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Knots and Links in Auras

John Starrett

New Mexico Institute of Mining and Technology

Department of Mathematics

When the local vector elds associated to a parameterized (oriented) knot or link

are globalized, the global vector eld can generate sets of stable auxiliary knots and

links. One particular class of knots and links, the torus knots, generates innite sets

of knots and links that can be completely characterized.

In recent years tools from knot theory have become important to dynamicists, especially

the template construction of Sullivan. By nding templates associated with a strange attrac-

tor, one can determine which knot and link types are represented by the unstable periodic

orbit set that forms the skeleton of the attractor. This analogy suggests a question: what

is the minimum set of periodic orbits (skeleton of knots) necessary to generate a strange

attractor?

The results below are unintended consequences of research on this question. Our attempts

to generate custom built strange attractors, designed to have certain built in sets of knots

and links among their solutions, led to the generation of interesting sets of auxiliary knots

and links. We start with a minimal set of simple parameterized knots and build differential

equations having those knots (approximately) as a solution by globalizing a local vector

eld. This process generate sets, sometimes innite sets, of additional knots and links which

we call the aura of the master knot or link.

The method is simple: we globalize a local vector eld by tting a set of basis functions

to the vector components of the tangent vector to a parameterized knot. In order to gain an

understanding of how the vector elds changed with the knot type and parameterization of

the periodic basis orbits, we tested many parameterized knots that were much simpler than

the knotted periodic orbits generated by a chaotic system.

Begin with a parameterized knot K or link L = j K j , which we call the master knot

or link . By tting global functions to the local vector eld (the tangent vectors to the

master knot or link), we construct a global vector eld, and thus a differential equation,

that generates the original knot (approximately) and usually additional knots or links as


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periodic solutions. These additional knots K and links L we call auxiliary knots and links,

and the link consisting of all the auxiliary knots we call the aura A of the master link. The

master link is always a particular parameterized knot or link, and its aura is always relative

to the particular basis functions and tting procedure.In contrast to the unstable periodic orbits of chaotic systems, the knots we investigate

are stable or Lyapunov stable [1]. While it is possible that there are additional unstable

periodic orbits for some of these systems, we did not attempt to nd them. For almost every

knot and link we tested, innite sets of auxiliary knots could be generated (and for some

parameterizations of unknots and simple links, strange attractors, which is the subject of

another paper).

I. ROLE OF THE BASIS FUNCTIONS

Given a particular parameterized knot K = f 1(t), f 2(t), f 3(t) , we would like to build a

global vector eld F 1(x,y,z ), F 2(x,y,z ), F 3(x,y,z ) , with F k (x,y,z ) = df kdt |(f 1 ,f 2 ,f 3 ) along K .We extend the local vector eld on K to the entire embedding space by expressing each F k

as the sum of basis functions i (x,y,z ), which may be orthogonal or orthonormal, chosen

from some complete set

{k (x,y,z )

}so that

F 1(x,y,z ) F 1(x,y,z ) =ni=1 a1i i (x,y,z )



.

The coefficients aki are chosen to minimize the error over each tangent vector component

x = m j =1df xdt |(x j ,y j ,z j )

ni=1 ax i i (x j , y j , z j )

y = m j =1 df y

dt |(x j ,y j ,z j ) ni=1 ay i i (x j , y j , z j )z = m j =1

df zdt |(x j ,y j ,z j )

ni=1 ax i i (x j , y j , z j )

.

over all m data points. We expect that in the limit as n, m we can approximate themaster link arbitrarily well, so that the global vector eld F restricted to K is identical withdKdt . Thus, for each complete set of basis functions { j }k , K or L has a unique limiting aura

A k . Of course, different parameterizations of the same topological link can have topologi-


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cally distinct aurae, as in the case of a link, e.g. reversing the orientation of a single link

component.

