research
Read More
This paper is a very brief introduction to knot theory. It describes knot coloring by quandles, the fundamental group of a knot complement, and handle-decompositions of knot complements.
This book is an introduction to hyperbolic geometry in dimension three, and its applications to knot theory and to geometric problems arising in knot theory. It has three parts. The first part covers basic tools in hyperbolic geometry and geometric structures on 3-manifolds. The second part focuses on families of knots and links that have been amenable to study via hyperbolic geometry, particularly twist knots, 2-bridge knots, and alternating knots. It also develops geometric techniques used to study these families, such as angle structures and normal surfaces. The third part gives more detail on three important knot invariants that come directly from hyperbolic geometry, namely volume, canonical polyhedra, and the A-polynomial.
The aim of this survey article is to highlight several notoriously intractable problems about knots and links, as well as to provide a brief discussion of what is known about them.
A Chebyshev curve C(a,b,c,phi) has a parametrization of the form x(t)=Ta(t); y(t)=T_b(t) ; z(t)= Tc(t + phi), where a,b,c are integers, Tn(t) is the Chebyshev polynomial of degree n and phi in RR. When C(a,b,c,phi) has no double points, it defines a polynomial knot. We determine all possible knots when a, b and c are given.
This survey article discusses three aspects of knot colorings. Fox colorings are assignments of labels to arcs, Dehn colorings are assignments of labels to regions, and Alexander-Briggs colorings assign labels to vertices. The labels are found among the integers modulo n. The choice of n depends upon the knot. Each type of coloring rules has an associated rule that must hold at each crossing. For the Alexander Briggs colorings, the rules hold around regions. The relationships among the colorings is explained.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
