arXiv subjects

Luke Conners

Publications and source records attributed to Luke Conners.

4 recordsLinked to original sources

arXiv2026

HOMFLY homology for oriented tangles

We introduce an extension of Khovanov and Rozansky's categorification of the HOMFLY polynomial to arbitrary oriented tangles. Unlike previous such extensions, ours is invariant under the full suite of oriented Reidemeister moves, including the "misoriented" Reidemeister IIb. For oriented links $\mathcal{L}$, our construction recovers Khovanov and Rozansky's construction while removing the preliminary step of presenting $\mathcal{L}$ as a braid closure.

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arXiv2025

Doubly-graded exponential growth of colored torus knot homology

We give an invariant construction of reduced HOMFLY homology for arbitrary links reduced at components of arbitrary color and prove some structural properties relating this invariant to unreduced HOMFLY homology. Combined with previous results, this gives a recursive formula for the reduced HOMFLY homology of colored positive torus knots and some colored positive torus links. We prove that after forgetting the quantum grading, the resulting doubly-graded invariant of positive torus knots grows exactly exponentially in the color, resolving a 2013 conjecture of Gorsky--Gukov--Stošić. Finally, we verify the ``color-shifting" conjecture of Gukov--Nawata--Saberi--Stošić--Sułkowski in many examples.

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arXiv2025

Fray functors and equivalence of colored HOMFLYPT homologies

We construct several families of functors on the homotopy category of singular Soergel bimodules that mimic cabling and insertion of column-colored projectors. We use these functors to identify the intrinsically-colored homology of Webster--Williamson and the projector-colored homology of Elias--Hogancamp for an arbitrary link, up to multiplication by a polynomial in the quantum degree $q$. Combined with the results of arXiv:2303.16271, this establishes parity results for the intrinsic column-colored homology of positive torus knots, partially resolving a conjecture of Hogancamp--Rose--Wedrich in arXiv:2107.09590.

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arXiv2024

Row-Column Mirror Symmetry for Colored Torus Knot Homology

We give a recursive construction of the categorified Young symmetrizer introduced by Abel-Hogancamp in arXiv:1510.05330 corresponding to the single-column partition. As a consequence, we obtain new expressions for the uncolored $y$-ified HOMFLYPT homology of positive torus links and the $y$-ified column-colored HOMFLYPT homology of positive torus knots. In the latter case, we compare with the row-colored homology of positive torus knots computed by Hogancamp-Mellit in arXiv:1909.00418, verifying the mirror symmetry conjectures of arXiv:1112.0030 and arXiv:1304.3481 in this case.

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