Open strings on knot complements

Abstract

Using skein valued holomorphic curve counting techniques, we give a flow loop formula for the skein valued partition function of the Lagrangian knot complement of a fibered knot (of the $A$-model open topological strings with Lagrangian $A$-branes wrapping the complement) in the cotangent bundle of the three-sphere and in the resolved conifold. For torus knots we show that the partition function in the cotangent bundle localizes on two or three holomorphic annuli and give a corresponding generalized quiver structure for the partition function in the resolved conifold. We connect the formula to the augmentation curve, the representation variety of the knot contact homology algebra of the knot, generated by Reeb chords of its Legendrian conormal and with differential given by holomorphic disks interpolating between words of Reeb chords. The curve admits a quantization as a $q$-difference equation for the generating function of symmetrically colored HOMFLYPT-polynomials of the knot or, geometrically, for the $U(1)$-partition function of the knot conormal. For $(2,2p+1)$-torus knots we show that, after a change of variables, the partition function of the knot complement also satisfies this $q$-difference equation. This gives another geometrically defined coordinate chart for the $D$-module defined by the quantized augmentation polynomial.

Figures (18)

• Figure 1: The complement of a torus knot $T_{2,2p+1}$ with $p=3$, shown with opposite orientation (negative crossings), and the Hopf link formed by its flow loops.
• Figure 2: Augmentation curves feature several large volume regions in correspondence of punctures.
• Figure 3: Fibration of the 3-sphere by 2-tori, with a torus knot.
• Figure 4: Examples of torus knot complements, respectively $T_{2,7}$ and $T_{3,5}$ and corresponding Morse flow loops. Blue loops are elliptic critical points (minima) of the Morse function, red loops are hyperbolic (saddles).
• Figure 5: The $T_{2,7}$ torus knot with loop flow lines for $\alpha$ forming a negative Hopf link, after the simplification described in Proposition \ref{['prp:2,2p+1torusknot']}. The blue line corresponds to the elliptic flow loop while the red line corresponds to the hyperbolic flow loop. Note that orientation of $M_K$ here agrees with that of $S^3$. In particular the orientation of $(\lambda,\mu)$ on the boundary torus gives a normal vector that points outward from $M_K$.

Theorems & Definitions (20)

• Theorem 1.1
• Corollary 1.2
• Remark 2.1: Annuli from elliptic flow loops
• proof : Proof of Theorem \ref{['t:skeinvalued knot complement']}
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Remark 3.3: Coalescence phenomena for flow loops
• Remark 3.4