Ph.D Thesis
Ph.D Student Michael Brandenbursky Knot Invariants and their Applications to Constructions of Quasi-Morphisms on Groups Department of Mathematics Full Professor Polyak Michael Full Professor Entov Michael

Abstract

In this work we study knot and link invariants and their applications to constructions of quasi-morphisms on braids groups and the group  of area-preserving compactly supported diffeomorphisms of a two-dimensional open disc.

In the first part of the thesis, which is independent from the subsequent chapters, we give a sufficient criterion for a knot/link invariant to define quasi-morphisms on braid groups, provided a certain way of closing braids into knots or links. We then discuss a generalized Gambaudo-Ghys construction which allows to build quasi-morphisms on . In particular, we study quasi-morphisms on braid groups and  defined in this way by knot and link invariants coming from the knot Floer homology and Khovanov-type link homology. We also compute the values of quasi-morphisms obtained by this construction on the time-one flow of a generic time-independent Hamiltonian  in terms of the Reeb graph of .

In the second part of the thesis we consider link invariants arising from the Conway and HOMFLYPT polynomials. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting (with signs and multiplicities) subdiagrams of certain geometric-combinatorial types. These formulas generalize the calculation of a linking number by counting signs of crossings in a link diagram. Until recently, explicit formulas of this type were known only for few invariants of low degrees. We generalize the result of Chmutov-CapKhoury-Rossi and present Gauss diagram formulas for all coefficients of , where  is a link with arbitrary number of components, and  is the Conway polynomial of . We discuss an interesting interpretation of these formulas in terms of counting surfaces of a certain genus and with one boundary component. We also present two different extensions of the Conway polynomial to long virtual links. We compare these extensions with the existing extensions of the Alexander and Conway polynomials and show that they are new.

In the remaining part of this work we modify Chmutov-CapKhoury-Rossi construction and present Gauss diagram formulas for the coefficients of the first partial derivative of the HOMFLYPT polynomial, with respect to the variable , evaluated at . These formulas are related, in a similar way, to a certain count of orientable surfaces with two boundary components. At the end we present a modification of these formulas in case of knots.