jones polynomial
Post on 23-Feb-2016
32 Views
Preview:
DESCRIPTION
TRANSCRIPT
JONES POLYNOMIALTy Callahan
Historical Background Lord Kelvin thought
that atoms could be knots
Mathematicians create table of knots
Organization sparks knot theory
Background Knot
A loop in R3
Unknot
Arc Portion of a knot
Diagram Depiction of a knot’s
projection to a plane
Diagram OK NOT OK
Equivalence 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
Figure 8 Knot
Orientation Choice of the sense in which a knot can
be traversed
Crossings Orientation results in two possible crossings
Right and Left
Jones Polynomial 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
€
t−1R[t] − tL[t] = (t12 − t
−12)Q[t]
Ex. Trefoil Knots
1) Skein Relation for Right Trefoil
€
t−1R1[t] − t = (t12 − t
−12)Q1[t]
€
R1[t] = (t32 − t
12)Q1[t]+ t
2
2) Skein Relation for Link
€
t−1R2[t] − tL2[t] = (t12 − t
−12)
€
R2[t] = t2L2[t]+ t
32 − t
12
3) Skein Relation for Twisted Unknot
€
t−1 − t = (t12 − t
−12)Q3[t]
€
t−12 − t
32 = (t −1)Q3[t]
€
(t −1)(−t12 − t
−12 ) = (t −1)Q3[t]
€
Q3[t] = −t12 − t
−12
4) Substitute and Simplify
€
L2[t] =Q3[t] = −t12 − t
−12
€
R2[t] = t2(−t
12 − t
−12) + t
32 − t
12
€
R2[t] = −t52 − t
32 + t
32 − t
12
€
R2[t] = −t52 − t
12
4) Continued..
€
Q1[t] = R2[t] = −t52 − t
12
€
R1[t] = (t32 − t
12)(−t
52 − t
12) + t 2
€
R1[t] = −t4 − t 2 + t 3 + t + t 2
€
R1[t] = −t4 + t 3 + t
5) Compare to Left Trefoil
€
R1[t] = −t−4 + t−3 + t−1
€
R1[t] = −t4 + t 3 + tRight
Left
Conclusion The Jones Polynomial of the Right Trefoil
knot does not equal that of the Left Trefoil knot
The knots aren’t isotopes
“KNOT” EQUAL!!
top related