Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A categorification for the partial-dual genus polynomial

Zhiyun Cheng, Ziyi Lei

Abstract

The partial-dual genus polynomial $^\partial\varepsilon_G(z)$ of a ribbon graph $G$ is the generating function that enumerates all partial duals of $G$. In this paper, we give a categorification for this polynomial. The key ingredient of the construction is an extended Frobenius algebra related to unoriented topological quantum field theory.

A categorification for the partial-dual genus polynomial

Abstract

The partial-dual genus polynomial of a ribbon graph is the generating function that enumerates all partial duals of . In this paper, we give a categorification for this polynomial. The key ingredient of the construction is an extended Frobenius algebra related to unoriented topological quantum field theory.
Paper Structure (22 sections, 10 theorems, 46 equations, 13 figures)

This paper contains 22 sections, 10 theorems, 46 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Ribbon graphs and partial-dual genus polynomial
  3. Ribbon graphs and partial dual
  4. Partial-dual genus polynomial
  5. Graded partial-dual genus polynomial
  6. A punctured (1+1)D-TQFT
  7. Commutative $n$-cube category and four cubes
  8. Commutative n-cube category
  9. The anti-commutative $n$-cube $(S, s)$
  10. The commutative $n$-cube $(F, f)$
  11. The commutative $n$-cube $(V, v)$
  12. The commutative $n$-cube $(W, w)$
  13. Cochain complex
  14. An example
  15. Some applications
  16. ...and 7 more sections

Key Result

Lemma 2.2

For an arbitrary ribbon graph $G$, we have where $A^c$ denotes $E(G)\setminus A$.

Figures (13)

  • Figure 1: Three examples of ribbon graphs
  • Figure 2: Partial dual with respect to one edge-ribbon
  • Figure 3: A ribbon graph (indexed by $11$) and its three proper subgraphs
  • Figure 4: Basic cobordisms and corresponding maps
  • Figure 5: A $S$-cube
  • ...and 8 more figures

Theorems & Definitions (37)

  • Definition 2.1: GMT2020
  • Lemma 2.2
  • Example 2.3
  • Definition 2.4
  • Example 2.5
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 4.1
  • Definition 4.2
  • ...and 27 more