The Jones polynomial of a knot Ty Callahan



Lord Kelvin thought
that atoms could be
knots
Mathematicians create
table of knots
Organization sparks
knot theory

Knot

A loop in R3

Unknot

Arc


Portion of a knot
Diagram

Depiction of a knot’s
projection to a plane

OK

NOT OK


Two knots are equivalent if there is an isotopy
that deforms one link into the other
Isotopy



Continuous deformation of ambient space
Able to distort one into the other without breaking
Nothing more than trial and error can
demonstrate equivalence

Can mathematically distinguish between
nonequivalence

Choice of the sense in which a knot can be
traversed

Orientation results in two possible crossings

Right and Left

Two Principles
1)
Assign a value of 1 to any diagram representing an
unknot
2)
Skein Relation: Whenever three oriented diagrams
differ at only one crossing, the Jones Polynomial is
governed by the following equation
1
t R[t]  tL[t]  (t
1
2
t
1
2
)Q[t]
1
t R1[t]  t  (t
R1[t]  (t
3
2
1
2
1
t
1
2
)Q1[t]
 t )Q1[t] t
2
2
1
t R2[t]  tL2[t]  (t
R2[t]  t L2[t] t
2
1
2
3
2
t
1
2
t
1
2
)
1
t  t  (t
t
1
2
t
(t 1)(t
1
2
3
2
1
2
t
1
2
)Q3[t]
 (t 1)Q3[t]
t
1
2
Q3[t]  t
)  (t 1)Q3[t]
1
2
t
1
2
L2[t]  Q3[t]  t
R2[t]  t (t
1
R2[t]  t
5
2
2
2
t
1
3
2
1
2
)t
t 2 t
R2[t]  t
5
2
t
3
t
2
1
2
3
1
2
2
t
t
1
2
1
2
Q1[t]  R2[t]  t
R1[t]  (t
3
2
1
 t )(t
2
5
5
2
2
t
1
2
1
2
t )t
R1[t]  t  t  t  t  t
4
2
3
R1[t]  t  t  t
4
3
2
2


Right
Left
R1[t]  t  t  t
4
4
3
3
R1[t]  t  t  t
1


The Jones Polynomial of the Right Trefoil knot
does not equal that of the Left Trefoil knot
The knots aren’t isotopes
“KNOT” EQUAL!!