Department of Mathematical Sciences

Mathematics

Quantum Topology

Current members of quantum topology research group:

  • Dr Bruce Bartlett

The last two decades or so have seen a remarkable resurgence in the interaction between pure mathematics and theoretical physics. For mathematics this has led to an influx of profound new ideas, ideas which oftentimes seem strange and bizarre from a traditional mathematical perspective, but which have nevertheless proved useful. Specifically, it has been the impact of ideas from quantum field theory on geometry which have been most useful; this has spawned a new field called “quantum topology” which tries to understand and place these ideas into a coherent mathematical framework. The main theme is trying to understand the physicist’s most powerful tool, the “path integral”, and to utilize the resulting structures to compute new invariants of geometric objects such as knots and manifolds. This endeavour draws on many parts of modern mathematics such as higher category theory, homotopy theory and representation theory.

To get an idea of what “quantum topology” is all about, consider the following two knots:

trefoils

Can the left hand knot (known as the “left-handed trefoil knot”) be continuously deformed into the right hand knot (known as the “right-handed trefoil knot”)? The rule is: you are allowed to continuously deform the strands of the knot in space, but you can’t let the strands pass through each other.

Maybe if we turn the left hand trefoil “around”? No – try it, you’ll get something which is not the same as the right hand trefoil.

So we suspect these two knots aren’t the same. But how can we prove this?

Well, one way to prove it is by calculating the Jones polynomial of each link. In 1984, Jones discovered a way of assigning polynomials to knots, in such a way that if two different knots get different Jones polynomials, then we know they can’t be deformed into each other!

It turns out that the Jones polynomial of the left hand trefoil is

-1/t^4 + 1/t^3 + 1/t

while the Jones polynomial of the right hand trefoil knot is

t + t^3 – t^4.

These two polynomials aren’t the same… so the knots can’t be deformed into each other!

To read up more on this sort of thing, try the following references:

Kwantum Topologie

Tans bestaan die kwantum topologie navorsingsgroep uit die volgende lede:
  • Dr Bruce Bartlett (Stellenbosch)
  • Gerrit Goosen (Stellenbocsh, North-West University)
Die laaste twee dekades of so het ‘n merkwaardige herlewing gesien in die interaksie tussen suiwer wiksunde en teoretiese fisika. Vir wiskunde het dit gelei
na ‘n instroming van nuwe en diepgaande idees. Alhoewel hierdie idees, gesien vanaf ‘n tradisionele wiskundige perspektief, vreemd en selfs bisarre lyk, het hulle
hul nuttigheid reeds bewys. Meer spesifiek is dit die impak van idees vanaf kwantum veld teorie op meetkunde wat die mees hulpvaardigste is; dit het die nuwe veld
“kwantum topologie” op die been gebring, wat probeer om hierdie idees te verstaan en binne ‘n samehangende wiskundige raamwerk te plaas. Die hoof tema is om die fisikus se kragtige ”padintegraal” te probeer verstaan, en om die resulterende strukture te gebruik om nuwe invariante van meetkundige voorwerpe soos knope en variëteite.
Hierdie poging trek inspirasie van verskeie dele van moderne wiskunde soos hoër kategorie teorie, homotopie teorie, en representasie teorie.
Om ‘n idee te kry waaroor “kwantum topologie” regtig gaan, beskou die volgende twee knope:
trefoils
Is dit moontlik om die linkerste knoop (bekend as die “linkerhandse trefoil”) op ‘n kontinu manier te vervorm sodar dit lyk soos die regterkantste knoop (bekend as die “regterhandse trefoil”). Ons mag die string kontinu rondskyf en vervorm, solank ons nie twee stringe “deur mekaar” beweeg nie.
Dalk as ons die linker knoop net “omdraai”? Nee – probeer self, dit lyk nie soos die regterkant nie.
Ons vermoed dus dat die knope nie dieselfde is nie. Maar, hoe kan ons dit bewys? Wel, een manier is om die sogenaamde Jones polinoom van elke knoop te bereken.
In 1984 het Jones ‘n nuwe manier ontdek om ‘n polinoom met elke knoop te assosiëer, op sodanige manier dat, indien twee knope se polinome nie dieselfde is nie, dan is dit onmoontlik om die een knoop te vervorm om soos die ander een te lyk!
Na berekeninge blyk dit dat die Jones polinoom van die “linkerhandse trefoil”
-1/t^4 + 1/t^3 + 1/t
is, terwyl die Jones polinoom van die “regterhandse trefoil”
t + t^3 – t^4
is. Aangesien die polinoom nie dieselfde is nie, kan die knope dus nie van die een tot die ander vervorm word nie!
Om meer hiervan te leer, probeer die volgende bronne: