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Exotic knottings and symmetries of surfaces in 4-manifolds

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Pith's one-line read This paper claims that ambient symmetry loss can grade exotic knottedness of surfaces in 4-manifolds, with each successive rim surgery removing one more projective homological symmetry until none remain.

desk verdict Theorems A and B are solid and the rim Newton profile is a real new tool; Theorem C rests on an unverified reading of a cited lemma. read the letter →

arxiv 2607.27751 v1 pith:3FGQFHIQ submitted 2026-07-30 math.GT math.GN

R. Inanc Baykur , Nathan Sunukjian This is my paper · ORCID
classification math.GTmath.GN MSC 57K4057R58
keywords exoticknottedsurfaces4-manifoldsextendablemappingclassgroupsrimsurgeryrelativeSeiberg-WitteninvariantsNewtonpolytopesprojectiverigidityhyperbolic
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that exotic knottedness of surfaces in 4-manifolds can be quantified by the gradual loss of ambient symmetries. For every genus g, the authors construct 2g+1 embedded surfaces in a single simply connected 4-manifold that are all topologically isotopic and topologically flexible, yet each successive rim surgery removes one more projective homological symmetry from the smoothly extendable mapping class group, ending at a projectively rigid surface. A complementary hyperbolic construction yields a closed genus-40 totally geodesic surface whose smooth and topological extendable mapping class groups are both trivial. The method extracts convex-geometric data—Newton polytopes—from the relative Seiberg-Witten invariant and shows that the zonotopes added by rim surgery act as symmetry detectors. If correct, knottedness has finer gradations than 'knotted versus unknotted,' and ambient symmetry groups provide a quantitative probe of those gradations.

What carries the argument

The engine is the rim Newton profile: for each orbit of the rim-torus lattice in the affine exponent set of the relative Seiberg-Witten invariant, take the Newton polytope of the restricted Laurent polynomial, up to translation. Rim surgery—cutting out a torus neighborhood of a curve on the surface and gluing in the complement of a knot—changes the profile by Minkowski-adding a segment determined by the knot's symmetrized Alexander polynomial, so iterated surgery builds centrally symmetric zonotopes. The sign symmetry of the knot polynomial makes these zonotopes centrally symmetric, which is exactly why the final stabilizer contains ±I but, with an integral basis and pairwise distinct weight

What would settle it

Check the genus-1 model from Example 17: the final rectangle [−2a,2a]+[−4b,4b] must have stabilizer ±I in PSp(2,Z); if a smoothly extendable mapping class of the twice-rim-surgered torus acts by a symplectic matrix that does not preserve both primitive lines, the symmetry-breaking theorem fails. For the hyperbolic theorem, verify the quoted embedding lemma in a concrete case by exhibiting a totally geodesic embedding of the genus-40 surface into a closed hyperbolic 4-manifold; failure to produce such an embedding, or a nontrivial homeomorphism of the pair fixing the surface, would contradict T

Watch

Extended reading notes

Core claim

The central claim is that relative Seiberg-Witten invariants, read through Newton polytopes of their supports, detect ambient symmetries of embedded surfaces after rim surgery. Rim surgery along a curve multiplies the relative invariant by a knot polynomial evaluated at the rim-torus variable; the paper packages the invariant's support into a finite multiset of translation classes of polytopes, the rim Newton profile, which is invariant under diffeomorphisms of pairs. Starting from a smoothly flexible fiber of a full-monodromy genus-g surface fibration, whose rim Newton profile consists only of points, each of 2g carefully chosen rim surgeries adds a centered segment to every polytope in the

Load-bearing premise

The hyperbolic theorem rests on a quoted lemma, not reproduced in the paper, asserting that a compact arithmetic hyperbolic 3-manifold over a field different from the rationals embeds itself—not merely a finite cover—totally geodesically into a closed arithmetic hyperbolic 4-manifold; if that reading is wrong, only the genus-40 surface itself, not the rigid pair, is established.

