Untangling Knot Theory

 

When I was a child, my grandfather’s house had a sun porch/library.  Since he was an amateur sailor, a lot of the books were nautical in nature, but one in particular drew my attention.  Called simply The Book of Knots by Clifford W. Ashley, it illustrated the tying and documented the history and uses of something like 4000 knots.  The book has passed through the generations, and I now have the book and still enjoy trying a new knot.

Within the pages of that book are a few cryptic comments, such as:

…The Law of the Common Divisor is quite simple.  For example, within the limits of twenty-four leads and twenty-four bights there are 576 combinations.  Of these combinations, 240 have a common divisor and cannot be tied as a Turk’s-Head, and 366 have no common divisor and can be tied.  If a knot is attempted in one cord with dimensions that posses a common divisor, the working end and the standing end will meet before the desired knot is complete…(Ashley, p.233)

These made me ponder the rules governing knots, and develop an interest in knot theory

My mother, also a math teacher, likes to quote “everything comes back to math.”  I don’t know if she’s right, but there is a branch of Mathematics, which deals with knots.  Knot theory, technically a branch of Topology, tries to understand what knots can be tied, if two knots are unique and all other questions related to knots in space.

 

History

 

The origins of knot theory go back to early attempts to understand atomic structure.  The School of Mathematics of the University of Wales, Bangor offers the following information on their website:

 

…A theory of the atom had to explain:

·        The stability of atoms.

·        The variety of atoms, as shown by the periodic table of elements.

·        The vibrational properties of atoms, as shown by their spectral lines.

 

Lord Kelvin had seen smoke rings of his physicist friend P.G. Tait, and was impressed by their stability, and vibrational properties. He had a vision of atoms as vortices in space. How to explain the variety of atoms?  In 1867, Kelvin presented a paper to the Royal Society of Edinburgh, in which he wrote: Models of knotted and linked vortex atoms were presented to the Society, the infinite variety of which is more than sufficient to explain the allotropies and affinities of all known matter.  So Tait set about preparing a list of knots, to see if there was a relation with the elements in the periodic table. The vortex theory of the atom soon disappeared, but Tait's 10 years of work on his list of knots of up to 10 crossings and the conjectures he made (some of which have been proved only recently) have been an inspiration ever since. Further, the idea of relations between knots and fundamental properties of matter is being shown to have a continuing force.

- http://www.bangor.ac.uk/cpm/exhib/aether.htm

 

It is interesting that Science, and Physics in particular, are now returning to this previously abandoned theory for explanations, not of atomic structure, but certain elements of string and quantum theory, statistical mechanics and DNA patterns.  We will discuss these applications presently.

 

Definitions

Knots, as perceived in knot theory, are defined a little differently than what might be shown in the Boy Scout Handbook.   A partial catalog of knots appears in Appendix A.

“A knot is a closed, one dimensional, and non-intersecting curve in three dimensional space. From a more mathematical and set-theoretic standpoint, a knot is a homeomorphism that maps a circle into three dimensional space and cannot be reduced to the unknot by an ambient isotopy.” http://library.thinkquest.org/12295/main.html

The “unknot”, referred to above, is the simplest of the items considered in knot theory.  It consists of a single loop, a string with the ends tied together.  In knot theory, the ends are generally tied together to avoid questions of what happens at the ends, and to ensure that parts of the knot don’t spill off the ends.

 

All illustrations from http://www.bangor.ac.uk/cpm/exhib/menu.htm unless otherwise noted.

 

Unknot        Trefoil        Figure 8

 

 

The next simplest knot is the Trefoil.  This three-leafed shape is simply the old Boy Scout “overhand knot”, again with the ends joined.  Taking a “figure-8” knot and joining the ends results in a closed figure called, appropriately, a figure-8 knot. 

 

Quantifying the differences between the unknot and the trefoil, or other knots, is one of the main objectives of knot theory.  The easiest way to do this is by counting the number of times that a strand crosses over another strand.  In the trefoil, there are three places where a strand passes over another.  Consequently the trefoil has a crossing number of three.  The figure-8 knot has a crossing number of 4.  Appendix A shows a catalog of knots up to 9 crossings.

 

We can demonstrate that these knots do exist, by a procedure called tri-coloration. If a knot is tri-colorable, then at every crossing no two strands share the same color.  The trefoil at the right is tri-colorable. Since the unknot is not tri-colorable, a theorem of knot theory assures us that the trefoil is unique from the unknot, therefore the set of knots has at least two elements.

Uniqueness Polynomials

 

Traditionally, it was very difficult to demonstrate that two knots were actually the same knot or unique.  The only real option was to take the knot in question, tied in string, and try to twist it, pull it or deform it into the other.  As an example of the potential complexity of this task, consider the knots below.  The first deforms into the unknot with the right combination of twists and inversions.  They are, therefore, the same knot.

