| This thesis explores some mathematical applications of string dualities in open topological string theory and presents some new techniques for studying and computing open Gromov-Witten invariants. First, we prove a mild generalization of the gamma class formula of [BCR13], and show that it applies in two novel examples: the quintic threefold Q with Lagrangian given by the real quintic QR Q, and for Lagrangians LK ? X = O P1 (--1, --1) obtained from the conormal bundles of (r, s) torus knots K ? S3 via the conifold transition. Disk enumeration on (Q, Q R ) was first considered in [PSW08], and disk enumeration for (X, LK) was studied in winding-1 in [DSV13]. The gamma class formula agrees with the results of [DSV13] and [PSW08], and we generalize the formula of [DSV13] to arbitrary winding.;Next we study a relationship between mirror symmetry and knot contact homology described in [AENV14, AV12]. For knots K ? S 3 , large-N duality relates open Gromov-Witten theory on (X, L_K ) to SU (N) Chern-Simons theory on (S3, K). We use the conjecture of [AV12] to compute open Gromov-Witten invariants of (X, L K) through mirror symmetry in many examples, including several non-toric knots. We also find further evidence for this conjecture: for ( r, s) torus knots, we find a formula for the genus-0, 1-boundary-component, degree-d, winding-w open Gromov-Witten invariants of (X, LK ) using localization. This formula agrees with the results of the mirror symmetry calculation. Moreover, using this formula, we describe a method for obtaining the augmentation polynomial of a knot K from the open Gromov-Witten invariants of ( X, LK ). This method is shown to correctly recover the augmentation polynomial for the unknot and (3, 2) torus knot. |