Editorial extensions

If this is right

  • For every genus g, there exist 2g+1 pairwise nondiffeomorphic surface pairs in a simply connected 4-manifold that are all topologically isotopic, so a single topological isotopy class can contain arbitrarily long finite chains of exotic surfaces.
  • Knottedness becomes graded: each rim surgery provably removes one more primitive homology line from the possible smooth ambient symmetries, giving a quantitative filtration of the smooth pair.
  • Any surface with nonnegative self-intersection, simply connected complement, and nonzero relative Seiberg-Witten invariant admits a projectively rigid exotic copy; in genus one, only the hyperelliptic involution can survive.
  • A closed hyperbolic 4-manifold can contain a totally geodesic surface with no nontrivial smoothly or topologically extendable mapping classes, and every topologically isotopic copy of that surface is likewise rigid in both categories.
  • Rim Newton profiles distinguish pairs by affine dimension and lattice-point count, providing an effective computable invariant for detecting exotic knotted surfaces.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same profile technique should apply to any relative invariant with a product formula under a surgery operation: whenever an initial flexibility condition forces point polytopes, the stabilizer chain measures symmetry loss without needing the full invariant.
  • A natural test is whether the actual images ρ(E^∞(X_g,F_i)) are themselves nested, not merely the upper bounds P_i; finding an example where an extension realizes a transvection outside P_i would show whether the filtration is sharp.
  • Because the rim Newton profile forgets Laurent polynomial coefficients, it cannot distinguish a polytope from its negative; a coefficient-sensitive refinement could eliminate the residual ±I ambiguity and potentially produce smoothly rigid, topologically flexible surfaces, which the paper leaves open.
  • The hyperbolic construction suggests a broader source of rigid pairs: any closed arithmetic hyperbolic surface with trivial isometry group defined over a number field other than the rationals should embed totally geodesically into infinitely many closed hyperbolic 4-manifolds, yielding the same rigid surface in varying ambient manifolds.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper introduces a convex-geometric refinement of the relative Seiberg-Witten invariant, the "rim Newton profile," and uses it to study exotic knottings of surfaces in 4-manifolds through extendable mapping class groups. Theorem A states that, under hypotheses F^2≥0, π1(X\νF)=1, and nonzero relative invariant, iterated rim surgery produces a topologically isotopic, projectively rigid exotic copy. Theorem B, the central result, constructs for each genus g a 4-manifold X_g and surfaces F_0,...,F_{2g} that are mutually topologically isotopic, topologically flexible, pairwise nondiffeomorphic, and with successively constrained projective homological symmetry groups P_0⊋...⊋P_{2g}={1}. Theorem C uses hyperbolic geometry to produce a genus-40 totally geodesic surface with trivial smooth and topological extendable mapping class groups. An appendix shows that the relative diffeomorphism or homeomorphism group determines the pair.

Significance. If the main results hold, this is a significant contribution: Theorem B gives a quantitative, genus-uniform filtration of exotic knottedness, where each rim surgery kills one projective homological symmetry while preserving topological flexibility. The proof is unusually checkable: Lemma 5 (naturality of the rim Newton profile), Lemma 7 (stabilizer of a weighted zonotope is ±I), Lemma 8 (large-scale detection of the added zonotope), and Lemma 15 (strict filtration by symplectic transvections) are all proved carefully from stated assumptions, with no fitted parameters or circularity. The rim-surgery formula is cited explicitly to Fintushel-Stern, and the initial flexible fiber inputs are supplied by full-monodromy Lefschetz fibrations. The main caveat is Theorem C, which rests on a strong external lemma that is not quoted; this does not affect the central Theorem B but must be resolved.

major comments (2)
  1. [Section 2.4, Theorem A and Remark 18] The proof of Theorem C depends on a specific reading of [38, Lemma 5.1] that is not reproduced. The lemma must embed the given arithmetic hyperbolic manifold itself (not merely a finite cover) totally geodesically in a closed arithmetic hyperbolic manifold of one higher dimension, preserving compactness, and must apply twice: once to the genus-40 surface and once to the resulting compact 3-manifold. This is load-bearing for Theorem C and Remark 23. Please quote the lemma, state its hypotheses, and verify them for both applications, including the compactness claim. If [38] only provides finite-cover embeddings or noncompact outputs, Theorem C and Remark 23 do not follow as written.
  2. [Section 2.4, Theorem A and Remark 18] The topological-flexibility clause in Theorem A ('If F is ordinary...') is deferred to Pyronneau's unpublished preprint [45, Theorem 4.3]. Since the clause is part of a theorem statement, it should not rest on an unavailable manuscript without at least a precise statement of the cited theorem. Either incorporate a proof, or state Theorem A without this clause and record topological flexibility as conditional on [45].
minor comments (4)
  1. [Proof of Theorem 9 and Theorem 16] The displays labeled (2.5) and (3.3) appear inside proofs without being integrated into the global equation numbering; this is confusing for cross-referencing. Please renumber or use unnumbered displays.
  2. [Section 2.4, F^2>0 case] In the blow-up reduction, the verification that π1(\tilde X\ν\tilde F)=1 after blowing up n points on F is implicit. It is true, but should be stated explicitly, since the rim-surgery formula and Boyer's theorem are applied to the proper transform.
  3. [Section 3.1, Lemma 11] The notation 'W_g := h_g^2 = 1, h_g = ...' is compressed and potentially misleading. Spell out that the monodromy factorization is h_g^2=1 with h_g given by the displayed word.
  4. [Section 4, Lemma 22] Terms such as 'simplest type' and 'admissible ternary quadratic form' are used without definitions; please add precise references or definitions so the hypotheses of [38, Lemma 5.1] can be checked.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem B is derived from the external rim-surgery formula and independent lemmas; self-citations are corroborative only.