We used several different sets of basis functions, including polynomials [ xyz]k where [xyz]k

indicates the usual set of products of powers of x,y,z of maximum degree k, and radial basis

functions of the form e (( x x i )2 +( y y i )2 +( z z i )2 ) and ( +( xxi )2 +( yyi )2 +( zzi )2)

1. The

radial basis functions {i}are local functions anchored at {(x i , yi , zi )}, where the anchorpoints are usually data points, but can be located anywhere in P 3. While polynomial basis

functions have the disadvantage of increasing divergences away from the master link with

the addition of higher order terms, they can be computed quickly and simplied models can

be easily extracted by inspection. Radial basis functions that go rapidly to zero away from

their centers do not have the divergence problems of polynomials, but their computation

is slower, and simple lower order models cannot be easily extracted. We set the total

number of sample points to 60 , 120 or 240, which could be distributed evenly, or unevenly

between the components of the links if we wish to emphasize the contribution of one or more

components over the others.

The nature of the vector elds generated is highly dependent on the type and number of

basis functions used, the density and distribution of the data points, as well as the relation

between the parameterization and orientation of the links, and the relation between the

parameterization and the coordinate system in which the basis is written. A single master

link can generate an innite number of different vector elds and thus, potentially, an innite

number of different sets of auxiliary knots and links when the number of basis functions and

data points is nite.

As an example, we can parameterize a circular unknot of radius 3 in the xy plane by

< 3sin t, 3cost, 0 > and t polynomial basis functions {1,x ,y,z }to obtain the differentialequations x = y, y =

x, z = 0, whose solution set is the set of circles of all radii, centered

x = y = 0 in the planes z = c for < c < . These knots are neutrally stable. When thesame knot is rotated out of the plane by / 4 about the y axis, and a global vector eld

is t to the basis functions {1,x ,y,z }then we get a set of attracting roughly circular cycleswith a small basin of attraction. If we build a vector eld from this same parameterized

circle using basis functions {1,x ,y,z,x 2, y2, z2,xy,xz,yz }, then we get a single attractingunknot with a small basin of attraction, and if the tilt about the y axis is / 5 rather than

/ 4 then there is an attracting xed point at the center of the knot. Doubling the number


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of data points in the same knot destabilizes the xed point. Some of these are shown in

gure (1). Obviously, there are no simple general criteria to indicate when a particular

parameterization will produce a particular aura, at least in this example.

FIG. 1: On the left, the aura of an xy planar circle from a degree one polynomial t using 120

data points. Second from the left, view down the z axis of the ow from the same master knot twith a second degree polynomial the aura is conned to a region near the z axis, and the restof the space is unstable. In the middle, the same as the second from the left except using 60 datapoints. Second from the right, a portion of the ow associated with the circle rotated about the yaxis by / 4 using 120 data points there is an attracting xed point surrounded by an attractinglimit cycle near the master knot, divided by an unstable limit cycle. On the right the same thingbut with 60 data points. Now we have a set of neutrally stable knots near the master in a smallbasin of stability.

II. ORGANIZATION OF THE PAPER

We begin with a recipe for nding the aura of a knot, then describe a few simple cases.

We then analyze a special class of knots that is especially stable, in the sense that changes

in the number of data points and number of basis functions used does not have too great an

effect on the type of aura produced. We show that, for a large number of different knots in

the class, and for a large range of parameters, the auras produced are very similar. We end

the paper with a short tour through a bestiary of aurae. This is experimental mathematicsguided largely by artistic considerations. Although there are important applications to the

general technique of tting vector elds and differential equations to data [ ? ] [? ] [? ]

[? ], we know of nopractical applications for the technique applied to parameterized knots,

other than the creation of interesting pictures to lure students from other departments.


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FIG. 2: A trefoil and its local vector eld on the left, and the same trefoil with its global vectoreld on the right.

(1, 0), (0, 1), (1, 1) and (1, 2) torus knot while generating auxiliary knots. The ( n, 1), n > 1

torus knot, while unknotted, did generate auxiliary knots, while giving a good approximation

to the master knot as a solution using polynomial basis functions. There is a qualitative

change in the character of the aura with increasing n for (n, 1) knots, which can be seen in

the gure (7). For polynomial ts of degrees two to ve, the aura of the (2 , 1), (3, 1) and

(4, 1) knots consists of knots on tori concentric with the master, while for (5 , 1) and higher,

the tori on which the knots live are concentric with the unknot at the center of the torus on

which the master resides.

