Pith. sign in

REVIEW 4 major objections 4 minor 45 references

Milnor Invariants --From classical links to surface-links, and beyond--

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Cut-diagrams carry Milnor invariants from classical links to surface-links and higher-dimensional links, yielding new concordance obstructions.

desk verdict A clear expository survey of Milnor invariants that advertises a new surface-link invariant but leaves all the new proofs in an unpublished preprint—worth reading as a map, not as a research paper. read the letter →

arxiv 2411.18032 v1 pith:MSH2FLIM submitted 2024-11-27 math.GT

Akira Yasuhara This is my paper · ORCID
classification math.GT MSC 57K1057K45
keywords Milnorinvariantssurface-linkscut-diagramsRosemanmoveslinkconcordanceweldedlinksMagnusexpansionhigher-dimensional
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 expository paper argues that Milnor invariants, classically defined for collections of circles in 3-space, can be extended to surface-links, meaning closed surfaces smoothly embedded in 4-space, and more generally to m-dimensional links embedded in (m+2)-space. The bridge is the cut-diagram, a kind of net of a link diagram obtained by cutting along under-intersections, which records the gluing data needed to recover the peripheral group. The paper defines Milnor invariants for cut-diagrams, transfers them to surface-links, and states that they are invariant under Roseman moves and under link concordance for every m at least 2. If correct, this gives a higher-dimensional analogue of classical Milnor invariants and a new concordance obstruction for knotted surfaces and beyond. The exposition also covers welded links, where Milnor invariants are completely characterized by the equivalence relations called Wk-concordance and self Wk-concordance.

What carries the argument

The central object is the cut-diagram, a net of a link diagram obtained by cutting along under-intersections. For surfaces it is a triple (Σ, D, f) consisting of closed surfaces Σ, a 1-dimensional diagram D on Σ whose arcs may end at branch-point markings, and a map f sending each arc of D to a region of Σ − D; the map f encodes which region lies on the other side of each under-intersection. From this data the paper defines a group G(C,p) with relations $zxz^{-1}y^{-1}$ at each arc, a peripheral system with meridian $a_{i0}$ and longitude set $\Lambda_i = \{w(l)\}$ for loops $l$ representing elements of $\pi_1(\Sigma_i, p_i)$, and then Milnor invariants by taking coefficients in the Magnus expansion of $w(l)$ and reducing modulo gcds of shorter coefficients. The Chen map $\eta_q$, defined inductively from the labels met along chosen curves, makes the invariants computable in principle: Theorem 4.6 identifies the nilpotent quotient $N_q(C,p)$ and shows that $\xi_q(w(l)) = \eta_q(w(l))$ modulo the relevant normal closure.

What would settle it

Compute the Chen-map invariant for the cut-diagram of a specific surface-link such as the spun trefoil, apply a single Roseman move to its diagram, and recompute: any change in $\nu$ would refute Theorem 4.5. Alternatively, find two m-dimensional links that are known to be link-concordant but whose cut-diagrams yield different Milnor invariants, which would contradict Corollary 5.3.

Watch

Extended reading notes

Core claim

The central claim is that Milnor invariants are not a 3-dimensional phenomenon. For a surface-link L in 4-space, one chooses a diagram and cuts it along its under-intersections to obtain a cut-diagram C = (Σ, D, f); the paper builds a group G(C,p) from the regions of Σ − D and the labels on the arcs of D, equips it with a peripheral system whose ith longitude set Λ_i consists of words w(l) associated to generators of π1(Σ_i, p_i), and reduces Magnus-expansion coefficients modulo gcds of shorter coefficients to obtain integers ν_C(I). These integers are shown to be independent of the chosen base point and of the chosen diagram, so they define invariants ν_L(I) of the surface-link; Theorem 4.5 states that ν_L(I) is unchanged by Roseman moves, the surface analogue of Reidemeister moves. The same construction works for m-dimensional cut-diagrams, and Corollary 5.3 states that ν_L(I) is a link-concordance invariant for every m ≥ 2. For m = 1 the construction recovers the classical and welded Milnor invariants, so the higher-dimensional invariants are presented as a continuation of the classical story rather than a separate theory.

Load-bearing premise

Everything in the higher-dimensional chapters rests on the definitions and proofs deferred to an unpublished companion preprint: if the gluing conditions that make an m-dimensional cut-diagram well-defined, or the proofs that $\nu_C(I)$ is invariant under Roseman moves and concordance, contain a flaw, the central claims of Chapters 4 and 5 collapse.

