Degree
Doctor of Philosophy (PhD)
Department
Department of Mathematics
Document Type
Dissertation
Abstract
We develop space-level refinements of two categorified invariants, one in low-dimensional topology and one in graph theory, by constructing stable homotopy types associated to annular links and to planar trivalent graphs with perfect matchings, respectively. In the first project, we develop a space-level formulation of Khovanov skein homology by constructing a stable homotopy type for annular links. We explicitly define a cover functor from the Khovanov skein flow category to the cube flow category, thereby establishing the Khovanov skein spectrum, $\mathcal{X}_{Sk}(L)$. This spectrum extends the framework of Lipshitz and Sarkar’s Khovanov spectrum $\mathcal{X}_{Kh}(L)$ and provides new avenues for understanding transverse link invariants in the annular setting. Furthermore, we establish a map from the Khovanov spectrum to the Khovanov skein spectrum, which, at specific gradings, recovers the cohomotopy transverse invariant. In the second project, we develop a space-level refinement of the $2$-factor homology by constructing a stable homotopy type associated to a certain family $ \mathscr{G} $ of planar trivalent graphs equipped with perfect matchings. Specifically, we define a cover functor from the $2$-factor flow category $\mathscr{C}(\Gamma_{M})$ to the cube flow category $\mathscr{C}_{C}(n)$, where $\Gamma_{M}$ denotes the perfect matching graph associated to a planar trivalent graph $G$ with perfect matching $M$, with $(G,M) \in \mathscr{G}$. By applying the Cohen–Jones–Segal realization to the $2$-factor flow category, we obtain the $2$-factor spectrum $\mathcal{X}(\Gamma_{M})$. This spectrum serves as a space-level version of the $2$-factor homology, analogous to the Lipshitz–Sarkar Khovanov spectrum $\mathcal{X}_{Kh}(L)$ for links. We show that the cohomology of $\mathcal{X}(\Gamma_{M})$ with $\mathbb{Z}_{2}$-coefficients is isomorphic to the $2$-factor homology as defined by Baldridge. We prove that the stable homotopy type of the $2$-factor spectrum is an invariant of planar trivalent graphs $G$ equipped with perfect matchings $M$, whenever $(G, M) \in \mathscr{G}$. Furthermore, we show that the closed webs obtained by performing flattenings at each crossing of an oriented link diagram in the context of $ \mathfrak{sl}_3 $ link homology belong to the family $\mathscr{G}$. Thus, we obtain a stable homotopy type invariant for closed webs.
Date
3-27-2026
Recommended Citation
Bhattacharyya, Nilangshu, "Stable Homotopy Type Invariant" (2026). LSU Doctoral Dissertations. 7056.
https://repository.lsu.edu/gradschool_dissertations/7056
Committee Chair
Prof. Scott Baldridge
LSU Acknowledgement
1
LSU Accessibility Acknowledgment
1