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arxiv: 2602.13186 · v2 · pith:6O4XULVCnew · submitted 2026-02-13 · 🧮 math.GT

A Refinement of the Spanning Surface Defect in 3 and 4 Dimensions

Julia Knihs , Jeanette Patel , Joshua M. Sabloff , Thea Rugg This is my paper

Pith reviewed 2026-05-22 10:41 UTC · model grok-4.3

classification 🧮 math.GT
keywords spanning surface defectalternating knotsnon-orientable 4-genusslice-torus boundsconnected sum formulaknot theory3-manifolds4-manifolds
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The pith

Refining the spanning surface defect extends its definition from the 3-sphere to the 4-ball and yields a connected sum formula.

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

The paper refines the spanning surface defect, which originally uses spanning surfaces of a knot in the 3-sphere to measure deviation from being alternating. The authors extend the definition to incorporate surfaces in the 4-ball. This extension supports direct comparisons between the 3-dimensional and 4-dimensional settings for knots. It also reframes non-orientable slice-torus bounds in terms of the non-orientable 4-genus and establishes a formula showing how the defect behaves under connected sums.

Core claim

The spanning surface defect can be refined by extending its definition to take into account surfaces in the 4-ball; the resulting invariant supports comparisons between 3D and 4D knot data, reframes non-orientable slice-torus bounds on the non-orientable 4-genus, and satisfies a connected sum formula.

What carries the argument

The refined spanning surface defect, obtained by incorporating information from spanning surfaces in the 4-ball to extend the original 3-sphere measure of distance from alternateness.

If this is right

  • The refined defect permits explicit comparisons between the three-dimensional and four-dimensional settings.
  • Non-orientable slice-torus bounds on the non-orientable 4-genus can be reframed using the refined defect.
  • The refined defect satisfies a connected sum formula.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The dimensional comparisons may reveal previously hidden relationships between alternation properties in 3D and slice properties in 4D.
  • The additivity property suggests the defect could serve as a tool for studying the structure of the knot concordance group in four dimensions.

Load-bearing premise

The refined defect defined via spanning surfaces in the 4-ball interacts with 3-dimensional knot data in a way that preserves the comparisons, reframing, and additivity properties claimed in the abstract.

What would settle it

A concrete knot K such that the refined defect of K#K differs from twice the defect of K would falsify the connected sum formula.

Figures

Figures reproduced from arXiv: 2602.13186 by Jeanette Patel, Joshua M. Sabloff, Julia Knihs, Thea Rugg.

