JONES POLYNOMIAL Ty Callahan
Dec 18, 2015
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
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t−1R1[t] − t = (t12 − t
−12)Q1[t]
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R1[t] = (t32 − t
12)Q1[t]+ t
2
2) Skein Relation for Link
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t−1R2[t] − tL2[t] = (t12 − t
−12)
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R2[t] = t2L2[t]+ t
32 − t
12
3) Skein Relation for Twisted Unknot
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t−1 − t = (t12 − t
−12)Q3[t]
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t−12 − t
32 = (t −1)Q3[t]
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(t −1)(−t12 − t
−12 ) = (t −1)Q3[t]
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Q3[t] = −t12 − t
−12
4) Substitute and Simplify
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L2[t] =Q3[t] = −t12 − t
−12
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R2[t] = t2(−t
12 − t
−12) + t
32 − t
12
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R2[t] = −t52 − t
32 + t
32 − t
12
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R2[t] = −t52 − t
12
4) Continued..
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Q1[t] = R2[t] = −t52 − t
12
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R1[t] = (t32 − t
12)(−t
52 − t
12) + t 2
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R1[t] = −t4 − t 2 + t 3 + t + t 2
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R1[t] = −t4 + t 3 + t
5) Compare to Left Trefoil
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R1[t] = −t−4 + t−3 + t−1
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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!!