arXiv subjects

Takahito Kuriya

Publications and source records attributed to Takahito Kuriya.

5 recordsLinked to original sources

arXiv2025

The LMO Spectrum: Factorization Homology and the E_3-Structure of the Jacobi Diagram Algebra

We define the LMO spectrum, a categorification of the Le-Murakami-Ohtsuki (LMO) invariant for 3-manifolds, using factorization homology. The theoretical foundation is our main algebraic result (Theorem A): the algebra of Jacobi diagrams, $\AJac$, possesses a homotopy $E_3$-algebra structure. This is a necessary condition for consistency within factorization homology, and the proof relies on the formality of the little 3-disks operad. A universal surgery formula is derived from the excision axiom (Theorem B), providing a computational basis independent of conjectural models. As an application (Theorem C), we construct an ``$H_1$-decorated LMO invariant'' that distinguishes the lens spaces $L(156, 5)$ and $L(156, 29)$, a pair that the classical LMO invariant fails to separate.

math.GT↗

arXiv2025

An Intrinsic $L_{\infty}$-Algebra on the Khovanov-Sano Complex

This paper reinterprets the symmetries of equivariant Khovanov homology, discovered by Khovanov and Sano, within the Batalin-Vilkovisky (BV) formalism. We identify the Shumakovitch operator $\hatν$ as a BV Laplacian whose nilpotency, a consequence of the algebra's defining relations, induces an $L_{\infty}$-algebra on homology. We prove this structure is non-trivial through explicit computations of higher brackets. Furthermore, we construct a dual $L_{\infty}$-structure, suggesting a unifying homotopy $\mathfrak{sl}_2$ symmetry. The main result of this paper is to lift this structure from homology to the chain level. Applying the Homotopy Transfer Theorem, we construct an intrinsic $L_{\infty}$-algebra on the Khovanov-Sano complex, whose $\infty$-quasi-isomorphism class is a canonical link invariant. This provides a new algebraic framework in which we conjecture the origin of Steenrod operations in knot homology.

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arXiv2010

The perturbative invariants of rational homology 3-spheres can be recovered from the LMO invariant

We show that the perturbative ${\frak g}$ invariant of rational homology 3-spheres can be recovered from the LMO invariant for any simple Lie algebra ${\frak g}$, i.e, the LMO invariant is universal among the perturbative invariants. This universality was conjectured in [25]. Since the perturbative invariants dominate the quantum invariants of integral homology 3-spheres [13,14,15], this implies that the LMO invariant dominates the quantum invariants of integral homology 3-spheres.

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arXiv2008

On a Lomonaco-Kauffman conjecture

Samuel J. Lomonaco Jr and Louis H. Kauffman conjectured that tame knot theory and knot mosaic theory are equivalent. We give a proof of the Lomonaco-Kauffman conjecture.

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arXiv2008

On the LMO conjecture

We give a proof of the LMO conjecture which say that for any simply connectd simple Lie group $G$, the LMO invariant of rational homology 3-spheres recovers the perturvative invariant $τ^{PG}$. By Habiro-Le theorem, this implies that the LMO invariant is the universal quantum invariant of integral homology 3-spheres.

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