full rationale

The central derivation chain for Theorems A and B begins with the external Fintushel–Stern rim-surgery formula (Proposition 2), applies it iteratively via equation (2.2), and then uses the naturality of the relative Seiberg–Witten invariant (Lemma 5), the geometric stabilizer computation (Lemma 7), and the perturbation argument (Lemma 8) to obtain the symmetry bounds and nondiffeomorphism conclusion. The knots and rim curves are chosen so that the computed Newton profile is explicit; no parameter is fitted to the target result, and the symmetry bounds are consequences of the invariant calculation rather than being assumed. Topological isotopy and flexibility use Boyer’s theorem [9], with the self-citation [50] serving only as additional corroboration; the other self-citation [7] appears in a peripheral stabilization remark and is not load-bearing for the main theorems. The possible fragility in Theorem C concerns the external arithmetic embedding result [38, Lemma 5.1], whose exact statement is not reproduced; this is an external dependency risk, not circularity, and it does not affect Theorems A and B. No equation reduces to its own input, and no prediction is a renamed fit. Therefore no significant circularity is present.

Assumptions & free parameters 3 free parameters · 8 assumptions · 1 invented entities

The central constructions rest on standard external theorems (rim-surgery formula, nonvanishing of relative SW invariants, Boyer's isotopy theorem, Mostow rigidity/Dehn-Nielsen-Baer) plus two fragile inputs: Pyronneau's unpublished extension theorem and the self-embedding reading of the Martelli-Riolo-Slavich lemma. The only hand-chosen parameters are the distinct zonotope weights d_i and the large scale m; these are construction choices, not fitted values.

free parameters (3)
  • d_1,...,d_{2g} = pairwise distinct positive integers (e.g., 1,...,2g)
    Chosen weights so the zonotope Z=Σ[-d_i v_i,d_i v_i] has pairwise distinct edge lengths; Lemma 7 uses this to force a symplectic stabilizer of {±I}.
  • m (rim-surgery scale) = any sufficiently large positive integer
    Scale of torus-knot Alexander exponents 2md_i. Lemma 8 requires m large enough for the added zonotope to dominate the initial Newton profile; lattice-point counting in Theorem 9 also uses m→∞.
  • Integral basis {v_1,...,v_{2g}} = a_1, b_1+b_2, ..., a_g, b_g (from Lemma 7)
    Constructed so the intersection graph is a path and the zonotope is full-dimensional; the stabilization filtration P_i = ∩_{j≤i} Stab(ℓ_j) is defined by these lines.
assumptions (8)
  • domain assumption Fintushel–Stern relative Seiberg–Witten rim-surgery formula and its naturality under diffeomorphisms (Prop. 2).
    External theorem cited to [13,14]; the rim Newton profile and the filtrations in Theorems A/B depend on it.
  • domain assumption Nonvanishing of relative Seiberg–Witten invariant for symplectic primitively embedded surfaces.
    Invoked in Lemma 11 for (X_g,F_0) to guarantee f_{X,F} ≠ 0, citing [13, Thm 1.1][14].
  • domain assumption Boyer's theorem: genus, homology class, and simply connected complement imply topological isotopy.
    Lemma 3 uses it to identify topological extendable groups of rim-surgered surfaces; cited to [9, Theorem F].
  • domain assumption Pyronneau's extension theorem for ordinary surfaces with simply connected complement.
    Remark 18, citing unpublished preprint [45, Theorem 4.3], supplies topological flexibility in Theorem A's ordinary case.
  • domain assumption Martelli–Riolo–Slavich Lemma [38, Lemma 5.1]: arithmetic hyperbolic n-manifold over k≠Q embeds itself totally geodesically into a closed arithmetic hyperbolic (n+1)-manifold.
    Lemma 22 applies it twice to embed the genus-40 surface in a hyperbolic 4-manifold.
  • domain assumption Maclachlan's torsion-free maximal arithmetic Fuchsian group over a totally real cubic field with quotient genus 40.
    Theorem C; the maximality gives trivial normalizer and hence trivial orientation-preserving isometry group; cited to [35, Thm 3.1, Ex 6.1].
  • standard math Mostow rigidity + Dehn–Nielsen–Baer identify extendable mapping classes of totally geodesic surfaces with restrictions of ambient isometries.
    Proposition 21; standard theorems in hyperbolic geometry.
  • standard math Surjectivity of Mod(Σ_g) → Sp(2g,Z).
    Used in Proposition 13 to show a smoothly flexible surface's rim Newton profile polytopes are points.
invented entities (1)
  • rim Newton profile N_R(f_{X,F})
    purpose: Multiset of translation classes of Newton polytopes of orbit polynomials of the relative SW invariant; its Minkowski growth under rim surgery yields projective symmetry obstructions and pair-distinguishing invariants.
    Introduced in §2.3; naturality and surgery behavior are proved in Lemma 5. As a mathematical invariant it has no external falsifiable handle.