Editorial extensions

If this is right

  • Milnor invariants $\nu_L(I)$ are invariants of surface-links: two surface-link diagrams related by Roseman moves have the same values (Theorem 4.5).
  • For every $m \geq 2$, $\nu_L(I)$ is a link-concordance invariant of m-dimensional links, so concordant links share all Milnor invariants (Corollary 5.3).
  • For $m = 1$ the cut-diagram definition reproduces the classical and welded Milnor invariants, so the higher-dimensional theory reduces to the known theory in the classical case.
  • The Chen map $\eta_q$ gives a concrete algorithm for computing $\nu_C(I)$ of cut-diagrams, extending the classical Magnus-expansion computation to surfaces.
  • Cut-concordance of cut-diagrams implies equality of $\nu$ for $m \geq 2$, so cut-diagrams form a combinatorial model in which concordance questions about higher-dimensional links can be studied (Proposition 5.1 and Theorem 5.2).

Reading between the lines

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

  • The paper leaves implicit that $\nu_L(I)$ is a probe of the fundamental group of the complement: it notes that the invariants vanish when the fundamental group is trivial, so the first interesting computations would come from knotted surfaces with nontrivial $\pi_1$, such as spun knots.
  • The welded-link machinery of Chapter 3, with W-trees and ascending presentations, suggests a testable higher-dimensional analogue: a characterization of when two surface-links have the same Milnor invariants in terms of some surface version of Wk-concordance.
  • A natural extension would define the same invariants for surface-links with boundary, mimicking string links, which would give a group structure and potentially a Habegger-Lin type classification in dimension two.
  • If Theorem 4.5 is sensitive enough, $\nu$ could obstruct unknottedness of surface-links in ways classical invariants cannot, because Roseman moves include birth-and-death and saddle-type changes that have no classical Reidemeister analogue.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. This paper is an English translation of an expository article on Milnor invariants, surveying the classical theory, the welded-link extension, and a proposed further extension to surface-links and m-dimensional links via cut-diagrams. The first three chapters present standard material on Milnor invariants for classical and welded links, including Habegger-Lin automorphisms, Chen-Milnor maps, and characterization results in terms of Wk-concordance and self Wk-concordance. The fourth and fifth chapters announce new definitions of Milnor invariants for 2-dimensional and m-dimensional cut-diagrams, claiming Roseman-move invariance (Theorem 4.5) and link-concordance invariance (Corollary 5.3). These central new results are stated without proof and are explicitly deferred to the unpublished preprint [3] by Audoux, Meilhan, and Yasuhara. The paper is openly expository and states that precision is sometimes sacrificed.

Significance. If the announced results are correct, they provide a genuine higher-dimensional analogue of classical Milnor invariants, which would be a valuable contribution to knot theory. The exposition of the classical and welded parts is sound and clearly organized, and the algorithmic treatment via the Chen-Milnor map is a useful feature. However, the significance of the new material cannot be assessed from the manuscript alone: the definitions of 2-dimensional and m-dimensional cut-diagrams rely on gluing conditions deferred to an unpublished source, and the theorems establishing invariance are quoted rather than proved. The paper is therefore best viewed as an expository announcement of results that should appear in a peer-reviewed research article before being relied upon. The manuscript honestly discloses its expository nature and the source of the new results, which is creditworthy, but that does not remove the verification gap.

