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Fixed-Level TQFT Transport of Marked Handleslide Coherence: Disjoint Cubes and a Three-Handle Pentagon

A mix of human-written and AI-generated textHuman understanding: some partsmath.GT — Geometric Topologymath.QA — Quantum Algebra

Contributed by Hao-Yu Sun ↗

Version 1 / Oct 16, 2026 / CC BY 4.0

Abstract

Fix an ordinary level K≥1K\geq1 and a fully specified family of pairwise disjoint positive handleslides on a compression body. The main question is how a literal geometric relation becomes a literal finite-level operator relation. We construct a finite reduced skein fibre and one normalized isomorphism $$ Q_{K,d}:\mathcal M_{K,d}\xrightarrow{\sim}\mathcal V^+_{K,d} $$ to the corresponding scalar-extended Walker--TQFT state space. Reusing this map at every vertex sends the supplied weight-zero slide cylinders to EKE_K-linear automorphisms: geometric squares become operator squares, finite slide orders give the same operator, and forward/reverse maps telescope for all declared phases and integer weights. The construction is fibrewise and makes no comparison between levels. These operator relations are transported images of the geometric relations proved in Papers~1 and~2; they are not new geometric coherence theorems. We then treat a fully specified positive three-handle pentagon. Its vertex-zero compression body admits a compatible fixed-level comparison core: a rational disk-kernel cap, a normalized complementary handlebody, a reduced spine model, and one universal-construction isomorphism QK,dq:MK,dq→∼VK,d+Q_{K,d}^{\mathfrak q}:\mathcal M_{K,d}^{\mathfrak q}\xrightarrow{\sim}\mathcal V^+_{K,d}. The geometric weight-zero Walker pentagon consequently has a literal operator image over EKE_K, with explicit inverse formulas and a fixed-basis matrix corollary. Different comparison cores are related by a typed comparison groupoid and give intertwined operator systems; no canonical comparison or literal equality between different coordinate choices is asserted.

Provenance statement

This results from a long-suspended (pre-COVID) project on how to construct the Z-hat invariant (by Gukov-Pei-Putrov-Vafa) via Heegaard splitting, and this manuscript lays one of several foundations. GPT-5.6, GPT-6, and GPT-6.1 were used. I used them by providing an initial roadmap on ChatGPT and constant guidance for over 10 days (accumulated runtime) via conversations through Codex.

Tools used

OpenAI
ChatGPT
OpenAI
Codex

References

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  2. Gilmer, Patrick M.; Masbaum, Gregor. Maslov index, lagrangians, mapping class groups and TQFT. Forum Mathematicum, vol. 25, no. 5, pp. 1067–1106. 2013. Equation and section locators refer to arXiv:0912.4706v3arXivDOI
  3. Masbaum, Gregor; Vogel, Pierre. 3-valent graphs and the Kauffman bracket. Pacific Journal of Mathematics, vol. 164, no. 2, pp. 361–381. 1994DOI
  4. Przytycki, Józef H. Fundamentals of Kauffman bracket skein modules. Kobe Journal of Mathematics, vol. 16, pp. 45–66. 1999arXivDOI
  5. Sun, Hao-Yu. Marked Positive Handleslide Pentagons in Compression Bodies: Relative Coherence. 2026. hexagon:2610.00227v1 (https://hexagonmath.org/2610.00227v1). Hexagon preprinthexagon
  6. Sun, Hao-Yu. Phase-Labeled Doubles and Disjoint Handleslide Cubes with Complete Markings. 2026. hexagon:2610.00251v1 (https://hexagonmath.org/2610.00251v1). Hexagon preprinthexagon
  7. Walker, Kevin. On Witten's 3-manifold invariants. 1991. Preliminary version no. 2 (1991); electronic version prepared 2001, minor changes 2003link

Version history

  1. v1Submitted by Hao-Yu SunInitial depositCurrentOct 15, 2026