HEXAGONbeta

Marked Heegaard Boundary Modules: Filling, Joint Completion, and Integral Lattices

Primarily AI-generated textHuman understanding: some partsmath.GT — Geometric Topologymath.QA — Quantum Algebra

Contributed by Hao-Yu Sun ↗

Version 1 / Oct 18, 2026 / CC BY 4.0

Abstract

A marked handlebody state must retain its filling relations and coefficient control before complementary capping or root specialization. We construct boundary modules and regulated integral lattices that keep the disk and cyclotomic residues of one finite-stage representative together. Standard handle-attachment and recoupling results supply the geometric inputs; the comparison constructs endpoint lattices, separating transport, raw and saturated quotient towers, and exact cap domains. These objects supply the companion rootwise theory, while the two completed topologies remain distinct and root specialization remains restricted to its declared domain.

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 in ChatGPT and giving constant guidance over 10 days (accumulated runtime) through Codex conversations. Wolfram MCP was used for symbolic computations.

Tools used

OpenAI
ChatGPT
OpenAI
Codex
Wolfram
Mathematica

References

  1. M. Aganagic and S. Shakirov, Knot homology and refined Chern--Simons index , Commun. Math. Phys. 333 (2015), no. 1, 187--228; doi:10.1007/s00220-014-2197-4 (https://doi.org/10.1007/s00220-014-2197-4); preprint Knot homology from refined Chern--Simons theory , arXiv:1105.5117 (https://arxiv.org/abs/1105.5117).arXivDOI
  2. G. Alagic and C. Lo, Quantum invariants of 3-manifolds and NP vs. #P , Quantum Inf. Comput. 17 (2017), nos. 1--2, 125--146; doi:10.26421/QIC17.1-2-8 (https://doi.org/10.26421/QIC17.1-2-8); arXiv:1411.6049 (https://arxiv.org/abs/1411.6049).arXivDOI
  3. S. Arthamonov and Sh. Shakirov, Refined Chern--Simons theory in genus two , J. Knot Theory Ramifications 29 (2020), no. 7, 2050044; doi:10.1142/S0218216520500443 (https://doi.org/10.1142/S0218216520500443); arXiv:1504.02620 (https://arxiv.org/abs/1504.02620).arXivDOI
  4. H. Baik, I. Choi, and D. M. Kim, On the surjectivity of the symplectic representation of the mapping class group , Topology Appl. 322 (2022), 108334; doi:10.1016/j.topol.2022.108334 (https://doi.org/10.1016/j.topol.2022.108334); arXiv:2008.10142 (https://arxiv.org/abs/2008.10142).arXivDOI
  5. R. P. Bakshi and J. H. Przytycki, Kauffman bracket skein module of the connected sum of handlebodies: a counterexample , Manuscripta Math. 167 (2022), nos. 3--4, 809--820; doi:10.1007/s00229-021-01288-5 (https://doi.org/10.1007/s00229-021-01288-5); arXiv:2005.07750 (https://arxiv.org/abs/2005.07750).arXivDOI
  6. C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel, Topological quantum field theories derived from the Kauffman bracket , Topology 34 (1995), no. 4, 883--927; doi:10.1016/0040-9383(94)00051-4 (https://doi.org/10.1016/0040-9383(94)00051-4).DOI
  7. D. Bullock, C. Frohman, and J. Kania-Bartoszyńska, The Yang--Mills measure in the Kauffman bracket skein module , Comment. Math. Helv. 78 (2003), no. 1, 1--17; doi:10.1007/s000140300000 (https://doi.org/10.1007/s000140300000); arXiv:math/0005218 (https://arxiv.org/abs/math/0005218).arXivDOI
  8. D. Cheptea, K. Habiro, and G. Massuyeau, A functorial LMO invariant for Lagrangian cobordisms , Geom. Topol. 12 (2008), no. 2, 1091--1170; doi:10.2140/gt.2008.12.1091 (https://doi.org/10.2140/gt.2008.12.1091); arXiv:math/0701277 (https://arxiv.org/abs/math/0701277).arXivDOI
