Pith Number

pith:W4PVK2NZ

pith:2026:W4PVK2NZJFSDLWNQT2A6MOB7YN

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Stratification of $\mathrm{AGL}_r(\mathbb{C})$-representation varieties of twisted Hopf links

\'Angel Molina-Navarro

AGL_r(C) representation varieties of twisted Hopf link complements can be stratified using corresponding GL_r(C) varieties.

arxiv:2605.13585 v1 · 2026-05-13 · math.GT

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Claims

C1strongest claim

We provide a stratification of the AGL_r(C)-representation variety of the fundamental group of the complement of a twisted Hopf link in terms of a stratification of the corresponding GL_r(C)-representation variety. For ranks 1 and 2, we explicitly describe this stratification and compute the motives of these varieties in terms of the Lefschetz motive q=[C] in the Grothendieck ring of complex algebraic varieties K_0(Var_C).

C2weakest assumption

The assumption that the AGL_r(C)-representation variety can be stratified directly in terms of the GL_r(C) one, which relies on the specific structure of the fundamental group of the twisted Hopf link complement and properties of algebraic group representations.

C3one line summary

The AGL_r(C) representation varieties for twisted Hopf links are stratified using GL_r(C) varieties, with explicit descriptions and motives computed for ranks 1 and 2 in the Grothendieck ring.

References

32 extracted · 32 resolved · 0 Pith anchors

[1] Burde,Darstellungen von Knotengruppen.Mathematische Annalen173(1967), 24-33 1967

[2] Calleja,Configuration spaces of orbits and their Sn-equivariant E-polynomials.arXiv preprint: arXiv:2403.07765v2, 2024 2024

[3] D. Cooper, M. Culler , H. Gillet, D. D. Long, and P. B. Shalen,Plane curves associated to character varieties of3-manifolds.Inventiones mathematicae118(1994), 47-84 1994

[4] P. R. Cromwell,Knots and Links, Cambridge University Press, 2004 2004

[5] M. Culler and P. B. Shalen,Varieties of group representations and splitting of3-manifolds.Annals of Mathematics117(1983), n. 1, 109-146 1983

Formal links

1 machine-checked theorem link

Receipt and verification

First computed	2026-05-18T02:44:23.169596Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

b71f5569b9496435d9b09e81e6383fc3468a6acc0a2cd9aee065a58ad92a7f7f

Aliases

arxiv: 2605.13585 · arxiv_version: 2605.13585v1 · doi: 10.48550/arxiv.2605.13585 · pith_short_12: W4PVK2NZJFSD · pith_short_16: W4PVK2NZJFSDLWNQ · pith_short_8: W4PVK2NZ

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/W4PVK2NZJFSDLWNQT2A6MOB7YN \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: b71f5569b9496435d9b09e81e6383fc3468a6acc0a2cd9aee065a58ad92a7f7f

Canonical record JSON

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    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.GT",
    "submitted_at": "2026-05-13T14:21:34Z",
    "title_canon_sha256": "814b55b522f430df7f5a3755dca8de01d67a2116353d2bb1a8613a9edcc8ba13"
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