In 1928, a J. Alexander developed a technique employing linear algebra and polynomials to further characterize knots.  Called the Alexander polynomial, this technique uses a matrix to represent the strands and crossings of the knot.  Each crossing is represented by a row, while columns represent strands.  Taking the determinant of the matrix, leaving out the last row and column, yields this polynomial.

 

Determining the elements of the matrix can be tricky, but I find the easiest way is to turn the diagram of the knot as if you were a car driving on the “road” of the knot.  This is referred to as orienting the knot.  Number the strands as you would come to them.  Every time you go over a “bridge”, number that as the next crossing and make a new row of the matrix.  For each crossing (row), the strand (column) that is crossing over is filled with 1-t.  The strand on the left gets a –1 and the strand on the right gets a t.

 

For example, let us begin with the trefoil.  Starting on the blue strand we would cross over at c1.  For the first row of the matrix, strand 1 is crossing over so the first column is filled with 1-t.  On the left is strand 2, so the second column is filled with –1.  On the right is strand 3, so column 3 gets a t.  As we continue “driving” around the knot, we cross under at c3, which we don’t consider, and are now on strand 2.  At c2 we form another row.  This continues until we arrive at the matrix shown below.  Strike the last row and column, and take the determinant.

 

 


                                                                                                        

The Alexander polynomial is useful for determining whether two knots are actually the same or different.  Calculating the Alexander polynomial on the knot below shows that it is a disguised trefoil.  With a piece of string this is easily seen for the Trefoil, but it can be much more difficult with more complicated knots.

 

         

 

 

Here is another form the trefoil can take.  Its polynomial should also work out to t2 – t + 1.

 

A theorem of knot theory states “If the Alexander polynomial for a knot is computed using two different sets of choices for diagrams and labelings, the two polynomials will differ by a multiple of ±tk, for some integer k. (Livingston, p. 50)”.  Since the three polynomials are the same, with the allowed multiple of a power of t, all three of these knots are different versions of the same knot.

 

Comparing these with the figure-8 knot, the next in complexity, the polynomial is different – not a ±tk multiple of the other polynomials.  This assures that the figure-8 knot is fundamentally different from the trefoil, and no amount of twisting or turning or pulling can transform one into the other.  This is also obvious since the figure-8 knot is not tri-colorable, while all versions of the trefoil knot are tri-colorable.

 

 

 

 

 

 

Algebra of knots

 

The algebraic concept of a group involves elements of a nonempty set and an operation. The elements of the set shall be the knots, some of which are illustrated here.  The set of knots has already been shown to be nonempty. The operation shall be “addition” as shown in the figure below. 

(It is understood the figures and comments below are not a proof of anything, but merely hand waving visualizations of one instance of the property.)  Herstein’s definition “a nonempty set G is said to be a group if in G there is defined an operation *”, requires the following four points:

a)      Closure - a,b Î G Þ a * b Î G.

If we join two or more knots as described above, it is unlikely we will end up with a car or a house or anything else – it’s probably going to be a knot.

 

b)      Associativity – Given a, b, c Î G, then a * (b * c) = (a * b) * c

 

c)      Existence of an identity - $ a special element e Î G | a * e = e * a = a " a Î G

The identity is the unknot.

K + 0 = K

 

d)      Existence of an inverse - " a Î G $ a special element b Î G | a * b = b * a = e.

Initially I believed this would be a triviality.  This necessary property, however, has turned out to be much more elusive than I imagined, even for the relatively simple trefoil.  I will not state that such a thing does not exist, but neither can I demonstrate even the simplest case. 

 

After some consideration (and much fiddling with string) I have come to doubt the existence of an inverse for any knot, much less each of them.  If an inverse existed, a knot and its inverse would “add” to the identity, the unknot.  It stands to reason, then, that I should be able to take an unknot and convert it back into the knot and its inverse.  This does not (seem to) happen.  QEHW (Quod erat hand waved)

If an inverse existed, we would have shown the set of knots to form a group under “knot addition.”  It would then be a simple matter to further show that this group would be abelian.  We would need to demonstrate that

e)      a * b = b * a " a, b Î G

to have an abelian group.

 

 

There are many sideshows in knot theory.  A synopsis of some of these follow.

 

Prime knots

 

In determining whether two knots are unique or not, it is necessary to consider prime knots.  Just as prime numbers are numbers which cannot be broken into smaller factors, prime knots are knots which cannot be formed by joining simpler knots.  The knot shown below is actually two trefoils joined together, thus it is not a prime knot.  The table of knots which concludes this paper shows all of the prime knots up to nine crossings.

 

Siefert surfaces

 

Imagining the surface formed by the edges of the knot leads directly into the questions addressed in topology.  If you mixed up a bubble solution with dish detergent, and dipped a knot formed by wire into it, a surface would form showing the Siefert surface of the knot.  (Supposedly a little glycerin will strengthen the bubble surface allowing it to withstand examination)

 

Links

 

Links are knots (including unknots) which are joined together as shown in the figure below.