major comments (4)
  1. [Section 4.4.2 and §4.5] The definition of a 2-dimensional cut-diagram as a triple (Σ, D, f) requires that f be defined so that 'local gluing can be performed' around each branch point and each triple point, but the actual condition is deferred to [3, Subsection 1.2.1]. All subsequent constructions, including the group G(C, p), the peripheral systems, and the invariants ν_C, depend on this condition. Without the precise gluing condition, the well-definedness of the central objects is not established within the manuscript. Please either include the gluing condition or explicitly state that the paper assumes the results of [3] and adjust the claims accordingly.
  2. [Section 4.6, Theorem 4.6(1)] The invariant ν_C is defined through the Magnus expansion of ξ_q(w(l)) in N_q(C, p), which by Theorem 4.6(1) is the quotient of the free group F by the relations [α_i, η_q(w(l_ij))] together with Γ_qF. This group is not a free nilpotent group, so the coefficients of the Magnus expansion of an element are not canonical unless a specific lift is chosen. The manuscript does not specify how the lift is chosen or prove that the coefficients are independent of that choice. This is a load-bearing well-definedness issue for the central definition, and Theorem 4.4, which asserts independence of all choices, is stated without proof.
  3. [Section 4.7] The Chen map η_q(C, p) depends on the choices of the curves γ_ij, as the paper itself notes. The claim that 'Lemma 2.5 (1) holds for η_q(C, p) as well' and Theorem 4.6(2), which identifies ξ_q(w(l)) with η_q(w(l)) modulo the normal closure W, are not proved in the text. Since the computational algorithm and the invariance of ν_C rely on these statements, the manuscript needs either proofs or a precise reference to a publicly available source. As written, the computational method and the invariants it produces are not verifiable from the manuscript.
  4. [Sections 5.1–5.2] The definition of m-dimensional cut-diagrams and cut-concordance uses a map f satisfying 'certain gluing conditions' deferred to [3], and Theorem 5.2 and Corollary 5.3 are quoted from the unpublished preprint [3]. Because the concordance invariance of ν_L for m-dimensional links is one of the paper's headline claims, the manuscript should either reproduce the relevant definitions and proofs or clearly label these as results proved elsewhere and provide a verifiable reference. As written, the central claims of Chapters 4 and 5 are a black box, and the reader cannot check the arguments.
minor comments (4)
  1. [Section 4.6] There is a typo in the opening sentence: 'In this sction' should be 'In this section'.
  2. [Section 5.2] There is a typo in the sentence introducing the definition for m = 1: 'Threfore' should be 'Therefore'.
  3. [Section 3.5.1] The notation ρRk(λk) uses the index k both for the component and for the level of the reduced group, which is confusing in statements such as Theorem 3.9. Consider using a different letter for the level, for example ρ_R^r(λ_i).
  4. [Section 4.5.1] In the definition of the group G(C, p), the relation zxz^{-1}y^{-1} is described verbally with reference to Figure 25, but the figure does not clarify which of the arcs x, y, z are labels and which are generators. Please make the convention explicit.

Circularity Check

3 steps flagged · score 5.0 of 10

Surface-link and higher-dimensional Milnor invariants are introduced as 'taken from [3]', with the key gluing definitions and all invariance proofs deferred to that unpublished same-author preprint; the central support is load-bearing self-citation, though the definitions themselves are concrete and not definitionally circular.

  1. self citation load bearing [Chapter 4, opening paragraph ('Milnor invariants for surface-links')]
    "The content of this and the next chapters is taken from [3]."

    This sentence announces that the paper's advertised new results—Milnor invariants for surface-links (Theorem 4.5) and for m-dimensional links (Theorem 5.2 and Corollary 5.3), together with the well-definedness of the invariants (Theorem 4.4)—are not derived in the text. Reference [3] is the unpublished preprint by Audoux, Meilhan and Yasuhara, whose author list overlaps with the present author. Since no proof of Theorems 4.4–4.6 or 5.2 is reproduced here, the derivation chain in this paper reduces to a citation of the authors' own unpublished work.

  2. self citation load bearing [Section 4.4.2, definition of 2-dimensional cut-diagrams]
    "Here, the map f is defined so that 'local gluing can be performed' around each branch point and each triple point. For a detailed definition, see [3, Subsection 1.2.1]."

    The well-definedness of a 2-dimensional cut-diagram—the object on which the surface-link invariant ν_C is defined—is delegated to [3, Subsection 1.2.1]. The invariance claims in Theorems 4.4 and 4.5 depend on this omitted gluing condition. Thus the central construction is not self-contained in the paper; its load-bearing definition is completed only by a citation to the same authors' preprint.

1 more flagged steps
  1. self citation load bearing [Section 5.1, definition of m-dimensional cut-diagrams and cut-concordance]
    "we consider a map f : { regions of Y } − → {regions of X − Y } that satisfies 'certain gluing conditions'. Here 'regions of Y ' correspond to arcs in diagrams in the case of 2-dimensional cut-diagrams. We call the triple ( X, Y, f) an m-dimensional cut-diagram. ... For the detailed definition, see [3]."

