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A supergroup series for knot complements

John Chae

TL;DR

This work defines a three-variable knot invariant $F_K(y,z,q)$ for plumbed knot complements associated with the Lie superalgebra $sl(2|1)$, extending the non-semisimple invariant $\hat{Z}_{b,c}$ for closed 3-manifolds and generalizing the GM two-variable knot invariant. A partial surgery framework ties $F_K$ to $\hat{Z}_{b,c}$ via chamber-dependent expansions, and explicit torus-knot calculations illustrate the structure and boundary symmetry $F_K(1/y,1/z,q)=-F_K(y,z,q)$. The paper also develops a solid-torus computation, analyzes boundary actions, and proposes a TQFT-like interpretation in terms of a boundary state space and gluing, yielding a Dehn-surgery formula that produces $q$-series rather than single terms. The results reveal qualitative differences from GM’s theory and provide a roadmap toward a three-dimensional non-semisimple Spin$^c$ decorated TQFT, with open problems focusing on closed forms, algorithms, and extensions to broader plumbed geometries.

Abstract

We introduce a three variable series invariant $F_K (y,z,q)$ for plumbed knot complements associated with a Lie superalgebra $sl(2|1)$. The invariant is a generalization of the $sl(2|1)$-series invariant $\hat{Z}(q)$ for closed 3-manifolds introduced by Ferrari and Putrov and an extension of the two variable series invariant defined by Gukov and Manolescu (GM) to the Lie superalgebra. We derive a surgery formula relating $F_K (y,z,q)$ to $\hat{Z}(q)$ invariant. We find appropriate expansion chambers for certain infinite families of torus knots and compute explicit examples. Furthermore, we provide evidence for a non semisimple $Spin^c$ decorated TQFT from the three variable series. We observe that the super $F_K (y,z,q)$ itself and its results exhibit distinctive features compared to the GM series.

A supergroup series for knot complements

TL;DR

This work defines a three-variable knot invariant for plumbed knot complements associated with the Lie superalgebra , extending the non-semisimple invariant for closed 3-manifolds and generalizing the GM two-variable knot invariant. A partial surgery framework ties to via chamber-dependent expansions, and explicit torus-knot calculations illustrate the structure and boundary symmetry . The paper also develops a solid-torus computation, analyzes boundary actions, and proposes a TQFT-like interpretation in terms of a boundary state space and gluing, yielding a Dehn-surgery formula that produces -series rather than single terms. The results reveal qualitative differences from GM’s theory and provide a roadmap toward a three-dimensional non-semisimple Spin decorated TQFT, with open problems focusing on closed forms, algorithms, and extensions to broader plumbed geometries.

Abstract

We introduce a three variable series invariant for plumbed knot complements associated with a Lie superalgebra . The invariant is a generalization of the -series invariant for closed 3-manifolds introduced by Ferrari and Putrov and an extension of the two variable series invariant defined by Gukov and Manolescu (GM) to the Lie superalgebra. We derive a surgery formula relating to invariant. We find appropriate expansion chambers for certain infinite families of torus knots and compute explicit examples. Furthermore, we provide evidence for a non semisimple decorated TQFT from the three variable series. We observe that the super itself and its results exhibit distinctive features compared to the GM series.

Paper Structure

This paper contains 25 sections, 9 theorems, 187 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Plumbed manifolds
  4. Plumbed knot complements
  5. Invariance
  6. $Spin^c$ Structures on knot complements
  7. A supergroup series invariant of plumbed knot complements
  8. A partial surgery formula
  9. The solid torus
  10. The boundary action
  11. The three variable knot invariant
  12. Torus knots
  13. Plumbing graphs
  14. Chambers
  15. Examples
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $Y_1$ and $Y_2$ be knot complements represented by (weakly) negative definite plumbing graphs and $Y=Y_1 \cup_{T^2} Y_2$ be the result of gluing them along their common torus boundary. Let also $(b_1 , c_1)$ and $(b_2 , c_2)$ be relative $Spin^c$ structures of $Y_1$ and $Y_2$, respectively, whic where for any choice of chamber $\alpha_i , i=\pm$.

Figures (9)

  • Figure 1: Kirby-Neumann moves on plumbing trees. Move 1: blow up/down (left), move 2: absorption/desorption (middle), move 3: fusion/fission (right).
  • Figure 2: A plumbing graph $\Gamma$ of a knot $\subset S^3$ (left) and corresponding surgery link $L(\Gamma)$. The linking between two link components is the Hopf link. This link diagram can be transformed into a knot diagram through the Kirby moves.
  • Figure 3: Plumbing graphs of the solid torus $S_{p/r}$. The distinguished vertex is the first vertex as shown by an open circle. The ellipsis indicates intermediate vertices on the leg whose framing coefficients are determined by the continued fraction expansion of $p/r$ in Section 3.1.
  • Figure 4: Plumbing graphs of $T(2,2n+1)$ (left), $T(3,3n+1)$ (right) and, $T(3,3n+2)$ (bottom). The ellipsis indicates intermediate vertices with weight $-2$ along the legs. Total number of $-2$ vertices in succession on the leg is $n-1$ for $T(2,2n+1), T(3,3n+1)$ and $T(3,3n+2)$.
  • Figure 5: Changing the plumbing graph of $T(s,t) \subset S^3$ to $T(s,t) \subset \mathbb{Z} HS^3$. The graph without the distinguished vertex corresponds to a plumbing graph of $\mathbb{Z} HS^3$. The ellipsis indicates intermediate vertices.
  • ...and 4 more figures

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Conjecture 1.4
  • Conjecture 1.5
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 3.1
  • ...and 20 more