  9. S. Chun, S. Gukov, S. Park, and N. Sopenko, 3d--3d correspondence for mapping tori , J. High Energy Phys. 2020 (2020), no. 9, 152; doi:10.1007/JHEP09(2020)152 (https://doi.org/10.1007/JHEP09(2020)152); arXiv:1911.08456 (https://arxiv.org/abs/1911.08456).arXivDOI
  10. T. D. Cochran and P. Melvin, Quantum cyclotomic orders of 3-manifolds , Topology 40 (2001), no. 1, 95--125; doi:10.1016/S0040-9383(99)00056-7 (https://doi.org/10.1016/S0040-9383(99)00056-7); arXiv:math/9809129 (https://arxiv.org/abs/math/9809129).arXivDOI
Show all 33 referencesShow fewer references
  1. F. Costantino, Integrality of Kauffman brackets of trivalent graphs , Quantum Topol. 5 (2014), no. 2, 143--184; doi:10.4171/QT/49 (https://doi.org/10.4171/QT/49); arXiv:0908.0542 (https://arxiv.org/abs/0908.0542).arXivDOI
  2. F. Costantino and T. T. Q. Lê, Stated skein modules of 3-manifolds and TQFT , J. Inst. Math. Jussieu 24 (2025), no. 3, 663--703; doi:10.1017/S1474748024000550 (https://doi.org/10.1017/S1474748024000550); arXiv:2206.10906 (https://arxiv.org/abs/2206.10906).arXivDOI
  3. S. Garoufalidis and T. T. Q. Lê, Is the Jones polynomial of a knot really a polynomial? , J. Knot Theory Ramifications 15 (2006), no. 8, 983--1000; doi:10.1142/S0218216506004919 (https://doi.org/10.1142/S0218216506004919); arXiv:math/0601139 (https://arxiv.org/abs/math/0601139).arXivDOI
  4. S. Garoufalidis and T. T. Q. Lê, From 3-dimensional skein theory to functions near Q\mathbb Q , Ann. Inst. Fourier (Grenoble) 76 (2026), no. 4, 1473--1493; doi:10.5802/aif.3740 (https://doi.org/10.5802/aif.3740); arXiv:2307.09135v2 (https://arxiv.org/abs/2307.09135v2).arXivDOI
  5. P. Ghiggini and P. Lisca, Open book decompositions versus prime factorizations of closed, oriented 3-manifolds , Geom. Topol. Monogr. 19 (2015), 145--155; doi:10.2140/gtm.2015.19.145 (https://doi.org/10.2140/gtm.2015.19.145); arXiv:1407.2148 (https://arxiv.org/abs/1407.2148).arXivDOI
  6. P. M. Gilmer and G. Masbaum, Integral lattices in TQFT , Ann. Sci. Éc. Norm. Supér. (4) 40 (2007), no. 5, 815--844; doi:10.1016/j.ansens.2007.07.002 (https://doi.org/10.1016/j.ansens.2007.07.002); arXiv:math/0411029 (https://arxiv.org/abs/math/0411029).arXivDOI
  7. P. M. Gilmer and G. Masbaum, Integral topological quantum field theory for a one-holed torus , Pacific J. Math. 252 (2011), no. 1, 93--112; doi:10.2140/pjm.2011.252.93 (https://doi.org/10.2140/pjm.2011.252.93); preprint Integral TQFT for a one-holed torus , arXiv:0908.2796 (https://arxiv.org/abs/0908.2796).arXivDOI
  8. S. Gukov and C. Manolescu, A two-variable series for knot complements , Quantum Topol. 12 (2021), no. 1, 1--109; doi:10.4171/QT/145 (https://doi.org/10.4171/QT/145); arXiv:1904.06057v2 (https://arxiv.org/abs/1904.06057v2).arXivDOI
  9. S. Gukov, D. Pei, P. Putrov, and C. Vafa, BPS spectra and 3-manifold invariants , J. Knot Theory Ramifications 29 (2020), no. 2, 2040003; doi:10.1142/S0218216520400039 (https://doi.org/10.1142/S0218216520400039); arXiv:1701.06567v2 (https://arxiv.org/abs/1701.06567v2).arXivDOI
  10. S. Gukov, P. Putrov, and C. Vafa, Fivebranes and 3-manifold homology , J. High Energy Phys. 2017 (2017), no. 7, 071; doi:10.1007/JHEP07(2017)071 (https://doi.org/10.1007/JHEP07(2017)071); arXiv:1602.05302v1 (https://arxiv.org/abs/1602.05302v1).arXivDOI