    For every m ≥ 2, the paper's higher-dimensional Milnor invariants are defined on m-dimensional cut-diagrams, but the defining condition on f is again only described as 'certain gluing conditions' and deferred to [3]. The cut-concordance invariance theorem (Theorem 5.2) and the link-concordance corollary (Corollary 5.3) are asserted without proof and rest on that deferred definition. This is another load-bearing self-citation to the authors' own unpublished preprint.

full rationale

The paper is an explicitly expository translation, so some reliance on prior literature is expected. Nevertheless, the advertised central results are not derived here: Chapter 4 begins with 'The content of this and the next chapters is taken from [3]', and [3] is an unpublished preprint co-authored by the present author. The gluing conditions needed for the cut-diagram objects are deferred to [3] in both the 2-dimensional and m-dimensional settings, and the well-definedness and invariance theorems (4.4, 4.5, 4.6, 5.2, Corollary 5.3) are stated without proof. Section 4.7 also explicitly notes that the Chen map η_q depends on choices of curves γ_ij, and the independence from those choices is exactly the content of the deferred Theorem 4.4. This is a load-bearing self-citation chain for the paper's main new claims. However, there is no visible reduction of a theorem to its own input by construction: the definitions of ν_C and ν_L are concrete and could be checked independently if [3] were supplied, and no fitted parameter is renamed as a prediction. Chapter 3's Theorem 3.9 is also attributed to [4], another same-author paper, but it is a supporting characterization rather than the paper's principal new target. For these reasons, the appropriate score is 5 rather than 0–2 (the central claims are not self-contained) and not 6–8 (no definitional tautology or fitted-input prediction is exhibited). The gap is a verification gap compounded by self-citation, not demonstrated circularity by equality.

Assumptions & free parameters 0 free parameters · 8 assumptions · 2 invented entities

The definitions in Chapters 1-3 rest on standard theorems in combinatorial group theory and knot theory; the genuinely new claims in Chapters 4-5 are not proven in this paper and are deferred to preprints, several of which are authored by the present author. No numerical parameters are fitted; the apparent choices of base points, regions, and curves are stated to be irrelevant for the resulting invariants.

assumptions (8)
  • standard math Magnus expansion properties: E is a bijection on the free group and E(gh)=E(g)E(h); g is in ΓqF iff the minimal degree of E(g)-1 is at least q (Proposition 1.1).
    Used throughout to extract integer coefficients from longitudes and to identify lower central series elements. Cited to [26] and [15].
  • standard math Stallings' theorem: inclusion of the boundary disk into the link complement induces an isomorphism on lower central series quotients of fundamental groups.
    Used in Section 1.3 to define the automorphism φq of F/ΓqF from a string link. Cited to [41, Theorem 5.1].
  • standard math Reidemeister's theorem: two classical links are equivalent iff their diagrams are equivalent under Reidemeister moves.
    Theorem 2.1, the foundation of the diagrammatic approach used in the whole paper.
  • standard math Welded-link equivalence theorem: two classical link diagrams are equivalent as classical links iff equivalent as welded links.
    Theorem 2.2, used to justify the 'embedding' of classical links into welded links. Cited to [42] and [19].
  • standard math Roseman's theorem: two surface-links are equivalent iff their diagrams are related by Roseman moves.
    Theorem 4.1, the surface-link analogue of Reidemeister's theorem, used to state Theorem 4.5. Cited to [38, 8, 9].
  • domain assumption Theorem 3.1 (Colombari): two based diagrams are Wk-concordant iff their Milnor invariants agree for all sequences of length at most k.
    The main characterization in Section 3.1, stated without proof and attributed to the preprint [13].
  • domain assumption Theorem 3.9 (Audoux-Bellingeri-Meilhan-Yasuhara): self Wk-concordance classification of based diagrams by Milnor invariants with r(I) ≤ k.
    Quoted from [4], which is listed as 'to appear'; the proof in Section 3.5.3 is only a sketch and the paper admits the most difficult direction.
  • domain assumption Theorems 4.4, 4.5, 4.6, 5.2 and Corollary 5.3: the invariants νC and νL are well-defined, Roseman-invariant, and cut-concordance invariant.
    The new higher-dimensional results are stated in the text but the proofs are deferred to the unpublished preprint [3] by the author and collaborators.
invented entities (2)
  • 2-dimensional cut-diagrams (Σ, D, f) independent evidence
    purpose: To define Milnor invariants for surface-links by encoding the gluing information of a surface-link diagram in a 'net' of closed surfaces with labeled curves.
    The definition is explicit (closed surfaces Σ with a 1-dimensional diagram D and label map f), and the Chen map algorithm in Section 4.7 gives a concrete computational handle. The claimed Roseman-invariance in Theorem 4.5 is a falsifiable statement that could in principle be checked on examples.
  • m-dimensional cut-diagrams (X, Y, f)
    purpose: To define Milnor invariants for m-dimensional links in (m+2)-space.
    The paper does not give a complete definition for general m; it refers to [3] for the detailed gluing conditions, so the reader cannot verify the object or its invariants from the text alone.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Milnor Invariants --From classical links to surface-links, and beyond--." pith.science (2026). https://pith.science/paper/MSH2FLIM

