Knot lattice homology and $q$-series invariants for plumbed knot complements

Abstract

We introduce an invariant of negative definite plumbed knot complements unifying knot lattice homology, due to Ozsváth, Stipsicz, and Szabó, and the BPS $q$-series of Gukov and Manolescu. This invariant is a natural extension of weighted graded roots of negative definite plumbed 3-manifolds introduced earlier by the first two authors and Krushkal. We prove a surgery formula relating our invariant with the weighted graded root of the surgered 3-manifold.

Figures (15)

• Figure 1: The weighted bigraded root of the trefoil at $t=1$ corresponding to the admissible family $\widehat{W}$ and $\varepsilon=1$.
• Figure 2: A plumbing $\Gamma$ and its associated framed link $\mathcal{L}(\Gamma)$. The $3$-manifold $Y(\Gamma)$ is the Brieskorn sphere $\Sigma(2,7,15)$.
• Figure 3: Two Neumann moves for negative definite plumbing graphs.
• Figure 4: A marked plumbing and its corresponding ambient and surgered plumbing graphs.
• Figure 5: Two Neumann moves involving the marked vertex for negative definite marked plumbing graphs.

Theorems & Definitions (76)

• Theorem 1.1: Invariance under Neumann moves; detailed version in Theorem \ref{['thm:invariance']}
• Theorem 1.2: Surgery formula; detailed version in Theorem \ref{['thm:p surgery']}
• Definition 2.1
• Remark 2.2
• Theorem 2.3: Neumann
• Remark 2.4
• Definition 2.5
• Definition 2.6
• Remark 2.7
• Remark 2.8