  11. K. Habiro, A unified Witten--Reshetikhin--Turaev invariant for integral homology spheres , Invent. Math. 171 (2008), no. 1, 1--81; doi:10.1007/s00222-007-0071-0 (https://doi.org/10.1007/s00222-007-0071-0); arXiv:math/0605314 (https://arxiv.org/abs/math/0605314).arXivDOI
  12. S. K. Hansen, Reshetikhin--Turaev invariants of Seifert 3-manifolds and a rational surgery formula , Algebr. Geom. Topol. 1 (2001), 627--686; doi:10.2140/agt.2001.1.627 (https://doi.org/10.2140/agt.2001.1.627); arXiv:math/0111057 (https://arxiv.org/abs/math/0111057).arXivDOI
  13. Y. Koda, Automorphisms of the 3-sphere that preserve spatial graphs and handlebody-knots , Math. Proc. Cambridge Philos. Soc. 159 (2015), no. 1, 1--22; doi:10.1017/S0305004114000723 (https://doi.org/10.1017/S0305004114000723); arXiv:1106.4777 (https://arxiv.org/abs/1106.4777).arXivDOI
  14. J. Korinman and J. Murakami, Relating quantum character varieties and skein modules , preprint (2022), arXiv:2211.04252 (https://arxiv.org/abs/2211.04252).arXiv
  15. T. Kuriya, T. T. Q. Lê, and T. Ohtsuki, The perturbative invariants of rational homology 3-spheres can be recovered from the LMO invariant , J. Topol. 5 (2012), no. 2, 458--484; doi:10.1112/jtopol/jts010 (https://doi.org/10.1112/jtopol/jts010); arXiv:1005.3895 (https://arxiv.org/abs/1005.3895).arXivDOI
  16. R. Lawrence and D. Zagier, Modular forms and quantum invariants of 3-manifolds , Asian J. Math. 3 (1999), no. 1, 93--108; doi:10.4310/AJM.1999.v3.n1.a5 (https://doi.org/10.4310/AJM.1999.v3.n1.a5).DOI
  17. T. T. Q. Lê, Quantum invariants of 3-manifolds: integrality, splitting, and perturbative expansion , Topology Appl. 127 (2003), nos. 1--2, 125--152; doi:10.1016/S0166-8641(02)00056-1 (https://doi.org/10.1016/S0166-8641(02)00056-1); arXiv:math/0004099 (https://arxiv.org/abs/math/0004099).arXivDOI
  18. J. Lovejoy, R. Osburn, and M. Storzer, Bailey pairs, Eichler integrals and unified Witten--Reshetikhin--Turaev invariants , preprint (2026), arXiv:2605.15740 (https://arxiv.org/abs/2605.15740).arXiv
  19. J. Marché, Introduction to quantum representations of mapping class groups , in Topology and Geometry: A Collection of Essays Dedicated to Vladimir G. Turaev (A. Papadopoulos, ed.), IRMA Lect. Math. Theor. Phys. 33 , EMS Press, Berlin, 2021, pp. 109--130; doi:10.4171/IRMA/33-1/7 (https://doi.org/10.4171/IRMA/33-1/7); arXiv:1812.03888 (https://arxiv.org/abs/1812.03888).arXivDOI
  20. G. Masbaum and P. Vogel, 3-valent graphs and the Kauffman bracket , Pacific J. Math. 164 (1994), no. 2, 361--381; doi:10.2140/pjm.1994.164.361 (https://doi.org/10.2140/pjm.1994.164.361).DOI
  21. Y. Murakami, Witten--Reshetikhin--Turaev invariants and homological blocks for plumbed homology spheres , Commun. Number Theory Phys. 18 (2024), no. 2, 371--403; doi:10.4310/CNTP.2024.v18.n2.a3 (https://doi.org/10.4310/CNTP.2024.v18.n2.a3); arXiv:2205.01282 (https://arxiv.org/abs/2205.01282).arXivDOI
  22. J. H. Przytycki, Fundamentals of Kauffman bracket skein modules , Kobe J. Math. 16 (1999), no. 1, 45--66; archived version of record (https://doi.org/10.24546/E0003677); arXiv:math/9809113 (https://arxiv.org/abs/math/9809113).arXivDOI
  23. H.-Y. Sun, Integral root fibers and closed factorizations: Pairing maps, Rees covariance, caps, and sewing , unpublished manuscript (2026).

Version history

  1. v1Submitted by Hao-Yu SunInitial depositCurrentOct 18, 2026