1 / 17

Jones Polynomial

Jones Polynomial. Ty 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 R 3 Unknot Arc Portion of a knot Diagram Depiction of a knot’s projection to a plane.

vita
Download Presentation

Jones Polynomial

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

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!!

More Related