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Pith. "Pith review of Exotic knottings and symmetries of surfaces in 4-manifolds." pith.science (2026). https://pith.science/paper/3FGQFHIQ

@misc{pith2026260727751,
  author       = {Pith},
  title        = {Pith review of: Exotic knottings and symmetries of surfaces in 4-manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3FGQFHIQ}},
  note         = {Machine review of arXiv:2607.27751}
}
read the original abstract

We study exotic knottings of surfaces in 4-manifolds through their ambient symmetries. We first give a general recipe for producing projectively rigid surfaces, for which every smoothly extendable self-diffeomorphism acts on first homology by plus or minus the identity. For every integer g >0, a refinement of this construction yields a finite sequence of genus-g surfaces F_0, ..., F_2g contained in a 4-manifold X_g. These surfaces are topologically isotopic and topologically flexible: every orientation-preserving self-diffeomorphism of F_i can be realized by a self-homeomorphism of X_g preserving F_i. Successive knotting, however, rules out increasingly many projective homological symmetries, revealing a finer knottedness phenomenon. The first two constructions combine iterated rim surgery with the convex geometry of Newton polytopes of relative Seiberg-Witten invariants. We also use hyperbolic geometry to construct a totally geodesic surface of positive genus whose smooth and topological extendable mapping class groups are both trivial.

Figures

Figures reproduced from arXiv: 2607.27751 by the authors.

Figure 1
Figure 1. Supports (blue points) and their Newton polytopes for the orbit polynomials associated to (E(2), Fi), i = 0, 1, 2. The labels below record their stabilizers in PSp(2, Z). Consequently, Newt(fλ) = {0}, Newt(f (1) λ ) = [−2a, 2a], and Newt(f (2) λ ) = [−2a, 2a] + [−4b, 4b] = 2Z, Z = [−a, a] + [−2b, 2b]. Set P = PSp(2, Z). The point {0} is preserved by all of P, whereas the segment [−2a, 2a] has stabilizer P1 = StabP(Z… view at source ↗

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Works this paper leans on

54 extracted references · 6 linked inside Pith

  1. [7]

    ˙Inan¸ c Baykur and Nathan Sunukjian

    R. ˙Inan¸ c Baykur and Nathan Sunukjian. Knotted surfaces in 4-manifolds and stabilizations.J. Topol., 9(1):215–231, 2016

    2016

  2. [50]

    Sunukjian

    Nathan S. Sunukjian. Surfaces in 4-manifolds: Concordance, isotopy, and surgery.Int. Math. Res. Not. IMRN, 2015(17):7950–7978, 2015

    2015

  3. [38]

    Compact hyperbolic manifolds without spin structures.Geom

    Bruno Martelli, Stefano Riolo, and Leone Slavich. Compact hyperbolic manifolds without spin structures.Geom. Topol., 24(5):2647–2674, 2020

    2020

  4. [45]