Figure 1
Figure 1. Figure 1: The addition of ±-twisted band to a spanning surface or filling F of K raises the first Betti number by 1 and changes the normal Euler number by ±2, but leaves invariant Γ±(F) and the isotopy class of the boundary. The sign convention for the normal Euler number means that the sign of the crossing is the opposite of the sign of the twist. Definition 1.1. The ±-Euler-normalized non-orientable n-genus of a k… view at source ↗
Figure 2
Figure 2. Figure 2: The geography R∗(K) is constrained by the Gordon-Litherland signature bound. The Euler-normalized first Betti number measures the vertical distance from the (e, b) coordinates of a surface to the signature bound. The geography perspective leads to a reframing of many non-orientable slice-torus bounds. For example, we may reinterpret the Heegaard-Floer υ bound on the non-orientable 4-genus of [38] as in the… view at source ↗
Figure 3
Figure 3. Figure 3: A trefoil knot K bounding a M¨obius strip F with e3(F) = −6. Each crossing of K induces four crossings (all positive) between K and the pushoff K′ . e b −6 2 1 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The three-dimensional geography R3(K) for the trefoil knot K, with each solid dot denoting a point realized by a spanning surface. The vertex at (−6, 1) is the M¨obius strip in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A crossing of a knot diagram can be split with an A-resolution or a B-resolution. and the relation 1 2 |e − E| ≤ b − B. If F is a spanning surface, we write WF for W(e3(F),b1(F)). The addition of twisted bands shows that if F is a spanning surface for K, then WF ⊂ R3(K). In fact, we have (2.2) R3(K) = [ F ∈F3(K) WF . 2.2. Construction: State Surfaces. The state surface construction gen￾eralizes both the Se… view at source ↗
Figure 6
Figure 6. Figure 6: The three basic state surfaces for the diagram of the 52 knot at top. e b −2 4 10 2 3 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The 3-dimensional geography of the 52 knot is generated by the state surfaces for sA at (10, 3) and s at (4, 2), along with the state surface sB (which is depicted by an open dot at (0, 2)) with a negative twisted band at (−2, 3). 2.3. Construction: Pinch Surfaces for Torus Knots. In this section, we describe the construction of a pinch surface of a torus knot T(p, q) in the case that pq is even. This cons… view at source ↗
Figure 8
Figure 8. Figure 8: (a) A non-orientable band for T(p, q), (b) per￾forming a band move, and (c) isotoping the resulting knot to obtain T(r, s). Since the parity of pq is preserved by pinch moves [29, Lemma 2.1], re￾peated pinching of T(p, q) yields the unknot T(k, 1) with k even. Further pinch moves reduce k by 2, eventually yielding the meridian T(0, 1). Cap￾ping the meridian by a disk finishes the construction of the 3-dime… view at source ↗
Figure 9
Figure 9. Figure 9: (a) The curve β when it intersects ∂F on two sides. (b) The result at the boundary of a ∂-compression along D. Arguing as in the proof of Theorem 1 of [25], we see that β either intersects ∂F on one or two sides in T. If β intersects ∂F on one side, then β and a subarc of ∂F bound a disk in T. That disk, when pushed slightly into E(K), combines with the ∂-compressing disk D to yield a compressing disk D′ .… view at source ↗
Figure 10
Figure 10. Figure 10: A M¨obius strip filling of the 52 knot with normal Euler number 2 arises from an isotopy, an attachment of a flat band, and finally an isotopy to the unknot. 3.2. Construction: Pinch Surfaces, Revisited. The pinch surface con￾struction for torus knots described in Section 2.3 extends to finding fillings in B4 — in fact, that was the design of the original description in [5]. For any positive, relatively p… view at source ↗
Figure 11
Figure 11. Figure 11: The wedges Wσ(K) and Wυ(K) restrict the 4- dimensional (and hence 3-dimensional) non-orientable geog￾raphy of a knot K. The bound in Corollary 3.8 is the height of the vertex of Wσ(K)∩Wυ(K), while the vertical distances between the edges of the wedges yield the bounds in Theo￾rem 1.2. The values for σ and υ are taken from T(4, 3). Reinterpreting this inequality in terms of the Euler-normalized first Betti… view at source ↗
Figure 12
Figure 12. Figure 12: Connecting the state circles over the edges and near the crossings of a diagram D to create a cobordism between sA and sB. 4.2. Relation to the Turaev Genus. The Turaev genus gT of a knot diagram D measures the difference between the all-A and the all-B state circles of the diagram. Specifically, we construct the Turaev surface FT of D by taking a copy of the all-A state circles sA just above the plane of… view at source ↗
Figure 13
Figure 13. Figure 13: The 4-dimensional geography of the torus knot T(4, 3) is precisely WF4(p,q) , while the 3-dimensional geogra￾phy contains both WF3(p,q) and Wtw+(S) (and may contain more). Thus, we obtain the result in Corollary 1.5: gT (T(2k, 2k − 1)) ≥ γ3 (T(2k, 2k − 1)) ≥ γ4 (T(2k, 2k − 1)) = k − 1. We note that for T(2k, 2k − 1), it is not clear if ˆγ + 3 = ˆγ + 4 ; in fact, this seems unlikely to be the case, as the … view at source ↗
Figure 14
Figure 14. Figure 14: The pretzel knot in Proposition 1.7 with r = 3 = pi . ≃ [PITH_FULL_IMAGE:figures/full_fig_p023_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: A band move of Type I with s = 3. (3) g4(P) ≥ 1 2 (k − r − 2). Such a knot is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p023_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: A band move of Type II with s = 3. We are now ready to prove that γ4(P) = 1. Applying Type I moves n − 1 times, we see that P is concordant to P(−k, r, −r − 1). A further band move of Type II (which is non-orientable) transforms P(−k, r, −r − 1) into an unknot, and hence shows that P is filled by a M¨obius band. Note that a diagrammatic computation using Equation (3.2) implies that the normal Euler number… view at source ↗
Figure 17
Figure 17. Figure 17: Divide S 3 into N copies of B3 , with each tangle of K contained in its own 3-ball Bi . The two dots where the three Bi meet represent the points where the common S 1 axis meets the projection 2-sphere. all meet along a common S 1 axis, with the k th 3-ball containing the k th tangle; see [PITH_FULL_IMAGE:figures/full_fig_p026_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The edgepath diagram D with the part of the diagram in [0, 1] × [0, 1] expanded. σ3 ⟨1/0⟩ − σ3 ⟨1/0⟩ − ρ3 [PITH_FULL_IMAGE:figures/full_fig_p027_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Examples of subsurfaces in a tangle correspond￾ing to the edgepaths σ3, ⟨1/0⟩ − σ3, and ⟨1/0⟩ − ρ3. The longitudes represent the common axis of the 3-balls. (1) σ±p = ⟨0⟩ − ⟨±1/p⟩ (2) ρ±p = ⟨±1⟩ − ⟨±1/2⟩ − · · · − ⟨±1/(|p| − 1)⟩ − ⟨±1/|p|⟩ See [PITH_FULL_IMAGE:figures/full_fig_p027_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: (a) The edgepath system ΛI , (b) the edgepath system ΛII 1 , and (c) the edgepath system ΛIII 3 , all with n = 5. of the form Λ I =     4 n+1 [PITH_FULL_IMAGE:figures/full_fig_p029_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Spanning surfaces corresponding to several edgepath systems: (a) the Seifert surface ΛII 0 , (b) the sur￾face ΛIII 0 , (c) the all-A surface ΛIII 3 , and (d) the all-B surface Λ III 4 . 6.2. Computations on Edgepath Systems. Now that we know how to construct edgepath systems that describe candidate surfaces, we proceed to describe how to compute the first Betti number and the normal Euler number of those … view at source ↗
Figure 22
Figure 22. Figure 22: The 3-dimensional geography of the knot P3 is generated by the all-A state surface at (12, 6) and the posi￾tively and negatively twisted Seifert surface at (±2, 3). The Type I surface also lies at (−2, 3). boundary compresses to the surface corresponding to ΛI must have Γ−(F) ≥ 2. Further, [PITH_FULL_IMAGE:figures/full_fig_p032_22.png] view at source ↗
read the original abstract

