Paper Citation Record · LEDGER

An Invariant for Triple-Point-Free Immersed Spheres

As of 12 August 2026, this Paper Citation Record lists 29 of 29 outbound references and 1 inbound Pith citation observation for arXiv:2506.21130.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
2506.21130 v2

Coverage vector

measured 29 of 29 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-06T22:45:40.559134Z

measured 30 of 30 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-12T06:34:41.77262+00:00

measured 1 of 1 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-08-06T22:00:37.914547Z

measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: pith, observed 2026-08-06T22:00:38.284919Z

Reference resolution

29 of 29 outbound references displayed

verified exact1
verified fuzzy14
unresolved14
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 015b23ec-de3f-4182-b13f-0ff55a34e3bb · outbound

This paper cites an unresolved cited work.

An Invariant for Triple-Point-Free Immersed Spheres Unresolved cited work

Reference 1

Resolution: unresolved; raw_fallback, observed 2026-08-06T22:45:42.716190Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation d24b6af1-42aa-4bcc-9bbc-86044cd6c9ed · outbound

This paper cites Triple points and singularities of projections of smoothly immersed surfaces.

An Invariant for Triple-Point-Free Immersed Spheres Triple points and singularities of projections of smoothly immersed surfaces

Reference 2

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:45:42.609946Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 74a4e1b9-b2d5-4236-911d-573d33a8178c · outbound

This paper cites Banchoff.

An Invariant for Triple-Point-Free Immersed Spheres Banchoff

Reference 3

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:45:42.482108Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 8736e971-e765-47f6-a572-f80f45e4604c · outbound

This paper cites A singular example for the W illmore flow.

An Invariant for Triple-Point-Free Immersed Spheres A singular example for the W illmore flow

Reference 4

Resolution: unresolved; no resolver link, observed 2026-08-06T22:45:38.376379Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 25234d25-f524-45a4-b85a-4c242bbaea38 · outbound

This paper cites On the self-intersections of an immersed sphere.

An Invariant for Triple-Point-Free Immersed Spheres On the self-intersections of an immersed sphere

Reference 5

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:45:42.378423Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 4653f40f-5e48-443b-9d82-92a3c1172358 · outbound

This paper cites an unresolved cited work.

An Invariant for Triple-Point-Free Immersed Spheres Unresolved cited work

Reference 6

Resolution: unresolved; no resolver link, observed 2026-08-06T22:45:38.655511Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 3e66baf8-42ce-425c-8d0a-18f9ce786fbd · outbound

This paper cites Chmutov, S.

An Invariant for Triple-Point-Free Immersed Spheres Chmutov, S

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:45:42.294956Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 4674b713-fd07-451e-94be-a61b1265f779 · outbound

This paper cites Goryunov.

An Invariant for Triple-Point-Free Immersed Spheres Goryunov

Reference 8

Resolution: unresolved; no resolver link, observed 2026-08-06T22:45:38.932689Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation e153a656-1420-402f-a949-d0e1b4c669b8 · outbound

This paper cites an unresolved cited work.

An Invariant for Triple-Point-Free Immersed Spheres Unresolved cited work

Reference 9

Resolution: unresolved; no resolver link, observed 2026-08-06T22:45:39.078719Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 21bd3824-743c-467a-9c41-bf270359ec3c · outbound

This paper cites an unresolved cited work.

An Invariant for Triple-Point-Free Immersed Spheres Unresolved cited work

Reference 10

Resolution: unresolved; raw_fallback, observed 2026-08-06T22:45:42.211226Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation e6abc707-bb06-4a71-a1e2-1556e551b394 · outbound

This paper cites an unresolved cited work.

An Invariant for Triple-Point-Free Immersed Spheres Unresolved cited work

Reference 11

Resolution: unresolved; no resolver link, observed 2026-08-06T22:45:39.344061Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation d794135d-55be-4702-8161-4903dece7d9f · outbound

This paper cites Interlacement of double curves of immersed spheres.

An Invariant for Triple-Point-Free Immersed Spheres Interlacement of double curves of immersed spheres

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:45:42.110051Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 043508cc-4d5d-4ca8-b22d-86811016f1ea · outbound

This paper cites On double points of immersions of spheres.

An Invariant for Triple-Point-Free Immersed Spheres On double points of immersions of spheres

Reference 13

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:45:42.006713Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 9ce6ab94-2a4f-4dd1-a17e-a84f446365b3 · outbound

This paper cites A new conformal invariant and its applications to the W illmore conjecture and the first eigenvalue of compact surfaces.

An Invariant for Triple-Point-Free Immersed Spheres A new conformal invariant and its applications to the W illmore conjecture and the first eigenvalue of compact surfaces

Reference 14

Resolution: unresolved; no resolver link, observed 2026-08-06T22:45:39.614824Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f3c93509-b5d7-41f5-88c8-fec727c3efe5 · outbound

This paper cites Every sphere eversion has a quadruple point.

