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Skein Theory and Algebraic Geometry for the Two-Variable Kauffman Invariant of Links

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Skein Theory and Algebraic Geometry for the Two-Variable Kauffman Invariant of Links University of Massachusetts Amherst ScholarWorks@UMass Amherst Doctoral Dissertations Dissertations and Theses November 2016 Skein Theory and Algebraic Geometry for the Two-Variable Kauffman Invariant of Links Thomas Shelly University of Massachusetts Amherst Follow this and additional works at: https://scholarworks.umass.edu/dissertations_2 Part of the Algebraic Geometry Commons Recommended Citation Shelly, Thomas, "Skein Theory and Algebraic Geometry for the Two-Variable Kauffman Invariant of Links" (2016). Doctoral Dissertations. 804. https://doi.org/10.7275/8997645.0 https://scholarworks.umass.edu/dissertations_2/804 This Open Access Dissertation is brought to you for free and open access by the Dissertations and Theses at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. SKEIN THEORY AND ALGEBRAIC GEOMETRY FOR THE TWO-VARIABLE KAUFFMAN INVARIANT OF LINKS A Dissertation Presented by THOMAS SHELLY Submitted to the Graduate School of the University of Massachusetts Amherst in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY September 2016 Department of Mathematics and Statistics c Copyright by Thomas Shelly 2016 All Rights Reserved SKEIN THEORY AND ALGEBRAIC GEOMETRY FOR THE TWO-VARIABLE KAUFFMAN INVARIANT OF LINKS A Dissertation Presented by THOMAS SHELLY Approved as to style and content by: Alexei Oblomkov, Chair Tom Braden, Member Inan¸cBaykur, Member Andrew McGregor Computer Science, Outside Member Farshid Hajir, Department Head Mathematics and Statistics DEDICATION To Nikki, of course. ACKNOWLEDGEMENTS I am greatly indebted to my advisor, Alexei Oblomkov, for his guidance, sup- port, and generosity. From him I have learned how to be a mathematician. For the many, many hours he has spent teaching me mathematics, and for the oppor- tunities he has given me to become a better researcher and teacher, I am extremely grateful. Without the love and support of my wife and partner in everything, Nikki, this thesis would not have been possible. She believes in me always, and inspires me to do my best in all of my endeavors. v ABSTRACT SKEIN THEORY AND ALGEBRAIC GEOMETRY FOR THE TWO-VARIABLE KAUFFMAN INVARIANT OF LINKS SEPTEMBER 2016 THOMAS SHELLY, B.S., SEATTLE UNIVERSITY M.S., PORTLAND STATE UNIVERSITY Ph.D., UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor Alexei Oblomkov We conjecture a relationship between the Hilbert schemes of points on a singular plane curve and the Kauffman invariant of the link associated to the singularity. Specifcally, we conjecture that the generating function of certain weighted Euler characteristics of the Hilbert schemes is given by a normalized specialization of the difference between the Kauffman and HOMFLY polynomials of the link. We prove the conjecture for torus knots. We also develop some skein theory for computing the Kauffman polynomial of links associated to singular points on plane curves. vi TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ::::::::::::::::::::::::::: v ABSTRACT ::::::::::::::::::::::::::::::::::: vi LIST OF FIGURES ::::::::::::::::::::::::::::::: viii CHAPTER 1. INTRODUCTION ::::::::::::::::::::::::::: 1 2. PLANE CURVE SINGULARITIES AND THE KAUFFMAN POLY- NOMIAL :::::::::::::::::::::::::::::::: 3 2.1 Punctual Hilbert schemes on singular plane curves :::::: 3 2.2 Conventions for the HOMFLY and Kauffman Polynomials :: 6 2.3 A Conjecture and Some Examples ::::::::::::::: 8 2.4 Conjecture 1 in context ::::::::::::::::::::: 12 2.5 Torus knots :::::::::::::::::::::::::::: 14 2.5.1 Computation of the Integral ::::::::::::::: 14 2.5.2 The difference of the Kauffman and HOMFLY polyno- mials for torus knots :::::::::::::::::: 20 3. SKEIN THEORY :::::::::::::::::::::::::::: 24 3.1 Definition of the BMW Algebra ::::::::::::::::: 25 3.2 Symmetrizers in the BMW Algebras :::::::::::::: 26 3.3 A New Recurrence for the Symmetrizers :::::::::::: 33 3.3.1 Symmetrizers in the Hecke Algebras of type A and the Temperley-Lieb Algebra :::::::::::::::: 39 3.4 The Two-Pointed Annulus :::::::::::::::::::: 39 3.5 The Structure of C0 :::::::::::::::::::::::: 44 3.6 Applications of the Two-Pointed Annulus Skein :::::::: 53 3.6.1 Closures and Traces of the Symmetrizers :::::::: 53 3.6.2 The Meridian Map and Decorated Hopf Links ::::: 59 BIBLIOGRAPHY :::::::::::::::::::::::::::::::: 66 vii LIST OF FIGURES Figure Page 1. Trefoil knot :::::::::::::::::::::::::::::::: 11 2. The Hopf link. ::::::::::::::::::::::::::::::: 12 3. Staircase representing a monomial ideal in the coordinate ring of Ck;n. 16 4. Staircase for the ideal (xy; x3) in the ring C[x; y]=(y2 − x3). :::::: 17 i 5. Tk ::::::::::::::::::::::::::::::::::::: 29 i 6. Wk ::::::::::::::::::::::::::::::::::::: 29 i i 7. WkPk;l ::::::::::::::::::::::::::::::::::: 29 8. A diagrammiatic view of equation (3.15) :::::::::::::::: 34 −1 9. A proof that hmσm−1 = hmhm−1σm . :::::::::::::::::: 35 10. Diagram for equation (3.18). ::::::::::::::::::::::: 36 11. Diagram for equation (3.19). ::::::::::::::::::::::: 36 12. New three-term recurrence for the symmetrizers in the BMW algebras. 