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Understanding Knot Theory: Historical Background and the Jones Polynomial

This exploration of knot theory delves into its historical roots, starting with Lord Kelvin's intriguing hypothesis that atoms may hold properties akin to knots. Through the contributions of mathematicians who formulated knotted structures, we will understand the concepts of isotopy and knot equivalency. Additionally, the Jones Polynomial is introduced, showcasing its significance in distinguishing knots via skein relations. By examining examples, such as right and left trefoil knots, we can grasp how different knots can be mathematically identified as non-equivalent, emphasizing the importance of these structures in both mathematics and science.

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Understanding Knot Theory: Historical Background and the Jones Polynomial

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Presentation Transcript

  1. Jones Polynomial Ty Callahan

  2. Historical Background • Lord Kelvin thought that atoms could be knots • Mathematicians create table of knots • Organization sparks knot theory

  3. Background • Knot • A loop in R3 • Unknot • Arc • Portion of a knot • Diagram • Depiction of a knot’s projection to a plane

  4. Diagram • OK • NOT OK

  5. 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

  6. Figure 8 Knot

  7. Orientation • Choice of the sense in which a knot can be traversed

  8. Crossings • Orientation results in two possible crossings • Right and Left

  9. Jones Polynomial • Two Principles • Assign a value of 1 to any diagram representing an unknot • Skein Relation: Whenever three oriented diagrams differ at only one crossing, the Jones Polynomial is governed by the following equation

  10. Ex. Trefoil Knots

  11. 1) Skein Relation for Right Trefoil

  12. 2) Skein Relation for Link

  13. 3) Skein Relation for Twisted Unknot

  14. 4) Substitute and Simplify

  15. 4) Continued..

  16. 5) Compare to Left Trefoil Right Left

  17. 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!!

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