Some other links are shown at the bottom of the table of knots.

 

Braids

 

Braiding patterns are of especial interest to the textile and manufacturing industries.

               

 

Pretzel knots

 

Pretzel knots are a special type of knot.

 

 

 

 

 

 

 

Torus knots

 

Torus knots are knots that may be tied around a torus.  A couple of theorems of torus knots are included below.

 

 

 

 

 

·        All the torus knots are prime knots.

·        You can make a torus knot T(p,q) with any numbers p, q provided they are coprime, i.e. have no common divisor - so (2,6) will not do. The pair (15,4) is interesting because it uses the three prime numbers 2, 3, 5.

 

Applications of knot theory

One of the web sites offers this information on knot theory:

When knot theory was first developed, its major appeal was in its possible applications to chemistry. Lord Kelvin and chemistry motivated the early development of knot theory in the 1880's. Kelvin hypothesized that a substance called ether was what the entire universe was made of and matter could be explained as knots in the ether; however, we now know that this is not true. This early interest gave knot theory the impetus that it needed in order to become a major field of mathematical study.

The first place for knot theory to find a home was in the biologist's toolkit with applications to DNA. In 1953 James Watson and Francis Crick discovered that the basic genetic material of life on earth took the shape of DNA's double helix from there the possibilities of the marriage of knot theory and Deoxyribonucleic Acid (DNA) were endless. What was also found was that DNA often becomes knotted, making it difficult for DNA to carry out its function. There are enzymes called topoisomerases that can perform topological manipulations on DNA. Scientists let these enzymes act on circular DNA performing actions like those illustrated here so that they can then study the function of the enzymes from the resulting knots in the circular DNA. The circular DNA is used because if open-ended DNA is used, the knots cannot be observed, as the ends are free. This is just one of the exciting applications of knot theory in the world of molecular biology.

-http://library.thinkquest.org/12295/main.html

 

Another web site, the Bangor site, referenced below includes some electron microscope images taken by N.Cozzarelli of these knotted DNA structures, and includes a more familiar diagram of the knot.

 

 

 

          

The Bangor site also reports the following regarding knot theory as related to string theory and theoretical physics:

 

Since 1984 a new link between knots and theoretical physics has begun to take form. The discovery of some new invariants in knot theory took place. These invariants were discovered by Vaughan Jones who was working in an area of mathematics closely related to physics. Combining them with aspects of "string theory", a branch of theoretical physics, has produced a very rich theory that may one day give a unified description of the four fundamental forces of nature, gravity, electro-magnetism, and strong and weak interactions between particles.

 

My mom likes to say, “everything comes back to math.”  Maybe she’s wrong.  Maybe everything comes back to knot theory.

To me the simple act of tying a knot is an adventure in unlimited space. A bit of string affords the dimensional latitude that is unique among the entities. For an uncomplicated strand is a palpable object that, for all practical purposes, possesses one dimension only. If we move a single strand out of the plane, interlacing at will, actual objects of beauty result in what is practically two dimensions; and if we choose to direct our strand out of this plane, another dimension is added which provides an opportunity that is limited only by the scope of our own imagery and the length of a ropemakers coil.

-The Book of Knots, Clifford W Ashley

  

 

References

 

·         Knot Theory   - Charles Livingston, Mathematical Association of America, 1993 -Winthrop Library QA 612.2.L58

·         Introduction to Knot Theory   - Crowell and Fox , Ginn and Company, 1963-Winthrop Library QA 614.5.K6.C74

·         Ashley Book of Knots  - Clifford W. Ashley – York County Library 623.88 – The knot bible, you ought to own a copy, at least go check it out!

 

Web Links

·         http://www.bangor.ac.uk/cpm/exhib/menu.htm  - Indispensable primer on knot theory

·         http://www.pims.math.ca/knotplot/  - Page with tons of figures, links, free downloadable software to draw knots.

·         http://www.c3.lanl.gov/mega-math/workbk/knot/knot.html - Kid level introduction to Knot Theory.  Some fun classroom activities.

 

Appendix A – Knots up to 9 crossings and some knots with links. 

Big number = # of crossings, subscript = version of same crossing #, superscript = # of links

From http://www.pims.math.ca/knotplot/zoo/zoo1.html

Appendix B

  

Geometry

 

The compass turns in circles

and the line runs straight,

like the turning of the years

and the road outside my gate.

The line beckons me onward,

the circles hold me tight.

Lives and paths may intersect,

but angles diverge forever.

White paper and pencil marks

whisper to me things I never knew

about myself and life and fate

and the paths that we must choose.

God is there in glory and might

His universe in grand array,

and yet he too is circumscribed

by the path that He has laid.

Two points do indeed determine a line.

Disdain not these figures and scribblings,

for Geometry is not study.

It is worship