## Parameterized Knots

### Polynomial Knots

A polynomial knot is the image of a function $$f : \mathbb{R} \rightarrow \mathbb{R}^3$$ with the following properties:

• $$f(t) = \langle \, x(t), \, y(t), \, z(t) \, \rangle$$ and the coordinate functions $$x, y,$$ and $$z$$ are polynomials,
• $$f(u) \neq f(v)$$ for any $$u$$ and $$v$$ with $$0 \leq u \lt v \lt 1$$ (the curve does not intersect itself),
• $$\displaystyle \frac{ df }{ dt } \neq \langle 0, 0, 0 \rangle$$ for any $$t \in [0, 1]$$ (there are nonzero tangent vectors at every point).

Polynomial knots are not knots in the standard sense (as defined in the introduction), as they are not closed. For this reason, polynomial knots are sometimes referred to as "open-ended knots", "long knots", or "non-compact knots" in the literature. A technique called "one-point compactification" may be applied to convert polynomial knots into standard knots. This process can be easily visualized in the case of plane curves using stereographic projection, as we illustrate below; the case of space curves is completely analogous (but more difficult to visualize).

Let $$\mathcal{P}$$ denote the $$xy$$-plane in $$\mathbb{R}^3$$, that is, the plane $$z = 0$$. Let $$\mathcal{S}$$ denote a sphere of radius 1 centered at (0,0,2). (We choose this point for the center of the sphere for ease of visualization; in practice, the sphere is usually centered at (0,0,0) for algebraic simplicity.) Let $$N$$ denote the north pole of the sphere. We define a function $$F : \mathcal{P} \rightarrow \mathcal{S} - \{ N \}$$ as follows: for any point $$A$$ in the plane $$\mathcal{P}$$, the line containing $$A$$ and $$N$$ intersects the sphere $$\mathcal{S}$$ at a point $$B \neq N$$; let $$F(A) = B$$. The function $$F$$ is a bijection, and is formally known as "inverse stereographic projection". This function is illustrated in the applets contained in Figures 5, 6, and 7.

In Figure 5, a line is drawn from the red point in the plane to the north pole of the sphere; the projection of this point onto the sphere is displayed in blue. The red point may be moved around the plane by clicking and dragging with the left mouse button; the projection of this point onto the sphere will be automatically adjusted.

Figure 5. Stereographic Projection of a Point

In Figure 6, a number of red points are drawn in the plane, arranged in a square, with one point in the center of the square. The projections of these points onto the sphere are drawn in blue. The red point in the center of the square may be moved around the plane by clicking and dragging with the left mouse button; the surrounding points and their projections onto the sphere will be automatically adjusted.

Figure 6, Stereographic Projection of a Square

In Figure 7, a set of red points satisfying the $$x$$ and $$y$$ equations of Shastri's trefoil polynomial knot are drawn in the plane, and the projections of these points onto the sphere are drawn in blue. If the mouse pointer is positioned over the applet, an animation will be displayed: the image will rotate in three dimensions, and the line connecting points in the plane to their projections onto the sphere will trace through the points on the planar graph. The rotation of the image can be stopped by clicking once within the applet area. The animation of the line can be stopped and restarted by double-clicking within the applet area. Figure 7 illustrates the following fact: as the distance between a point $$A$$ on the plane and the origin of the plane increases, the distance between the projection of $$A$$ onto the sphere and the north pole of the sphere decreases. As the curve extends infinitely far away from the origin in the plane, the corresponding parts of the projection of the curve approach the north pole of the sphere. The union of the projection of the curve onto the sphere together with the north pole of the sphere is a closed curve; this result is called the "one-point compactification" of the curve.

Figure 7. Stereographic Projection of a Knot

Polynomial knots were introduced by Shastri in [15]. The knot-type of a polynomial knot is defined to be the knot-type of the one-point compactification of the polynomial knot (which is a knot in the standard sense). Using Weierstrass' approximation theorem, Shastri proved that for every knot type, there is a polynomial knot equivalent to it (after the one-point compactification process). As mentioned in the introduction, the minimal degree sequence for a polynomial knot is an invariant of interest. These values also yield bounds of other knot invariants, such as crossing number, bridge number, and superbridge number [6]. There is a concrete algorithm for finding polynomial knots equivalent to any given knot-type, although the degrees of the polynomials produced by this algorithm might not be minimal [12]. However, for polynomial knots that are equivalent to torus knots (defined in the next section), minimal degree sequences have been determined [13].

In the figures that follow in "Gallery I," we include interactive three-dimensional graphics for five different polynomial knots. The polynomial knot equations were obtained from articles by Shastri [15] and Brown [5], with some coefficients scaled and/or slightly altered to produce visually appealing graphs. Polynomial knots are identified by their common name (when it exists) as well as by Alexander-Briggs notation $$C_N$$, where $$C$$ denotes the minimal crossing number in any projection of the knot, and $$N$$ is an index number. A description of the basic applet controls are given in the previous section "Knot Software and LiveGraphics3D."