FIG. 3: On the left, the 2 , 1 torus knot and its approximation by the ow of a polynomial vectoreld of maximum degree 2. In the middle, the 2 , 1 torus knot and several of its auxiliary knots.The master knot is mostly hidden by the innermost knot. On the right, the (5 , 1) knot, like all(n, 1) knots, n 5, has a qualitatively different aura.


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FIG. 4: On the left, the 1 , 3 torus knot and several of its auxiliary knots, in the middle the 1 , 4torus knot, on the right the 1 , 7 torus knot and several of its auxiliary knots with the same initialconditions.

V. KNOTS GENERATED BY HIGHER ORDER TORUS KNOTS

The rst nontrivial knot that usually comes to mind is the trefoil. We nd that the 3 , 2

trefoil knot follows the general pattern for the n, 1 knots of the previous section, in that it

generates auxiliary knots that live on nested tori inside the torus of the master knot.

A least squares t to the x,y,z components of the tangent vector to the trefoil

x = cos( 23 t)(3 + cos( t))

y = sin( 23 t)(3 + cos( t))z = sin( t)

using polynomials in x,y,z of maximum degree 2 and 3 can be found in the appendix.

When we truncate by throwing out terms with small coefficients we nd (approximately)

this simple set of differential equations for both

x = 4y 2xzy = 4 x 2yzz = (x2 + y2 + z2) 10

. (1)

When these equations are numerically integrated the solutions appear to reside on a set of


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nested tori, see gure (5). More generally, m, n torus knots generate equations of the form

x = ay bxzy = ax

byz

z = c(x2 + y2 + z2) d, (2)

where the relations of the coefficients depend on K . Numerical experiments indicate that

there is a torroidal region inside of which orbits on a set of nested tori are stable in the sense

that a small perturbation of the orbit places it on a torus nearby.

FIG. 5: Some nested torus knots and the z isosphere on which orbits must exit and enter parallelto the xy plane.

We can justify some of the experimental evidence by doing a qualitative analysis of

the vector eld. First, there is a constant rotation about the z axis. From (1) we can

see that there are two xed points, a complex saddle with repelling direction along the z

axis at (0 , 0,

d/c ) and a complex saddle with an attracting direction along the z axis at

(0, 0, d/c ). Also, we have z = c(x2 + y2 + z2) d, so the z component of the vector eldis 0 on a sphere of radius d/c (the z isosurface , as well as the x and y isosurfaces can beseen in gure (V).

When z = 0, the x and y components of the vector eld are x = ay, y = ax , so thereis an unknot K = d/c cos t, d/c sin t, 0 in the xy plane. Inside the sphere, the vectoreld has a negative z component, and outside a positive z component. The ux of the vector

eld is everywhere inward on the upper half of the sphere and outward on the lower half,


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FIG. 6: Isosurfaces indicate the sets on which the x,y,z components of the vector eld vanish,colored red, blue and green respectively. The rst four are isosurfaces for (2 , 1), (3, 1), (4, 1) and(5, 1) torus knots t with a second degree polynomial, while the next three are a (7 , 4) torus knot

t with second, third and fourth degree polynomials. The nal gure is a truncated model of theform of (1) for a (2, 1) torus knot.


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so orbits beginning inside the sphere will move downward, exiting through the lower half of

the sphere parallel to the xy plane, then move upward. Orbits near enough the outside of

the sphere in the upper half space will enter the sphere parallel to the xy plane, then move

downward. Because of the symmetry of the vector eld, and the upward ow outside the

sphere, orbits leaving the bottom half of the sphere must reenter the top half of the sphere,

see gure (7).