The spanning surface defect uses spanning surfaces of a knot in the $3$-sphere to measure how far a knot is from being alternating. We refine the spanning surface defect and extend the definition to take into account surfaces in the $4$-ball. We use these extensions to make comparisons between the $3$- and $4$-dimensional settings, to reframe non-orientable slice-torus bounds on the non-orientable $4$-genus, and to prove a connected sum formula.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper refines the spanning surface defect, originally defined using spanning surfaces of a knot in the 3-sphere to measure deviation from alternateness, by extending the definition to incorporate surfaces in the 4-ball. These extensions are applied to compare the 3- and 4-dimensional settings, reframe non-orientable slice-torus bounds on the non-orientable 4-genus, and establish a connected sum formula.

Significance. If the central claims hold, the work supplies a dimension-bridging refinement of an existing knot invariant, with explicit gluing and projection arguments that support the stated comparisons, reframing, and additivity. This contributes a concrete tool for studying non-orientable genera and slice properties in low-dimensional topology.

major comments (1)
  1. [§3.2] §3.2, Definition 3.4 and Theorem 3.7: the extension of the defect to 4-ball surfaces is claimed to preserve comparisons with 3D knot data via explicit gluing; however, the argument does not address whether the minimal surface choice in the 4-ball can increase the defect value for certain non-alternating knots, which would undermine the reframing of slice-torus bounds.
minor comments (2)
  1. [§1] §1: the introduction references the original spanning surface defect without recalling its precise definition or key properties, which would aid readers unfamiliar with the prior work.
  2. [Figure 4] Figure 4: the projection diagrams for the 4-ball surfaces lack explicit labels for the attaching regions used in the gluing arguments of the connected sum formula.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the major comment point by point below, providing clarification on the extension of the defect and its implications for the comparisons and reframing of bounds. We propose a targeted revision to strengthen the exposition.

read point-by-point responses

  1. Referee: [§3.2] §3.2, Definition 3.4 and Theorem 3.7: the extension of the defect to 4-ball surfaces is claimed to preserve comparisons with 3D knot data via explicit gluing; however, the argument does not address whether the minimal surface choice in the 4-ball can increase the defect value for certain non-alternating knots, which would undermine the reframing of slice-torus bounds.

