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. 

With Tobias Ekholm, Pietro Longhi

ARXIV 2026

We relate the stability of knot invariants under twisting a pair of strands to the stability of symmetric quivers under unlinking (or linking) operation. Starting from the HOMFLY-PT skein relations, we confirm the stable growth of $Sym^r$-coloured HOMFLY-PT polynomials under the addition of a full twist to the knot. On the other hand, we show that symmetric quivers exhibit analogous stable growth under unlinking or linking of the quiver augmented with the extra node; in some cases this augmented quiver captures the spectrum of motivic Donaldson-Thomas invariants of all quivers in the sequence. Combining these two versions of the stable growth, we conjecture that performing a full twist on any knot corresponds to appropriate unlinking or linking of the corresponding augmented quiver -- this statement is an important step towards a direct definition of the knot-quiver correspondence based on the knot diagram. We confirm the conjecture for all twist knots, $(2,2p+1)$ torus knots, and all pretzel knots up to 15 crossings with an odd number of twists in each twist region.

With Piotr Kucharski, Dmitry Noshchenko, Ramadevi Pichai, Vivek Kumar Singh, Marko Stošić

ARXIV 2025