Satanic Cocircularities of Knots

G. Flowers

University of Victoria — Victoria, British Columbia

Introduction

The problem of classifying and distinguishing various knots lies at the heart of knottheory. Even determining if a given knot is non-trivial (i.e. isotopic to the standard

embedding of the circle in the 3-sphere S3) is known to be an NP-hard problem[1].Thus, we rely on knot invariants to aid us in deciding if two knots are equivalent

(isotopic). A numerical knot invariant assigns to each knot a number in such amanner so that two equivalent knots are assigned the same number.

In the 1990s, Victor Vassiliev developed a series of numerical knot invariants, now

known as Vassiliev invariants. These invariants are quite powerful in their abilityto distinguish knots, and have generated a great deal of interest in the community

of knot theorists. The computation of the invariants is largely algebraic, and untilrecently, the resulting numerical values were not known to describe any geometric

properties of the knots. By examining occurances of five points on a knot that lie ona common circle, we have established a geometric interpretation of the second Vas-

siliev invariant v2. Our knots will be interchangeably referred to as an embedding

f : S1 → S3, or as the image K of such an embedding.

Configuration Spaces

Consider the space consisting of n distinct points on a manifold. Such a space is

known as a configuration space. For our purposes, we consider only the configurationspaces of S1 and S3:

Cn(S3) := {(x1, ..., xn) ∈ (S3)n : xi 6= xj (for all i 6= j)}

Cn(S1) := {(t1, ..., tn) ∈ (S1)n : ti 6= tj, and t1, ..., tn are in counterclockwise order}

Note the additional requirement that the coordinates of a point in Cn(S1) must be incounterclockwise order.

Figure 1: The point (s1, ..., s5) is not an element of C5(S1): the coordinates are not

in counterclockwise order. The point (t1, ..., t5) is an element of C5(S1).

Given a knot f : S1 → S3, we extend f to the evaluation map Cn( f ) : Cn(S1) →Cn(S3) that sends (t1, ..., tn) to ( f (t1), ..., f (tn)). The image of this map is the spaceof n distinct points on the knot K, appearing in the order prescribed by the map f ,

and is denoted Cn(K).

Figure 2: The point(t1, ..., t5) maps into C5(K) via the evaluation map C5( f ).

Satanic Circles

A point (x1, ..., x5) ∈ C5(S3) is considered satanic if the following two conditions are

met:

1. x1, ..., x5 lie on a common circle in S3.

2. The points are arranged in a ’star-shaped’ pattern. That is, xi is not adjacent to

xi+1 on their common circle, with subscripts working modulo 5.

Denote the space of all satanic points in C5(S3) by S. This space is a submanifold

of dimension 11 in the manifold C5(S3). The space C5(S

3) is 15-dimensional, andC5(K) is five-dimensional. Therefore, we would expect the intersection of C5(K)with S in C5(S

3) to be a 11 + 5 − 15 = 1-dimensional manifold. Indeed, this is thecase for general knots, and in fact, the 1-manifold is a collection of closed curves in

C5(S3).

Let S|K = S ∩ C5(K). So S|K consists of points (x1, ..., x5) ∈ C5(S3) such that each

xi lies on the knot K (in order), and (x1, ..., x5) is satanic.

Figure 3: The point (t1, ..., t5) ∈ C5(S1) maps to a satanic point ( f (t1), ..., f (t5)) in

S|K. The circle containing f (t1), ..., f (t5) may appear slightly distorted, but this is

merely a result of projecting the knot down onto the plane.

Figure 4: A three dimensional rendering of the same knot. There are three satanic

’circles’ intersecting the darker dot on the knot.

It can be shown that the preimage of S|K under the evaluation map C5( f ) is also

a collection of closed curves in C5(S1). By an abuse of notation, we will simply let

f−1(S) denote this preimage: C5( f )−1(S|K).

Bibliography

[1]: Masao Hara, Seiichi Tani and Makoto Yamamoto. Unknotting is in AM ∩ coAM.Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, 2005.

Winding Numbers

Consider the curves f−1(S) for the knot f that has been used in our previous exam-ples (this knot is refered to as the 51 knot). We may project C5(S

1) down onto C2(S1)

by simply ’forgetting’ the last three coordinates. The space C2(S1) is diffeomorphic

to S1 × (0, 1). Below is a projection of the curve f−1(S) ⊂ C5(S1) onto C2(S

1):

Figure 5: A projection of f−1(S) into two-dimensional space for the 51 knot. Theamount of red, green and blue on the strand encodes the positions of the ’forgotten’

third, fourth and fifth coordinates on S1 (respectively). The black dot in the centeris the origin.

While the curve f−1(S) does not contain any self-intersections, the projection may

have instances of self-intersection (like in the figure above). Regardless, notice how

this curve ’wraps’ around the origin three times.

For a general knot f , a similar phenomenon takes place: the curve f−1(S), whenprojected, winds around the origin a particular number of times. The direction and

amount of this winding is an invariant of the knot. If two knots f1, f2 are equivalent,

the curves f−11 (S) and f−1

2 (S) may appear as radically different curves in C5(S1);

however, when projected, they wind around the origin the same number of times inthe same direction.

The direction of the winding is considered positive if it is counterclockwise, and

negative otherwise. This ’winding number’ is exactly the second Vassiliev invariant

v2. For the 51 knot, the second Vassiliev invariant is equal to 3, and the curve in theabove figure winds around the origin three times (in the counterclockwise direction,

although this is not apparent from the figure).

As another example, we examine a knot familiar to climbers and sailors: the grannyknot.

Figure 6: On the left is a depiction of the granny knot, with two satanic circles. On

the right is the projection of the satanic points described at the start of this section.The gaps in the strands are the result of techniques used to render the curve. In

reality, the gaps are nonexistent. The v2 invariant of the granny knot is 2.