Narrow Results

For a polygonal knot K, it is shown that a tube of radius R(K), the polygonal thickness radius, is an embedded torus. Given a thick configuration K, perturbations of size r < R(K) define satellite structures, or local knotting. We explore knotting within these tubes both theoretically and numerically. We provide bounds on perturbation radii for which one can obtain small trefoil and figure-eight summands and use Monte Carlo simulations to estimate the relative probabilities of these structures as a function of the number of edges.

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 cubic lattice stick index of a knot type is the least number of sticks glued end-to-end that are necessary to construct the knot type in the 3-dimensional cubic lattice. We present the cubic lattice stick index of various knots and links, including all (p, p + 1)-torus knots, and show how composing and taking satellites can be used to obtain the cubic lattice stick index for a relatively large infinite class of knots. Additionally, we present several bounds relating cubic lattice stick index to other known invariants.

In this paper, I present a new family of knots in the solid torus called lassos, and their properties. Given a knot $K$ with Alexander polynomial ${\mathrm{\Delta }}_K(t)$, I then use these lassos as patterns to construct families of satellite knots that have Alexander polynomial ${\mathrm{\Delta }}_K(t^d)$ where $d\in \mathbb{N}\cup \{0\}$. In particular, I prove that if $d\in \{0,1,2,3\}$ these satellite knots have different Jones polynomials.

We study the structure of the augmented fundamental quandle of a knot whose complement contains an incompressible torus. We obtain the relationship between the fundamental quandle of a satellite knot and the fundamental quandles/groups of its companion and pattern knots. General presentations of the fundamental quandles of a link in a solid torus, a link in a lens space and a satellite knot are described. In the last part of this paper, an algebraic approach to the study of affine quandles is presented and some known results about the Alexander module and quandle colorings are obtained.

In this paper, we conjecture a connection between the $A$-polynomial of a knot in ${𝕊}^3$ and the hyperbolic volume of its exterior ${{ℳ}}_K$: the knots with zero hyperbolic volume are exactly the knots with an $A$-polynomial where every irreducible factor is the sum of two monomials in $L$ and $M$. Herein, we show the forward implication and examine cases that suggest the converse may also be true. Since the $A$-polynomial of hyperbolic knots are known to have at least one irreducible factor which is not the sum of two monomials in $L$ and $M$, this paper considers satellite knots which are graph knots and some with positive hyperbolic volume.

A Gaussian random walk is a random walk in which each step is a vector whose coordinates are Gaussian random variables. In 3-space, if a Gaussian random walk of n steps begins and ends at the origin, then we can join successive points by straight line segments to get a knot. It is known that if n is large, then the knot is non-trivial with high probability. We give a new proof of this fact. Our proof shows in addition that with high probability the knot is contained as an essential loop in a fat, knotted, solid torus. Therefore the knot is a satellite knot and cannot be unknotted by any small perturbation.