FIG. 7: On the left, a generic cross section (a slice by a plane containing the z axis) of the vectoreld (light and dashed arrows) and the z isosphere (dark). On the right, a side view of somesolutions to the differential equation (1)

In order to determine more about the nature of the sphere piercing orbits, we will nd

a solution to a nearby dynamical system that is identical to equation (2) on the unknot

d/c cos t, d/c sin t, 0 . We write the DE in toral coordinates

x = ( R + r cos )cos

y = ( R + r cos )sin

z = r sin

where R is the major radius, r the minor radius, the azimuthal angle and the meridional

angle. Solving to get differential equations in the new coordinates, we nd

r = sin( )(c(R2 + r 2) d + rR (2c b)cos() br2 cos()2) = Rb + (( R2c d)/r + r (c + b))cos( ) + R(2c b)cos()2 br cos()3 = a

. (3)


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We see that there is a constant rotation around the z axis, and that the minor radius and

meridional angle have no dependence on the azimuthal angle. Orbits that pierce the sphere

must wrap around the unknot x2 + y2 = d/c, z = 0, and their distance from the unknot isindependent of the azimuthal angle.

For small r , r is approximately constant, and with that assumption the system simplies

tor = 0

= b(R + r cos )

= a

,

with solutionr = r 0

= 2 arctan R + rR r tan b2 R2 r 2(t + 0) = at + 0

. (4)

This system describes dynamics on a torus of major radius R, and for any particular minor

radius r , there is constant rotation in the direction and (for r = R) nonconstant rotation

in the direction. At least for small r , our assumptions appear to have been justied.

Solutions to the system (3) live on nested tori, and wind about these tori forming torus

knots or dense, non-periodic orbits. Setting 0 = 0 and taking the limit as r

0, we nd

= Rbt while in the limit as r R, = 2 arctan Rbt. If the master knot is parameterizedso that it is traversed for 0 t 1, then each knot in the aura will also be traversed for0 t 1. The minimum slope q/p for a ( p, q) torus knot is Rb and the maximum slope is2 arctan Rb 2 arctan 0 = 2 arctan Rb.

Thus, for xed R, r , a and b we can determine the knot and link types generated by the

system to rst order: Every knot is a torus knot, and every link is a link of torus knots

so that those residing on tori of smaller radii have smaller slopes than those residing on

tori of larger radii. For knots, the slope will be a ratio q/p .

Referring to gure (8), we see on the right a sampling of the nested torus knots generated

by differential equation (1); note that the slope decreases as the radius decreases. The gure

on the left shows the variation of the angle with time (equation (4)) for several different

values of minor radius r . Figure (9) shows how we may manipulate sets of nested torus

knots to form a chain.


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FIG. 8: The graph of versus time for three values of r with R = 3, b = 1 / 3. The dot-dash linecorresponds to r = 0, the dashed line to r = 2 and the solid line to r = R. On the right, numericalsolutions to differential equation (1) show how the slope of inner knots is less than that of the outerknots.

FIG. 9: Peeling open nested torus knots: we can arrange nested torus knots so that a portion of each knot is wrapped around its torus, and a portion is attened. Then we can arrange the toralportions in a circular chain so that only the attened portions of all the other knots in the linkpass through the toral portions.

VI. BESTIARY OF VARIOUS KNOTTY BEHAVIOR

Except for simply parameterized torus master knots using polynomial basis functions, we

have been unable to nd any unifying theme to the appearance of the aura. The aurae of

non-torus knots can be intriguing, though, so we have assembled a little zoo generated by

some simple master knots and links.


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When we use radial basis functions rather than polynomial bases, the auras generated are

frequently cablings of the master knot, as demonstrated by the contrasting auras of gure

(10).

FIG. 10: On the left, three knots (red, blue and green) generated from a linked trefoil and un-knot (black) using polynomial basis functions, and on the right three knots (red, blue and green)generated from a 3 , 2 torus knot (black) using radial basis functions.