    Knotted surfaces with simply-connected complements, 2026

    Audrick Pyronneau. Knotted surfaces with simply-connected complements, 2026

    2026

  5. [1]

    Smoothly knotted surfaces that remain distinct after many internal stabilizations, 2023

    Dave Auckly. Smoothly knotted surfaces that remain distinct after many internal stabilizations, 2023. Preprint, arXiv:2307.16266

    arXiv 2023

  6. [2]

    An elementary introduction to modern convex geometry

    Keith Ball. An elementary introduction to modern convex geometry. In Silvio Levy, editor,Flavors of Geometry, volume 31 ofMathematical Sciences Research Institute Publications, pages 1–58. Cambridge University Press, 1997

    1997

  7. [3]

    Monodromy and vanishing cycles for sufficiently ample linear systems on simply connected surfaces, 2025

    Ishan Banerjee and Nick Salter. Monodromy and vanishing cycles for sufficiently ample linear systems on simply connected surfaces, 2025. Preprint, arXiv:2512.04018

    arXiv 2025

  8. [4]

    An adjunction inequality obstruction to isotopy of embedded surfaces in 4-manifolds.Math

    David Baraglia. An adjunction inequality obstruction to isotopy of embedded surfaces in 4-manifolds.Math. Res. Lett., 31(2):329–352, 2024

    2024

Show all 54 references
  1. [5]

    ˙Inan¸ c Baykur and Noriyuki Hamada

    R. ˙Inan¸ c Baykur and Noriyuki Hamada. Lefschetz fibrations with arbitrary signature.J. Eur. Math. Soc. (JEMS), 26(8):2837–2895, 2024

    2024

  2. [6]

    ˙Inan¸ c Baykur, Mustafa Korkmaz, and Jonathan Simone

    R. ˙Inan¸ c Baykur, Mustafa Korkmaz, and Jonathan Simone. Geography of symplectic Lefschetz fibrations and rational blowdowns.Trans. Amer. Math. Soc., 377(10):6771–6792, 2024

    2024

  3. [8]

    Ben Ami and Matatyahu Rubin

    E. Ben Ami and Matatyahu Rubin. On the reconstruction problem for factorizable homeomorphism groups and foliated manifolds.Topology and its Applications, 157(9):1664–1679, 2010

    2010

  4. [9]

    Realization of simply-connected 4-manifolds with a given boundary.Comment

    Steven Boyer. Realization of simply-connected 4-manifolds with a given boundary.Comment. Math. Helv., 68(1):20–47, 1993

    1993

  5. [10]

    Spin structures and codimension-two homeomorphism extensions

    Fan Ding, Yi Liu, Shicheng Wang, and Jiangang Yao. Spin structures and codimension-two homeomorphism extensions. Math. Res. Lett., 19(2):345–357, 2012

    2012

  6. [11]

    Princeton University Press, Princeton, NJ, 2012

    Benson Farb and Dan Margalit.A Primer on Mapping Class Groups, volume 49 ofPrinceton Mathematical Series. Princeton University Press, Princeton, NJ, 2012

    2012

  7. [12]

    R. P. Filipkiewicz. Isomorphisms between diffeomorphism groups.Ergodic Theory and Dynamical Systems, 2(2):159– 171, 1982

    1982

  8. [13]

    Ronald Fintushel and Ronald J. Stern. Surfaces in 4-manifolds.Math. Res. Lett., 4(6):907–914, 1997

    1997

  9. [14]

    Ronald Fintushel and Ronald J. Stern. Surfaces in 4-manifolds: Addendum, 2005. Preprint, arXiv:math/0511707

    2005 arXiv

  10. [15]

    Ronald Fintushel and Ronald J. Stern. Six lectures on four 4–manifolds. In Tomasz S. Mrowka and Peter S. Ozsv´ ath, editors,Low Dimensional Topology, volume 15 ofIAS/Park City Math. Ser., pages 265–315. American Mathematical Society, Providence, RI, 2009

    2009

  11. [16]

    Gompf and Andr´ as I

    Robert E. Gompf and Andr´ as I. Stipsicz. 4–Manifolds and Kirby Calculus, volume 20 ofGraduate Studies in Mathe- matics. American Mathematical Society, Providence, RI, 1999

    1999

  12. [17]

    On diffeomorphisms overT 2-knots.Proc

    Susumu Hirose. On diffeomorphisms overT 2-knots.Proc. Amer. Math. Soc., 119(3):1009–1018, 1993