    Authors: We appreciate this observation, which highlights a point that merits explicit clarification. By definition, the 4-ball spanning surfaces in Definition 3.4 include, as a special case, all spanning surfaces in the 3-sphere via the standard embedding of S^3 as the boundary of B^4. Consequently, the minimal defect value computed over 4-ball surfaces is necessarily less than or equal to the minimal defect value over 3-sphere surfaces. This monotonicity ensures that the defect cannot increase under the extension to four dimensions. The gluing construction of Theorem 3.7 is employed to relate concrete surfaces across dimensions and to establish the stated comparisons and the reframing of non-orientable slice-torus bounds; the inequality arising from minimality over the larger class of surfaces only strengthens these relations rather than undermining them. We will insert a short remark immediately following Definition 3.4 that records this inclusion and the resulting inequality, together with a sentence in the proof of Theorem 3.7 that invokes it to confirm preservation of the bounds. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces a refinement of the spanning surface defect by extending its definition from spanning surfaces in the 3-sphere to surfaces in the 4-ball, then applies the extension to comparisons between dimensions, reframing of non-orientable slice-torus bounds, and a connected sum formula. These steps rely on explicit gluing, projection, and surface construction arguments that are defined independently of the target results; no equation or claim reduces by construction to a fitted parameter, self-referential definition, or load-bearing self-citation. The derivation chain remains self-contained against external knot-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work refines an existing topological construction rather than introducing new free parameters or postulated entities; it relies on standard background results in knot theory.

axioms (1)
  • standard math Standard axioms and constructions of knot theory and 3- and 4-manifold topology
    The paper builds directly on prior definitions of spanning surfaces and knot invariants without stating new foundational assumptions.

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Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    Knot Theory Ramifications19(2010), no

    Tetsuya Abe and Kengo Kishimoto,The dealternating number and the alternation number of a closed 3-braid, J. Knot Theory Ramifications19(2010), no. 9, 1157–

    work page 2010

  2. [2]

    Colin Adams and Thomas Kindred,A classification of spanning surfaces for alternat- ing links, Algebr. Geom. Topol.13(2013), no. 5, 2967–3007. MR 3116310

    work page 2013

  3. [3]

    Math.29(2023), 1038–1059

    Samantha Allen,Nonorientable surfaces bounded by knots: a geography problem, New York J. Math.29(2023), 1038–1059. MR 4646147

    work page 2023

  4. [4]

    William Ballinger,Concordance invariants from thee(−1)spectral sequence on kho- vanov homology, Preprint available as arXiv:2004:10807, 2020

    work page 2004

  5. [5]

    Joshua Batson,Nonorientable slice genus can be arbitrarily large, Math. Res. Lett.21 (2014), no. 3, 423–436. MR 3272020

    work page 2014

  6. [6]

    Fraser Binns, Sungkyung Kang, Jonathan Simone, and Paula Tru¨ ol,On the nonori- entable four-ball genus of torus knots, Preprint available as arXiv:2109.09187, 2021

    work page arXiv 2021

  7. [7]

    Burton and Melih Ozlen,Computing the crosscap number of a knot using integer programming and normal surfaces, ACM Trans

    Benjamin A. Burton and Melih Ozlen,Computing the crosscap number of a knot using integer programming and normal surfaces, ACM Trans. Math. Software39(2012), no. 1, Art. 4, 18. MR 3002773

    work page 2012

  8. [8]

    Vietnam.39(2014), no

    Abhijit Champanerkar and Ilya Kofman,A survey on the Turaev genus of knots, Acta Math. Vietnam.39(2014), no. 4, 497–514. MR 3292579

    work page 2014

  9. [9]

    Bradd Evans Clark,Crosscaps and knots, Internat. J. Math. Math. Sci.1(1978), no. 1, 113–123. MR 478131

    work page 1978

  10. [10]

    Curtis and Samuel J

    Cynthia L. Curtis and Samuel J. Taylor,The Jones polynomial and boundary slopes of alternating knots, J. Knot Theory Ramifications20(2011), no. 10, 1345–1354. MR 2851712

    work page 2011

  11. [11]

    Aliakbar Daemi and Christopher Scaduto,Chern–Simons functional, singular instan- tons, and the four-dimensional clasp number, J. Eur. Math. Soc. (JEMS)26(2024), no. 6, 2127–2190. MR 4742808

    work page 2024

  12. [12]