One interesting oddball among the torus knot aurae with polynomial tting was the (2 , 3)

torus knot, which had in its aura, in addition to a conventional set of nested torus knots,

an additional set of nested torus knots cabling the outer most torus of the inner set (see

gure (VI)).

FIG. 11: The inner set of nested tori are cabled by an outer set for the (2 , 3) torus knot.


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FIG. 12: A sequence of solution paths residing on six nested tori inside the torus of the masterknot, and then four more solutions on knotted tori intersecting the master torus.


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The Borromean rings provide and interesting example. For this parameterization,

x = cos(0 / 3) + 7/ 4cos t y = sin(0 / 3) + 7/ 4sin t z = 1 / 2sin3t



and for exponential radial basis functions with a scaling of = 1 / 4, we found that all the

rings were stable and that there was one additional stable component that linked each pair

of original rings, wrapping once around the central axis. Using a polynomial t, we nd the

original link components are unstable, but that there is again an additional link component

generated by the vector eld that again links the original components in pairs, but wraps

twice around the central axis.

FIG. 13: Stable auxiliary knot generated from Borromean rings using radial basis functions on theleft and polynomial basis functions on the right. Several initial conditions, looking somewhat likea comb, are shown nearby the attracting knot .

FIG. 14: Left and center, two views of stable knots generated by a trefoil (2 , 3 torus knot) usingradial basis functions. On the right, stable knots (black) generated from linked rings.


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FIG. 15: From left to right, a torus knot cabling a trefoil master using radial basis functions, thetrefoil torus after a boundary crisis, where the radius of the cable increased to the point that thetorus self intersected, forming a standardly embedded torus (we see its knot) and the pair togetheron the right.

VII. SUMMARY

We have generated vector elds and their differential equations from knots and links given

by explicit parameterizations. We found that almost all of the master knots and links we

studied generated innite sets of Lyapunov stable auxiliary knots and links. We call the link

consisting of all the auxiliary knots the aura of the master knot or link, relative to a basis.

We found that relative to a manageable maximum degree polynomial basis (from degree one

to ve) there is a large set of simply parameterized torus knots that generates basically the

same vector eld. We analyzed the vector eld to show that it generated an aura of nested

torus knots, and gave a strict lower bound on the slopes of the innermost torus knots.

VIII. APPENDIX

A. 3, 2 torus knot second and third degree polynomial 240 points

Second degree: x = 3313487970 10 7

0.437559199610 8

x12.53299941y+0 .439520882010 7z0.586581421410

8x20.109331621110 8y20.321948554710

8z2 +0 .2818014241xy6.322816491xz + 0 .1597052374 10

7yz

y = 0 .1291558711 10 6 + 12 .53299938x + 0 .5989061164 10

8y 0.1037896974 10 7z +

0.1409007050x20.1409007207y20.833024218510 7z20.251932396210

9xy+0 .659443579010 9xz 6.322816480yz


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z = 31.41592799 + 0 .1147722353 10 7x + 0 .2385523291 10

9y 0.3895622312 10 8z +

3.141592750x2 +3 .141592793y2 +3 .141593132z20.616855254710 8xy +0 .127001895210

7xz +

0.5823540194 10 7yz

Third degree:

x = 0 .197901301510 80.184192943710

6x12.19880927y+0 .703500429410 4z

0.223161371910 7x2 0.535521417510 8y2 + 0 .238088860810

6z2 + 0 .7110229813xy 6.879396147xz + 0 .600305329010

7yz + 0 .193461591610 7x3 + 0 .714160510710

1y3 +

0.6265728764 10 4z3 + 0 .7142165242 10

1x2y 0.1627898450x2z + 0 .3015185438 10 8y2x + 0 .1627603199y2z + 0 .486485849910

