Calculating with Knots
in memoriam V. F. R Jones
Raphael J. F. Berger
September 13, 2020
A simplistic way to define a mathematical knot is as a ”normal knot” that is knotted
into a closed loop, or equivalently as a ”normal knot” where both ends of the rope are
connected. Sir Vaughan F. R. Jones was a mathematician who won the Fields medal
for a knot theoretical result that he more or less obtained by incident when workig on
Von-Neumann algebras. He passed away one week ago on the 6th of September. Since
I always had a certain affinity to knot theory, I decided to write this little text.
Knots are manifestations of inherently 2-dimensional notations, which are in con-
trast to the usual one-dimensional notations in mathematics not very frequent. Other
examples would include Feynman diagrams, Penrose notations or the commutative
diagrams in homological algebra.[1]
Figure 1: Three different drawings/diagrams of the trefoil knot.
One of the most important questions in knot diagrams, like those in Fig. 1 is if they
depict the same1 knot or if they don’t. In this case they indeed show the same knot.
The question is now how to arrive at this conclusion. Mathematicians have come up
1 Inmathematical knot theory there is a very clear concept of what the “same” means. The term
is homotopy, but for our purposes we can assume the intuitive idea of identity between concrete
knots in a rope with closed ends
1
with certain “invariants” for knots, these are algebraic objects (objects with a one-
dimensional notation) which can be assigned uniquely to a knot diagram, and if the
diagram shows the same knot the invariant will be algebraically identical. Now this in
turn unfortunately does not imply that if two diagrams show the same invariant, that
the knots neccessarily have to be identical. But the known invariants are pretty good,
such that one can say in a certain sense that if two diagrams have the same invariant,
they are in many cases identical and only in some execeptional cases this does not
hold. But in fact it’s an open question if there exist unique such invariants!
Jones contribution was to find such a close-to-perfect invariant, in particular his
invariant, the so called Jones polynomial in addition considers a orientation of the
diagram. That means the rope comes with a direction (like a rain worm for example,
which can be swallowed only in one direction). In such a setting the knodes are in
general not identical with their mirror image, they become chiral.
The rules for computing the Jones polynomial for a knot diagram are actually pretty
simple, such that one can easily play around with them which is quite fun. We write
the Jones polynomial of the knot diagram K like VK . In general we will use the variable
t for expressing the polynomials, then the rules are
V / =1 (1)
√
t−1 V O − tV O = ( t − √1 )V O (2)
t
/
Here the oriented (from here on assumed everywhere) knot diagram is just a
closed loop which is the simplest or the trivial knot which is also called the “unknot”.
The Jones polynomial of the unknot according to equation (1) is simply 1 (which is
t0 )2 . So how does it work if we want to compute V of another knot diagram? Let’s
o
go the possibly next simplest diagramm, two disconnected unknots / , lets call that
diagram C. Since this falls not under eq. (1) we have to apply eq. (2), which can be
seen as a reciepe to reduce recursively the polynomial of a complicated knot diagram
by manipulating a single crossing (or uncrossing like on the right hand side of (2)) to
obtain polynomials of (possibly simpler) knot diagrams that can be added (or sub-
stracted) from each other.
How does this work? We see at the right hand side of (2) two uncrossed lines passing
each others and we notice that this motive is present in C, now we just write down (2)
2A very simple simple question that is for example also open, is if the unknot is the only knot with
V = 1.
2
for the whole diagram which looks like
√
t−1 V O − tV O = ( t − √1 )V O (3)
t
We more or less immediately notice that the two diagrams on the left of (8) represent
exactly the unknot, just plotted in a “twisted way”. Since we know3 that the same
polynomial is obtained for any diagram respresenting the physcially “same” knot.
Then we can rewrite and rearrange (8) using O for the unknot as
√ 1
t−1 VO − tVO = ( t − √ )VC (4)
t
t−1 · 1 − tV · 1
VC = √ (5)
( t − √1t )
1 √
VC = −( √ + t) (6)
t
At this stage you should be able to compute V of two interconnected rings O which
is also called Hopf link or simply H. The result should be
1
VH = −( + t) (7)
t
Similarly for the trefoil knot T from Fig 1 we obtain for all three diagrams (a), (b)
and (c)
VT = t + t3 − t4 (8)
Since I am a chemist one could ask how is this relevant to Chemistry? It is quite
relevant, but in some non-obvious ways. I might write about this at some later stage.
I want to close this text with an overview of the Jones polynomials we have been able
to derive and encourage you to compute it for more examples.
3 We actually have to assume here that the rules (1) and (2) are working, that means that they govern
existing polynomials for every possible diagram and that they indeed yield the same polynomial
for the “same” knots.
3
Figure 2: Four different oriented knots and their Jones polynomial.
Finally I shall refer to a much better introduction into mathematical knot theory
from Jones himself, you can find it here[2].
References
[1] https://terrytao.wordpress.com/2020/09/09/vaughan-jones/
[2] https://math.berkeley.edu/~vfr/jonesakl.pdf