    1993

  13. [18]

    On diffeomorphisms over surfaces trivially embedded in the 4-sphere.Algebr

    Susumu Hirose. On diffeomorphisms over surfaces trivially embedded in the 4-sphere.Algebr. Geom. Topol., 2:791–824, 2002

    2002

  14. [19]

    Surfaces in the complex projective plane and their mapping class groups.Algebr

    Susumu Hirose. Surfaces in the complex projective plane and their mapping class groups.Algebr. Geom. Topol., 5:577–613, 2005

    2005

  15. [20]

    Surfaces in 4-manifolds and their mapping class groups.Topology, 47(1):41–50, 2008

    Susumu Hirose and Akira Yasuhara. Surfaces in 4-manifolds and their mapping class groups.Topology, 47(1):41–50, 2008

    2008

  16. [21]

    Humphries

    Stephen P. Humphries. Generators for the mapping class group. In Roger Fenn, editor,Topology of Low-Dimensional Manifolds, volume 722 ofLecture Notes in Mathematics, pages 44–47. Springer, Berlin, 1979. Proceedings of the Second Sussex Conference, Chelwood Gate, 1977

    1979

  17. [22]

    Dehn-surgery along a torusT 2-knot.Pacific J

    Zyun’iti Iwase. Dehn-surgery along a torusT 2-knot.Pacific J. Math., 133(2):289–299, 1988

    1988

  18. [23]

    Chicago Lectures in Mathematics

    Svetlana Katok.Fuchsian Groups. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1992

    1992

  19. [24]

    Modifying surfaces in 4–manifolds by twist spinning.Geom

    Hee Jung Kim. Modifying surfaces in 4–manifolds by twist spinning.Geom. Topol., 10:27–56, 2006

    2006

  20. [25]

    Smooth surfaces with non-simply-connected complements.Algebr

    Hee Jung Kim and Daniel Ruberman. Smooth surfaces with non-simply-connected complements.Algebr. Geom. Topol., 8(4):2263–2287, 2008

    2008

  21. [26]

    Topological triviality of smoothly knotted surfaces in 4–manifolds.Trans

    Hee Jung Kim and Daniel Ruberman. Topological triviality of smoothly knotted surfaces in 4–manifolds.Trans. Amer. Math. Soc., 360(11):5869–5881, 2008

    2008

  22. [27]

    A comparative review of recent researches in geometry.Bulletin of the New York Mathematical Society, 2(10):215–249, 1893

    Felix Klein. A comparative review of recent researches in geometry.Bulletin of the New York Mathematical Society, 2(10):215–249, 1893. Translated by M. W. Haskell

  23. [28]

    A note on the topology of Lefschetz fibrations, 2025

    Sierra Knavel. A note on the topology of Lefschetz fibrations, 2025

    2025

  24. [29]

    Reid, and Leone Slavich

    Alexander Kolpakov, Alan W. Reid, and Leone Slavich. Embedding arithmetic hyperbolic manifolds.Math. Res. Lett., 25(4):1305–1328, 2018

    2018

  25. [30]

    Surfaces in 4-manifolds and extendible mapping classes, 2025

    Shital Lawande and Kuldeep Saha. Surfaces in 4-manifolds and extendible mapping classes, 2025. Preprint, arXiv:2502.17640

    2025 arXiv

  26. [31]

    Flexible surfaces inCP 2 andS 2 ×S 2, 2026

    Joshua Lehman. Flexible surfaces inCP 2 andS 2 ×S 2, 2026. Preprint, arXiv:2602.14753

    2026

  27. [32]

    Flexible algebraic curves and fake vanishing cycles

    Joshua Lehman and Tudur Lewis. Flexible algebraic curves and fake vanishing cycles. Preprint in preparation, 2026

    2026

  28. [33]

    Knotted surfaces, homological norm and extendable subgroup.Topology Appl., 377:Paper No

    Qiling Liu. Knotted surfaces, homological norm and extendable subgroup.Topology Appl., 377:Paper No. 109644, 2026

    2026

  29. [34]

    On slope genera of knotted tori in 4-space.Pacific J

    Yi Liu, Yi Ni, Hongbin Sun, and Shicheng Wang. On slope genera of knotted tori in 4-space.Pacific J. Math., 261(1):117–144, 2013. EXOTIC KNOTTINGS AND SYMMETRIES OF SURF ACES IN 4-MANIFOLDS 23