7z2x 2.013561010z2y 0.455666025310 7xyz

y = 0 .3525418676

10 6 + 12 .19880898x + 0 .1585391326

10 6y

0.4265001669

10 4z + 0 .3555115748x2 0.3555115270y2 0.1051076016 10 5z2 0.1150993864

10 7xy +0 .404989155610 7xz 6.879396274yz 0.714174750510

1x3 +0 .837919392110 6y3 0.382999153410

4z3 0.256615415510 5x2y + 0 .900122760710

5x2z 0.7141741300 10

1y2x + 0 .8958689286 10 5y2z + 2 .013561648z2x 0.4030863902

10 7z2y + 0 .3255502338xyz

z = 31.41592780+0.706758566610 7x+0 .105228191610

6y0.203814950310 4z+

3.141592747x2 + 3 .141592879y2 + 3 .141592038z2

0.2695439506

10 7xy + 0 .5313692275

10 7xz + 0 .2013151850 10 6yz 0.2385730437 10

8x3 + 0 .3924500226 10 6y3

0.181783612410 4z3 0.122523004210

5x2y+0 .429519105710 5x2z +0 .6968966996

10 7y2x + 0 .426094852610 5y2z 0.388182862110

6z2x + 0 .343251594210 7z2y

0.104892642110 6xyz

B. 4, 7 torus knot second and third degree polynomial 240 points

Second degreex = 0 .309019556010

6 +0 .119620367810 8x43.98229723y+0 .251510396810

7z0.102570284410

7x2 0.117025149610 7y2 0.391312687310

6z2 + 0 .120815166110 8xy8.151159325xz 0.378750952210

8yz y = 0 .131579998110 7 +43 .98229719x +

0.177006545410 8y 0.227376710010

7z 0.723448066810 9x2 0.5036705302

10 8y2 + 0 .3934667805 10 7z2 + 0 .2221743140 10

8xy 0.1004534098 10 7xz

8.151159327yz z = 41.88790177 + 0.1775531389 10 8x + 0 .1645520580 10

8y +


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0.181368860510 8z + 4 .188790181x2 + 4 .188790176y2 + 4 .188790201z2 + 0 .9697088513

10 10xy 0.169882105010 9xz + 0 .216375256910

8yz

Third degree

x = 0 .3090195554

10 6 +0 .8039299028

10 7x

43.98229698y+0 .5854449484

10 6z

0.102570284610 7x2 0.117025150010

7y2 0.391312685410 6z2 + 0 .1208151647

10 8xy8.151159325xz 0.378750919810 8yz 0.432717805510

8x3 0.258312545610 7y3 0.351840645210

6z3 0.226492658410 7x2y 0.337787289010

7x2z 0.718012761510

8y2x0.303106600010 7y2z0.445652862710

7z2x +0 .779050911610 7z2y + 0 .492683364610

8xyz

y = 0 .131579994910 7 +43 .98229730x +0 .318808070710

7y0.326733674910 6z

0.7234480581

10 9x2

0.5036705278

10 8y2 + 0 .3934667839

10 7z2 + 0 .2221743142

10 8xy0.100453409310 7xz 8.151159327yz 0.916800369610

8x3 0.141328041410 8y3 + 0 .451670638010

6z3 0.462414077910 8x2y 0.784174595210

8x2z 0.210762624310

8y2x+0 .326724012910 9y2z0.421067595310

7z2x0.992292589210 8z2y + 0 .714871226010

8xyz

z = 41.88790177+0.503571567910 7x+0 .128911052910

6y+0 .163900372410 7z+

4.188790181x2 + 4 .188790176y2 + 4 .188790201z2 + 0 .969709021910 10xy 0.1698821316

10 9xz + 0 .2163752566

10 8yz

0.3169504911

10 8x3

0.5923762730

10 8y3 +

0.710496634910 9z3 0.734294821610

8x2y0.334394496110 8x2z0.2731587718

10 8y2x + 0 .770863415710 10y2z 0.280854411610

7z2x 0.114288370510 6z2y +

0.115887136310 8xyz

[1] An orbit x (t) of a dynamical system is said to be stable under the ow if all sufficiently nearby

initial conditions are asymptotic to x (t), while an orbit is Lyapunov stable if all sufficiently

nearby initial conditions x (t0) result in orbits x (t) remain nearby x

de from knot revised 1

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