Browsing by Subject "INVARIANTS"

Now showing 1 - 5 of 5
  • Thumbnail Image
    ItemOpen Access
    A slicing obstruction from the $\frac {10}{8}$ theorem
    (Proceedings of the American Mathematical Society, 2016-08-29) Donald, A; Vafaee, F
    © 2016 American Mathematical Society. From Furuta’s 10/8 theorem, we derive a smooth slicing obstruction for knots in S3 using a spin 4-manifold whose boundary is 0-surgery on a knot. We show that this obstruction is able to detect torsion elements in the smooth concordance group and find topologically slice knots which are not smoothly slice.
  • Thumbnail Image
    ItemOpen Access
    Berge–Gabai knots and L–space satellite operations
    (Algebraic & Geometric Topology, 2015-01-15) Hom, J; Lidman, T; Vafaee, F
    © 2014 Mathematical Sciences Publishers. All rights reserved. Let P(K) be a satellite knot where the pattern P is a Berge–Gabai knot (ie a knot in the solid torus with a nontrivial solid torus Dehn surgery) and the companion K is a nontrivial knot in S3. We prove that P(K) is an L–space knot if and only if K is an L–space knot and P is sufficiently positively twisted relative to the genus of K. This generalizes the result for cables due to Hedden [13] and Hom [17].
  • Thumbnail Image
    ItemOpen Access
    On the Knot Floer Homology of Twisted Torus Knots
    (International Mathematics Research Notices, 2015) Vafaee, F
    © 2014 The Author(s) 2014. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com. In this paper, we study the knot Floer homology of a subfamily of twisted (p,q) torus knots where q=plusmn;1 (mod p). Specifically, we classify the knots in this subfamily that admit L-space surgeries. To do calculations, we use the fact that these knots are (1,1) knots and, therefore, admit a genus one Heegaard diagram.
  • Thumbnail Image
    ItemOpen Access
    Satellites of Legendrian knots and representations of the Chekanov–Eliashberg algebra
    (Algebraic & Geometric Topology, 2013-08-01) Ng, L; Rutherford, D
    We develop a close relation between satellites of Legendrian knots in ℝ3and the Chekanov-Eliashberg differential graded algebra of the knot. In particular, we generalize the well-known correspondence between rulings of a Legendrian knot in ℝ3and augmentations of its DGA by showing that the DGA has finite-dimensional representations if and only if there exist certain rulings of satellites of the knot. We derive several consequences of this result, notably that the question of existence of ungraded finite-dimensional representations for the DGA of a Legendrian knot depends only on the topological type and Thurston-Bennequin number of the knot.
  • Thumbnail Image
    ItemOpen Access
    Topological strings, D-model, and knot contact homology
    (Advances in Theoretical and Mathematical Physics, 2014) Aganagic, M; Ekholm, T; Ng, L; Vafa, C
    © 2014 International Press. We study the connection between topological strings and contact homology recently proposed in the context of knot invariants. In particular, we establish the proposed relation between the Gromov- Witten disk amplitudes of a Lagrangian associated to a knot and augmentations of its contact homology algebra. This also implies the equality between the Q-deformed A-polynomial and the augmentation polynomial of knot contact homology (in the irreducible case). We also generalize this relation to the case of links and to higher rank representations for knots. The generalization involves a study of the quantum moduli space of special Lagrangian branes with higher Betti numbers probing the Calabi-Yau. This leads to an extension of SYZ, and a new notion of mirror symmetry, involving higher dimensional mirrors. The mirror theory is a topological string, related to D-modules, which we call the "D-model". In the present setting, the mirror manifold is the augmentation variety of the link. Connecting further to contact geometry, we study intersection properties of branches of the augmentation variety guided by the relation to D-modules. This study leads us to propose concrete geometric constructions of Lagrangian fillings for links. We also relate the augmentation variety with the large N limit of the colored HOMFLY, which we conjecture to be related to a Qdeformation of the extension of A-polynomials associated with the link complement.