37 13. The product xy in C0 ::::::::::::::::::::::::::: 40 14. Maps from B to C0 :::::::::::::::::::::::::::: 41 15. The identity e, and the elements t and t−1 of C0. :::::::::::: 42 16. A proof of Lemma 11. :::::::::::::::::::::::::: 42 17. A proof of (3.25) from Lemma 12. :::::::::::::::::::: 43 0 −i 18. A proof that ∆l(Wi) = t . :::::::::::::::::::::::: 43 19. The map ∆ : Bn !C taking X 2 Bn to its closure ∆(X) 2 C. :::: 45 20. The diagram of e ∗ ∆(X). :::::::::::::::::::::::: 46 viii 0 21. The diagram of Θ(∆l(X)) in the annulus C ::::::::::::::: 46 22. Turaev's generators of C. ::::::::::::::::::::::::: 46 0 −1 23. ∆l(σ1) :::::::::::::::::::::::::::::::::: 48 0 −1 24. A diagram of ∆l(σ1) after applying the skein relation to the crossing. 48 25. Diagram of (X ⊗ 1k+1)hn−k+j−1hn−k+j−2 ··· hn−kσn−k−1σn−k−2 ··· σ1 : 51 0 0 26. Recurrence relation for the ∆l(ai) 2 C ::::::::::::::::: 54 27. A proof of (3.29). ::::::::::::::::::::::::::::: 55 28. A proof of (3.30). ::::::::::::::::::::::::::::: 55 29. A proof of (3.31). ::::::::::::::::::::::::::::: 55 30. Diagram of ∆−(X) in C. ::::::::::::::::::::::::: 56 31. The map Θ applied to ai+1 :::::::::::::::::::::::: 57 0 32. The map Θ applied to ∆l(ai ⊗ 11). ::::::::::::::::::: 57 0 33. The map Θ applied to ∆u(1 ⊗ ai). :::::::::::::::::::: 57 34. A diagram of Γ(∆(X))) in C. :::::::::::::::::::::: 60 35. A diagram of the skein relation (3.42). ::::::::::::::::: 60 36. Basic skein relation in C0. ::::::::::::::::::::::::: 61 37. The symmetrizers are eigenvectors of the meridian map. ::::::: 62 ix C H A P T E R 1 INTRODUCTION In the study of link invariants, two central problems are that of computing the invariants and that of interpreting what the invariants may tell us about the link or about the mathematical objects from which the link arises. Some invariants, such as the fundamental group of the link, are inherently topological, whereas other more combinatorially presented invariants can have a more obscure meaning. Two link invariants which fall into the latter category are the HOMFLY polynomial and Kauffman polynomial. Each of these invariants (for suitable choices of variables) is an element of Z[a±; q±; (q − q−1)±] for which a topological interpretation in the general setting is not currently known. In this thesis, we make a contribution to both the computation and the interpretation of the Kauffman polynomial, with a particular emphasis on trying to understand the Kauffman polynomial of links that arise from singular points on plane curves. In algebraic geometry, knots and links arise in the neighborhood of singular points on complex hypersurfaces. In particular, one obtains a link by intersecting a curve with a small 3-sphere centered at a singular point of the curve. The topology of such links was studied in deatail by Milnor in [14]. Recent work [3] by Campillo, Delgado, and Gusein-Zade computes the Alexander polynomial of the link of a singular point from the ring of functions on the curve. In [20], Oblomkov and 1 Shende conjectured a relationship between the HOMFLY polynomial of the link of a singularity and certain topological invariants of algebro-geometric spaces associated to the singularity known as punctual Hilbert Schemes. In [13], Maulik proved the conjecture of Oblomkov and Shende. It is this proof which is the motivation for the present work. The central problem of this thesis is how to compute the Kauffman polynomial of the link of a singular plane curve from the algebraic geometry of the curve. The approach we take is two-fold, tackling the problem both from the point of view of the geometry of the curve and from the computational skein-theoretic framework of the knot invariants. On the algebraic geometry side of things, the bulk of the effort is in finding the appropriate spaces and the appropriate invariants of those spaces which will give data about the polynomial invariants of the link. To this end, and motivated by the results of [20], we study certain numerical invariants of Hilbert schemes of points on singular plane curves. This is the subject of Chapter 2 of this thesis. In Chapter 3, we move to the skein-theoretic viewpoint and develop for the Kauffman polynomial some of the analogous results from the theory of the HOMFLY polynomial that were needed in Maulik's proof of the conjecture of Oblomkov and Shende. 2 C H A P T E R 2 PLANE CURVE SINGULARITIES AND THE KAUFFMAN POLYNOMIAL In this chapter we conjecture a relationship between certain topological invari- ants of punctual Hilbert schemes on a singular plane curve and the Kauffman polynomial of the link associated to the singularity of the curve. We prove the conjecture for a class of knots known as torus knots. 2.1 Punctual Hilbert schemes on singular plane curves For the remainder we consider an algebraic plane curve C and a point p 2 C. The point p may be smooth, but the interesting geometry will arise in the case that C is singular at p, in which case we will take p to be the unique singular point on C.
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