Short Accessible Descriptions of Preprints

The symplectic isotopy conjecture is a longstanding open problem that posits every smooth symplectic curve in the complex projective plane CP^2 is isotopic to an algebraic curve. The conjecture has been established for smooth symplectic curves of degrees 1 and 2 by Gromov, degree 3 by Sikorav, degree up to 6 by Shevchishin, and degree up to 17 by Siebert and Tian, all using Gromov's theory of pseudoholomorphic curves. The importance of the symplectic isotopy conjecture stems from the work of Auroux and Auroux-Katzarkov. The upshot of these works is that in principle, the classification of closed symplectic 4-manifolds reduces to the isotopy problem for symplectic curves in CP^2 with only nodal and cuspidal singularities. Furthermore, the classification of which closed symplectic 4-manifolds are complex projective surfaces reduces to the problem of which symplectic curves in CP^2 (with only nodal and cuspidal singularities) are isotopic to algebraic curves.

In my paper, I develop a new theory in the braid groups to completely classify isotopy classes of degree three symplectic curves in CP^2 with A_n-singularities (an A_n-singularity is locally modelled by the equation y^{n+1} = x^2 - e.g., an A_1-singularity is a node and an A_2-singularity is a cusp). The isotopy class of a symplectic curve in CP^2 with A_n-singularities is completely determined by its braid monodromy, which can be represented as a factorization of the full twist in the braid group into positive half-twists. My theory is the first instance of using purely algebraic techniques in the braid groups to establish results on the symplectic isotopy problem in CP^2, unifies and extends known results on isotopy classes of degree three symplectic curves with singularities by several authors (with a single method of proof), and entirely avoids Gromov's theory of pseudoholomorphic curves.
 

The HOMFLY-PT polynomial is a fundamental polynomial invariant of links in S^3  - the Alexander and Jones polynomials are specializations of the HOMFLY-PT polynomial. However, computing the HOMFLY-PT polynomial of a link directly in terms of a link diagram is very difficult in general because computing the HOMFLY-PT polynomial of a link with n crossings requires the application of up to 2^n Skein relations.

In my paper, I use the explicit characterization of the entries of Burau matrices of 4-braids I developed in paper 3. in order to establish the first simple explicit formulas for the coefficients of the HOMFLY-PT polynomial (and, in particular, the Alexander polynomial and Jones polynomial) for all closed 3-braids and generic closed 4-braids. The formulas lead to new infinite families of detection results for classical knot polynomials restricted to closed 4-braids and new results on the Fox trapezoidal conjecture (a longstanding problem since the 1960s that the coefficients of the Alexander polynomial of an alternating knot are trapezoidal). In particular, I establish that the Jones polynomial restricted to an explicit generic family of closed 4-braids detects the unknot (it is a well-known conjecture whether the Jones polynomial of a knot detects the unknot, and this is a novel approach to the conjecture via a deeper study of braid group representations).
 

The question of whether or not the Burau representation of the braid group B_4 is faithful is a longstanding open problem since the 1930s. The question is equivalent to whether B_4 is a group of 3 x 3 matrices over the real numbers. The faithfulness question is also closely related to the well-known open problem of whether the Jones polynomial of a knot detects the unknot (a negative answer to the faithfulness question would imply a negative answer to this question). The Burau representation is fundamental in low-dimensional topology: the classical Alexander polynomial can be defined completely in terms of the Burau representation and studying isotopy classes of curves in punctured disks is closely linked to the Burau representation.

In my paper, I develop a new theory to establish a strong generic faithfulness result for the Burau representation of the braid group B_4. I use my theory to broadly understand the Burau representation and obtain precise characterizations of the entries of Burau matrices. The theory represents the first major progress on the faithfulness question and explicitly characterizing the Burau representation of B_4 in the literature.
 

The Alexander and Jones polynomials are classical invariants to distinguish knots and links in 3-space. A natural question is to what extent these invariants fail to distinguish non-isotopic knots and links.

In my paper, I prove that the class numbers of quadratic number fields precisely measure this failure for links of braid index 3, and in doing so, I establish a concrete and novel connection between low-dimensional topology and algebraic number theory. In particular, the fact that the class number of Q(√-3) is 1 implies that the Jones polynomial detects the unknot for 3-braid links, and (a special case of) Minkowski's theorem on the finiteness of class groups implies that that there are finitely many 3-braid links with a given Jones polynomial!
 

The conjugacy problem in a group asks for an efficient algorithm to distinguish conjugacy classes in the group. For example, in the group of n x n invertible matrices over the complex numbers, the set of n eigenvalues together with their geometric multiplicities is a nice complete numerical invariant that distinguishes conjugacy classes. The conjugacy problem in the braid groups is fundamental in low-dimensional topology since it is closely intertwined with the isotopy problem for links in 3-space.

In my paper, I prove that continued fractions furnish a nice complete numerical invariant to distinguish conjugacy classes in the braid group B_3. I associate an infinite purely periodic continued fraction to each element of B_3 and I show that the period (up to cyclic reordering) is a complete conjugacy class invariant. A consequence is a linear time solution to the conjugacy problem in the braid group B_3.
 

In my PhD thesis, I use my theory (referred to in paper 1. above) to establish a strong (generic) constraint on the kernel of the Burau representation of the braid group B_4. The main result of my thesis is different to the main result in 1. and the proof uses different techniques.