An Invariant for Triple-Point-Free Immersed Spheres Every sphere eversion has a quadruple point

Reference 15

Resolution: unresolved; no resolver link, observed 2026-08-06T22:45:39.718110Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 139ec1cd-17db-409e-b512-d60a2b29885a · outbound

This paper cites The Willmore Energy Landscape of Spheres and Avoidable Singularities of the Willmore Flow.

An Invariant for Triple-Point-Free Immersed Spheres The Willmore Energy Landscape of Spheres and Avoidable Singularities of the Willmore Flow

Reference 16

Resolution: verified exact; local_arxiv, observed 2026-08-06T22:45:40.691505Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 68d325ff-428c-453b-88de-8df11a3eaf8b · outbound

This paper cites On the classification of germs of maps from R ^2 to R ^3.

An Invariant for Triple-Point-Free Immersed Spheres On the classification of germs of maps from R ^2 to R ^3

Reference 17

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:45:41.889039Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 3548b5f0-52a4-4a74-9a00-ef85d5796c2b · outbound

This paper cites Mayer and Gieri Simonett.

An Invariant for Triple-Point-Free Immersed Spheres Mayer and Gieri Simonett

Reference 18

Resolution: unresolved; no resolver link, observed 2026-08-06T22:45:39.906581Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 772d7c77-e278-4cb1-983f-e31328ae6b92 · outbound

This paper cites Quadruple points of regular homotopies of surfaces in 3-manifolds.

An Invariant for Triple-Point-Free Immersed Spheres Quadruple points of regular homotopies of surfaces in 3-manifolds

Reference 19

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:45:41.752873Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 8ed485fa-a69f-482f-bbcc-9dd9f97caf62 · outbound

This paper cites Automorphisms and embeddings of surfaces and quadruple points of regular homotopies.

An Invariant for Triple-Point-Free Immersed Spheres Automorphisms and embeddings of surfaces and quadruple points of regular homotopies

Reference 20

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:45:41.652822Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation b3609481-fae9-4f95-bfa3-831d22d15482 · outbound

This paper cites Finite order q -invariants of immersions of surfaces into 3-space.

An Invariant for Triple-Point-Free Immersed Spheres Finite order q -invariants of immersions of surfaces into 3-space

Reference 21

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:45:41.552321Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 2e6a63ff-d2a1-4b01-81dc-59f73640efa0 · outbound

This paper cites Order one invariants of immersions of surfaces into 3-space.

An Invariant for Triple-Point-Free Immersed Spheres Order one invariants of immersions of surfaces into 3-space

Reference 22

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:45:41.413591Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation af811ab0-67ca-49b0-8af8-400eeda48b71 · outbound

This paper cites Higher-order invariants of immersions of surfaces into 3-space.

An Invariant for Triple-Point-Free Immersed Spheres Higher-order invariants of immersions of surfaces into 3-space

Reference 23

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:45:41.347583Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 1e420d64-3371-472b-9dc7-074ddc34ffca · outbound

This paper cites Dissecting the 2-sphere by immersions.

An Invariant for Triple-Point-Free Immersed Spheres Dissecting the 2-sphere by immersions

Reference 24

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:45:41.233946Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 5a9c192b-7707-446a-b53c-9cc7f0f8c168 · outbound

This paper cites Oset Sinha and R.

An Invariant for Triple-Point-Free Immersed Spheres Oset Sinha and R

Reference 25

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:45:41.105850Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation a0dccd8b-05db-4864-bdee-a953307c641e · outbound

This paper cites an unresolved cited work.

An Invariant for Triple-Point-Free Immersed Spheres Unresolved cited work

Reference 26

Resolution: unresolved; raw_fallback, observed 2026-08-06T22:45:40.963846Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 4910778d-7278-49ff-ae4a-20534d051e34 · outbound

This paper cites A classification of immersions of the two-sphere.

An Invariant for Triple-Point-Free Immersed Spheres A classification of immersions of the two-sphere

Reference 27

Resolution: unresolved; no resolver link, observed 2026-08-06T22:45:40.398727Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 6995a482-653d-4b93-bb1f-edc5aaa22fe0 · outbound

This paper cites Diffeomorphisms of the 2 -sphere.

An Invariant for Triple-Point-Free Immersed Spheres Diffeomorphisms of the 2 -sphere

Reference 28

Resolution: unresolved; no resolver link, observed 2026-08-06T22:45:40.458479Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 4aa40c76-4d04-4d9e-bdb6-46583a37032b · outbound

This paper cites an unresolved cited work.

An Invariant for Triple-Point-Free Immersed Spheres Unresolved cited work

Reference 29

Resolution: unresolved; raw_fallback, observed 2026-08-06T22:45:40.842369Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Pith citing papers

Observation 6b6f57ea-ffa9-4fcd-bc11-ed7256a0e842 · inbound

The Willmore Energy Landscape of Spheres and Avoidable Singularities of the Willmore Flow cites this paper.

The Willmore Energy Landscape of Spheres and Avoidable Singularities of the Willmore Flow An Invariant for Triple-Point-Free Immersed Spheres

Reference 62

Resolution: verified exact; local_arxiv, observed 2026-08-06T22:00:38.352454Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.