Trunk of Satellite and Companion Knots

TL;DR

This paper investigates the relationship between the trunk invariant of satellite knots and their companion knots, establishing bounds involving winding and wrapping numbers, and extends these results to more general cases.

Contribution

It provides new inequalities relating the trunk of satellite knots to that of their companions, incorporating winding and wrapping numbers, and discusses potential generalizations.

Findings

Established that trunk(K) ≥ n · trunk(J) for satellite knot K with companion J and winding number n.

Showed that trunk(K) > (1/2) m · trunk(J) when considering wrapping number m.

Discussed possible extensions of the main inequalities to broader contexts.

Abstract

We study the knot invariant called trunk, as defined by Ozawa, and the relation of the trunk of a satellite knot with the trunk of its companion knot. Our first result is ${\rm trunk}(K) \geq n \cdot {\rm trunk}(J)$ where ${\rm trunk}(\cdot)$ denotes the trunk of a knot, $K$ is a satellite knot with companion $J$, and $n$ is the winding number of $K$. To upgrade winding number to wrapping number, which we denote by $m$, we must include an extra factor of $\frac{1}{2}$ in our second result ${\rm trunk}(K) > \frac{1}{2} m\cdot {\rm trunk}(J)$ since $m \geq n$. We also discuss generalizations of the second result.

Equations

$$k:S^1\hookrightarrow S^3$$

$$h\circ k:S^1\to \mathbb{R}.$$

$$c_1<r_1<c_2<r_2<\dots <c_{s-1}<r_{s-1}<c_s.$$

$$\omega (k)=\sum _{i=1}^{s-1}{w}_i.$$

Full text

1 Preliminaries

We will start with some necessary definitions.

Definition 1.1**.**

A knot is a smooth embedding

[TABLE]

where $S^{1}$ is the unit circle in $\mathbb{R}^{2}$ and $S^{3}$ is the unit sphere in $\mathbb{R}^{4}$.

Definition 1.2**.**

A knot class is a set of knots that are all isotopic to each other. See [rolfsen1976knots] for the definition of an isotopy.

We shall fix a Morse function throughout the paper. We consider $h:S^{3}\rightarrow\mathbb{R}$ to be the standard height function $h(x,y,z,w)=w$ restricted to the unit sphere $S^{3}\subset\mathbb{R}^{4}$. The pre-images of $\pm 1$ are denoted by $\pm\infty$.

Definition 1.3**.**

With the above notation, a knot $k$ is called Morse if the following composition is a Morse function:

[TABLE]

Convention 1.4**.**

The following conventions will be used throughout the paper:

(1) We will only consider knots that are Morse and whose critical points are all at different levels.

(2) Knots are denoted by a lowercase letter like $k$, while knot classes are generally denoted by a capital letter like $K$.

(3) By a knot we can either mean the embedding $S^{1}\rightarrow S^{3}$ or the image of the embedding. We do not distinguish between them.

Notation 1.5**.**

Let $k$ be a knot in $S^{3}$. Denote the critical levels of $k$ by $c_{i}$, and pick regular levels $r_{i}$ between two consecutive critical levels $c_{i}$ and $c_{i+1}$, so that:

[TABLE]

For each regular level $r_{i}$, we define $w_{i}=|h^{-1}(r_{i})\cap k|$, that is, $w_{i}$ is the number of intersections of this regular level with $k$.

1.1 Trunk and Width of Knots

Now we will define two invariants, trunk and width, for knots and knot classes.

Definition 1.6**.**

• •

The trunk number of a knot $k$ is given by $\operatorname{trunk}(k)=\displaystyle{\max_{1\leq i\leq s-1}w_{i}(k)}$.

• •

The width number of a knot $k$ is given by

[TABLE]

Definition 1.7**.**

We can extend the definitions of the trunk and width numbers of a knot to also apply to knot classes:

• •

The trunk number of a knot class $K$ is given by $\operatorname{trunk}(K)=\displaystyle{\min_{k\in K}\operatorname{trunk}(k)}$.

• •

The width number of a knot class $K$ is given by $\omega(K)=\displaystyle{\min_{k\in K}\omega(k)}$.

Example 1.8**.**

Suppose $K$ is the trefoil knot. Let $k$ be the particular embedding as in Figure 1. Its width and trunk happen to be those of the trefoil knot class.

Definition 1.9**.**

Let $T^{2}=S^{1}\times S^{1}$ be a two dimensional torus. A curve $\alpha\subseteq T^{2}$ is inessential if $\alpha=\partial D$ for some $D\subseteq T^{2}$ and is essential otherwise.

1.2 Satellite and Companion Knots

Here, we will define the process of forming a satellite knot, which is one of the main ways to construct complicated knots from simple ones. We also define two invariants pertaining to satellite knots inside a solid torus.

Definition 1.10**.**

Let $\hat{V}=S^{1}\times D^{2},$ and $\hat{j}=S^{1}\times 0.$ A meridian disk of $\hat{V}$ is a properly embedded disk $D$ whose boundary $\partial{D}\subset\partial{\hat{V}}$ is essential on $\partial{\hat{V}}$.

Definition 1.11**.**

Let $\hat{k}$ be a knot inside $\hat{V}$ such that $\hat{k}$ intersects any meridian disk in $\hat{V}$. Let $f$ be a smooth embedding from $\hat{V}$ to $S^{3}$ and let $f(\hat{j})=j$ and $f(\hat{k})=k$. The knot $k$ is a satellite knot with companion $j$.

Definition 1.12**.**

The winding number $n$ is the absolute value of the sum of all algebraic intersections of any fixed meridian disk with $\hat{k}$. The wrapping number $m$ is the minimal geometric intersection number of $k$ with any meridian disk.