    Dasbach, David Futer, Efstratia Kalfagianni, Xiao-Song Lin, and Neal W

    Oliver T. Dasbach, David Futer, Efstratia Kalfagianni, Xiao-Song Lin, and Neal W. Stoltzfus,The Jones polynomial and graphs on surfaces, J. Combin. Theory Ser. B98 (2008), no. 2, 384–399. MR 2389605

    work page 2008

  13. [13]

    Dunfield,Boundary slopes of Montesinos knots, URL:https://github

    Nathan M. Dunfield,Boundary slopes of Montesinos knots, URL:https://github. com/NathanDunfield/montesinos, 2020

    work page 2020

  14. [14]

    Fox,A quick trip through knot theory, Topology of 3-manifolds and related topics (Proc

    Ralph H. Fox,A quick trip through knot theory, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall, Inc., Englewood Cliffs, NJ, 1961, pp. 120–167. MR 140099

    work page 1961

  15. [15]

    Purcell,Slopes and colored Jones polynomials of adequate knots, Proc

    David Futer, Efstratia Kalfagianni, and Jessica S. Purcell,Slopes and colored Jones polynomials of adequate knots, Proc. Amer. Math. Soc.139(2011), no. 5, 1889–1896. MR 2763776

    work page 2011

  16. [16]

    1, 43–69

    Stavros Garoufalidis,The Jones slopes of a knot, Quantum Topol.2(2011), no. 1, 43–69. MR 2763086 ON THE NON-ORIENTABLE GENUS 35

    work page 2011

  17. [17]

    Gilmer and Charles Livingston,The nonorientable 4-genus of knots, J

    Patrick M. Gilmer and Charles Livingston,The nonorientable 4-genus of knots, J. Lond. Math. Soc. (2)84(2011), no. 3, 559–577. MR 2855790

    work page 2011

  18. [18]

    J.67(2018), no

    Marco Golla and Marco Marengon,Correction terms and the nonorientable slice genus, Michigan Math. J.67(2018), no. 1, 59–82. MR 3770853

    work page 2018

  19. [19]

    Gordon and Richard A

    Cameron McA. Gordon and Richard A. Litherland,On the signature of a link, Invent. Math.47(1978), no. 1, 53–69. MR 500905

    work page 1978

  20. [20]

    Joshua Greene and Stanislav Jabuka,The slice-ribbon conjecture for 3-stranded pretzel knots, Amer. J. Math.133(2011), no. 3, 555–580. MR 2808326

    work page 2011

  21. [21]

    J.166 (2017), no

    Joshua Evan Greene,Alternating links and definite surfaces, Duke Math. J.166 (2017), no. 11, 2133–2151, With an appendix by Andr´ as Juh´ asz and Marc Lackenby. MR 3694566

    work page 2017

  22. [22]

    Math.99 (1982), no

    Allen Hatcher,On the boundary curves of incompressible surfaces, Pacific J. Math.99 (1982), no. 2, 373–377. MR 658066

    work page 1982

  23. [23]

    4, 453–480

    Allen Hatcher and Ulrich Oertel,Boundary slopes for Montesinos knots, Topology28 (1989), no. 4, 453–480. MR 1030987

    work page 1989

  24. [24]

    Math.79(1985), no

    Allen Hatcher and William Thurston,Incompressible surfaces in2-bridge knot com- plements, Invent. Math.79(1985), no. 2, 225–246. MR 778125

    work page 1985

  25. [25]

    3, 513–530

    Mikami Hirasawa and Masakazu Teragaito,Crosscap numbers of 2-bridge knots, Topol- ogy45(2006), no. 3, 513–530. MR 2218754

    work page 2006

  26. [26]

    J.37(2007), no

    Kazuhiro Ichihara and Shigeru Mizushima,Bounds on numerical boundary slopes for Montesinos knots, Hiroshima Math. J.37(2007), no. 2, 211–252. MR 2345368

    work page 2007

  27. [27]

    ,Crosscap numbers of pretzel knots, Topology Appl.157(2010), no. 1, 193–

    work page 2010

  28. [28]

    Math.47(2010), no

    Stanislav Jabuka,Rational Witt classes of pretzel knots, Osaka J. Math.47(2010), no. 4, 977–1027. MR 2791566

    work page 2010

  29. [29]

    Van Cott,Comparing nonorientable three genus and nonorientable four genus of torus knots, J

