Mini Project - Knot Invariants and Braid Groups

Weekly reading plan

1. Week 1

   References

   • Adams, The Knot Book, Section 1.1 - 1.4
   • Lickorish, An Introduction to Knot Theory, Chapter 1
   • Kassel, Quantum Groups, Section X.1 - X.3

   Goal

   • Definition of knots, links, diagrams (projections). Relationship between links and diagrams? → Reidemeister theorem → Reidemeister moves
   • Basic examples of links and some basic link invariants → e.g. linking number

2. Week 2

   References

   • Adams, The Knot Book, Section 6.1, Section 6.3
   • Lickorish, An Introduction to Knot Theory, Chapter 3
   • Kassel, Quantum Groups, Section X.4

   Goal

   • Local operations → Determine coefficients → Kauffman bracket → Writhe
   • Skein relation → Jones polynomials → HOMFLYPT

3. Week 3

   References

   • Lickorish, An Introduction to Knot Theory, Chapter 3
   • Reshetikhin, lecture notes I Page 1 - 18

   Goal

   • Reidemeister move I → Framed links → Relation with writhe → Repeat all with framing

4. Week 4

   References

   • Adams, The Knot Book, Section 5.4
   • Kassel--Turaev, Braid Groups, Section 1.1 - 1.2, Section 2.1 - 2.5
   • Kassel, Quantum Groups, Section X.6
   • Reshetikhin, lecture notes I Page 19 - 25
   • Reshetikhin, lecture notes II Page 1 - 9

   Goal

   • Alexander Theorem → Geometric braids → Braid groups
   • Isotopic links → Markov Theorem → Construct link invariants from braids

5. Week 5

   References

   • Kassel--Turaev, Braid Groups, Section 4.1 - 4.4
   • Reshetikhin, lecture notes II Page 7 - 11

   Goal

   • Hecke algebra behind HOMFLYPT → Link invariants and trace → Algebraic construction of HOMFLY

6. Week 6

   References

   • Kassel, Quantum Groups, Section X.5, Chapter XI, Chapter XII
   • Reshetikhin, lecture notes III, IV

   Goal

   • Local computation → Tangles → the category of tangles
   • Monoidal category, generators and relations

7. Week 7

   References

   • Kassel, Quantum Groups, Chapter XIII

   Goal

   • Braided categories
   • R-matrices

8. Week 8

   References

   • Kassel, Quantum Groups, Section X.8, Section XIV.1 - XIV.3
   • Reshetikhin, lecture notes V, VI

   Goal

   • Ribbons → the category of ribbons
   • Ribbon category

9. Week 9

   References

   • Kassel, Quantum Groups, Section III.1 - III.5, Section VIII.1 - VIII.2, Section XIV.6
   • Reshetikhin, lecture notes VI

   Goal

   • Category of representations → Hopf algebra → Ribbon Hopf algebra

10. Week 10

    References

    • Kassel, Quantum Groups, Section VI.1 - VI.3, Section VII.1, Section XVII.1 - XVII.3

    Goal

    • Quantum groups

11. Week 11

    References

    • Kassel, Quantum Groups, Section XII.5, Section XIV.4 - XIV.5

    Goal

    • Representation category of quantum groups → Quantum Schur--Weyl duality
    • Quantum trace → Link invariants from quantum groups

12. Week ?

    References

    • Adams, The Knot Book, Chapter 9
    • Lickorish, An Introduction to Knot Theory, Chapter 12
    • Bakalov--Kirillov, Lecturs on Tensor Categories and Modular Functors, Chapter 3 - 4

    Goal

    • Further on this topic: Quantum groups at root of unity → Modular category
    • 3-manifold from links → Reshetikhin--Turaev construction