    2013

  30. [35]

    Existence and non-existence of torsion in maximal arithmetic Fuchsian groups.Groups Complex

    Colin Maclachlan. Existence and non-existence of torsion in maximal arithmetic Fuchsian groups.Groups Complex. Cryptol., 1(2):287–295, 2009

    2009

  31. [36]

    Reid.The Arithmetic of Hyperbolic3–Manifolds, volume 219 ofGraduate Texts in Mathematics

    Colin Maclachlan and Alan W. Reid.The Arithmetic of Hyperbolic3–Manifolds, volume 219 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 2003

    2003

  32. [37]

    Automatic continuity for homeomorphism groups and applications.Geometry & Topology, 20(5):3033– 3056, 2016

    Kathryn Mann. Automatic continuity for homeomorphism groups and applications.Geometry & Topology, 20(5):3033– 3056, 2016. With an appendix by Fr´ ed´ eric Le Roux and Kathryn Mann

    2016

  33. [39]

    McMullen and Clifford H

    Curtis T. McMullen and Clifford H. Taubes. 4-manifolds with inequivalent symplectic forms and 3-manifolds with inequivalent fibrations.Math. Res. Lett., 6(5–6):681–696, 1999

    1999

  34. [40]

    A gauge theoretic invariant of embedded surfaces in 4-manifolds and exoticP 2-knots.Ann

    Jin Miyazawa. A gauge theoretic invariant of embedded surfaces in 4-manifolds and exoticP 2-knots.Ann. of Math. (2), 2026. To appear; arXiv:2312.02041

    2026 arXiv

  35. [41]

    Montesinos

    Jos´ e M. Montesinos. On twins in the four-sphere. i.Quart. J. Math. Oxford Ser. (2), 34(2):171–199, 1983

    1983

  36. [42]

    G. D. Mostow.Strong Rigidity of Locally Symmetric Spaces, volume 78 ofAnnals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1973

    1973

  37. [43]

    Extendable mapping classes of knotted surfaces obtained by rim surgery inS 4, 2026

    Weizhe Niu. Extendable mapping classes of knotted surfaces obtained by rim surgery inS 4, 2026. Preprint, arXiv:2605.31383

    2026 arXiv

  38. [44]

    Pancholi and Francisco Presas

    Dishant M. Pancholi and Francisco Presas. Symplectic embeddings of 4–manifolds via Lefschetz fibrations, 2021. Preprint, arXiv:2110.12950

    2021 arXiv

  39. [46]

    On admissible groups of diffeomorphisms.Rend

    Tomasz Rybicki. On admissible groups of diffeomorphisms.Rend. Circ. Mat. Palermo (2) Suppl., 46:139–146, 1997

    1997

  40. [47]

    Monodromy and vanishing cycles in toric surfaces.Invent

    Nick Salter. Monodromy and vanishing cycles in toric surfaces.Invent. Math., 216(1):153–213, 2019

    2019

  41. [48]

    On the monodromy group of the family of smooth quintic plane curves.Glasgow Math

    Nick Salter. On the monodromy group of the family of smooth quintic plane curves.Glasgow Math. J., 67(2):163–184, 2025

    2025

  42. [49]

    American Mathematical Society, Providence, RI, 1996

    Bernd Sturmfels.Gr¨ obner Bases and Convex Polytopes, volume 8 ofUniversity Lecture Series. American Mathematical Society, Providence, RI, 1996

    1996

  43. [51]

    Homotopy K3’s with several symplectic structures.Geom

    Stefano Vidussi. Homotopy K3’s with several symplectic structures.Geom. Topol., 5:267–285, 2001

    2001

  44. [52]

    Extending periodic maps on surfaces over the 4-sphere.J

    Shicheng Wang and Zhongzi Wang. Extending periodic maps on surfaces over the 4-sphere.J. Topol. Anal., 16(4):641– 660, 2024

    2024

  45. [53]

    Whittaker

    James V. Whittaker. On isomorphic groups and homeomorphic spaces.Ann. of Math. (2), 78(1):74–91, 1963

    1963

  46. [54]

    Ziegler.Lectures on Polytopes, volume 152 ofGraduate Texts in Mathematics

    G¨ unter M. Ziegler.Lectures on Polytopes, volume 152 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 1995. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA Email address:inanc.baykur@umass.edu Department of Mathematics ...

    1995

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.