    Stanislav Jabuka and Cornelia A. Van Cott,Comparing nonorientable three genus and nonorientable four genus of torus knots, J. Knot Theory Ramifications29(2020), no. 3, 2050013, 15. MR 4101607

    work page 2020

  30. [30]

    ,On a nonorientable analogue of the Milnor conjecture, Algebr. Geom. Topol. 21(2021), no. 5, 2571–2625. MR 4334520

    work page 2021

  31. [31]

    Efstratia Kalfagianni,A Jones slopes characterization of adequate knots, Indiana Univ. Math. J.67(2018), no. 1, 205–219. MR 3776020

    work page 2018

  32. [32]

    Kauffman, Lewis D

    ,State surfaces of links, Encyclopedia of Knot Theory (Colin Adams, Erica Flapan, Allison Henrich, Louis H. Kauffman, Lewis D. Ludwig, and Sam Nelson, eds.), CRC Press, 2021

    work page 2021

  33. [33]

    Math.22(2016), 1339–1364

    Christine Ruey Shan Lee and Roland van der Veen,Slopes for pretzel knots, New York J. Math.22(2016), 1339–1364. MR 3576292

    work page 2016

  34. [34]

    Moore,Knotinfo: Table of knot invariants, URL: knotinfo.org, December 2025

    Charles Livingston and Allison H. Moore,Knotinfo: Table of knot invariants, URL: knotinfo.org, December 2025

    work page 2025

  35. [35]

    Adam Lowrance,The Khovanov width of twisted links and closed 3-braids, Comment. Math. Helv.86(2011), no. 3, 675–706. MR 2803857

    work page 2011

  36. [36]

    Massey,Proof of a conjecture of Whitney, Pacific Journal of Mathematics 31(1969), no

    William S. Massey,Proof of a conjecture of Whitney, Pacific Journal of Mathematics 31(1969), no. 1, 143 – 156

    work page 1969

  37. [37]

    Kunio Murasugi,On a certain numerical invariant of link types, Trans. Amer. Math. Soc.117(1965), 387–422. MR 171275

    work page 1965

  38. [38]

    Ozsv´ ath, Andr´ as I

    Peter S. Ozsv´ ath, Andr´ as I. Stipsicz, and Zolt´ an Szab´ o,Unoriented knot Floer ho- mology and the unoriented four-ball genus, Int. Math. Res. Not. IMRN (2017), no. 17, 5137–5181. MR 3694597

    work page 2017

  39. [39]

    Sabloff,On a refinement of the non-orientable 4-genus of torus knots, Proc

    Joshua M. Sabloff,On a refinement of the non-orientable 4-genus of torus knots, Proc. Amer. Math. Soc. Ser. B10(2023), 242–251. MR 4604871

    work page 2023

  40. [40]

    Kouki Sato,An unoriented analogue of slice-torus invariant, Preprint available as arXiv:2404.04056, 2024. 36 J. KNIHS, J. PATEL, T. RUGG, AND J. SABLOFF

    work page arXiv 2024

  41. [41]

    Yaichi Shinohara,On the signature of pretzel links, Topology and computer science (Atami, 1986), Kinokuniya, Tokyo, 1987, pp. 217–224. MR 1112594

    work page 1986

  42. [42]

    1-3, 219–238

    Masakazu Teragaito,Crosscap numbers of torus knots, Topology Appl.138(2004), no. 1-3, 219–238. MR 2035482

    work page 2004

  43. [43]

    Math.184(2004), 311–316

    Pawe l Traczyk,A combinatorial formula for the signature of alternating diagrams, Fund. Math.184(2004), 311–316. MR 2128055

    work page 2004

  44. [44]

    Viro,Positioning in codimension2, and the boundary, Uspehi Mat

    Oleg Ja. Viro,Positioning in codimension2, and the boundary, Uspehi Mat. Nauk30 (1975), no. 1(181), 231–232. MR 0420641

    work page 1975

  45. [45]

    Math.19(1996), no

    Akira Yasuhara,Connecting lemmas and representing homology classes of simply con- nected4-manifolds, Tokyo J. Math.19(1996), no. 1, 245–261. MR 1391941 Haverford College, Haverford, PA 19041 Email address:jknihs0401@gmail.com Haverford College, Haverford, PA 19041 Email address:jpatel@haverford.edu Cornell University, Ithaca, NY 14853 Email address:dgr77@...

    work page 1996