@misc{pith2026241118032,
  author       = {Pith},
  title        = {Pith review of: Milnor Invariants --From classical links to surface-links, and beyond--},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MSH2FLIM}},
  note         = {Machine review of arXiv:2411.18032}
}
read the original abstract

This is an English translation of the expository article written by the author in Japanese for publication in {\em Sugaku}. The author will explain Milnor invariants from the viewpoint of his research.

Figures

Figures reproduced from arXiv: 2411.18032 by the authors.

Figure 1
Figure 1. Meridian and longitude of a link Since G(L) is non-commutative, classifying peripheral systems is as difficult as classi￾fying links. Therefore we consider the natural projection ρ2 : G(L) −→ N2(L) = G(L)/Γ2G(L), 1While we skip the detailed definition, in this article it is enough to know that meridians and longitudes are special elements in G(L) that depend on the link [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Meridian and longitude for a string link 1.3. Automorphisms of nilpotent groups and Milnor invariants. Milnor invariants for string links are defined by N. Habegger and X.S. Lin [20]. Their definition is different from that in the previous section. In this section, we present the idea of the definition by Habegger and Lin. Set Dε = D2 × {ε} (ε ∈ {0, 1}). For an n-string link L ⊂ D2 × [0, 1], by Stallings Theorem [41… view at source ↗
Figure 3
Figure 3. Crossing Two (string) link diagrams are equivalent if one is deformed into the other by a com￾bination of continuous deformations (fixing the boundary) and the three local moves R1, R2, R3 in [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (23 more)
Figure 4
Figure 4. Figure 4: Reidemeister moves 4This follows from [20, Theorem 1.7], which is not stated in terms of Milnor invariants [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: are called virtual Reidemeister moves (VR1,VR2,VR3,VR4), and the last move is called OC move (Overcrossings Commute). The residue classes of the set of virtual link diagrams modulo continuous deformations, Reidemeister moves, virtual Reidemeister moves and OC moves is …
Figure 6
Figure 6. Figure 6: A base point passing through a virtual crossing 2.4. Peripheral systems of based welded links. A diagram of a (based/string) welded link means a representative of the welded link, i.e., a virtual link diagram. However, since it is tedious to distinguish them, the welde…
Figure 7
Figure 7. Figure 7: Note that airi is the ingoing arc to pi The group G(L, p) of (L, p) is the quotient group of the free group Fe with generating set {aij}i,j modulo the following relations Rij = ai(j−1)u ε(ij) ij a −1 ij u −ε(ij) ij (1 ≤ i ≤ n, 1 ≤ j ≤ ri), where uij denote the arc cont…
Figure 7
Figure 7. Figure 7: Arcs of Ki For each i, the two elements ai0 and λi = a −wi i0 u ε(i1) i1 u ε(i2) i2 · · · u ε(iri) iri (where wi is the sum of signs ε(il) for all uil ⊂ Ki) of G(L, p) are called the ith meridian and the ith longitude respectively. As in the case of classical links, th…
Figure 8
Figure 8. Figure 8: A union of a W3-tree, a W2-tree and two W-arrows W-trees act as ‘guides’ for transforming diagrams locally. Surgery along a W-tree is a local move on a diagram using the W-tree as a guide, as described below. Before describing surgery along a W-tree, we need to explain…
Figure 10
Figure 10. Figure 10: An arrow presentation for a diagram L is a pair (V, A) of a diagram V without classical crossings and a union of W-arrows A for V , such that VA is equivalent to the diagram L. Any diagram admits an arrow presentation since all classical crossings can be replaced with…
Figure 9
Figure 9. Figure 9: Surgery along W-arrow (i) A A LA A L LA A LA [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 11
Figure 11. Figure 11: A classical crossing can be replaced by a virtual crossing and a W-arrow 3.2.3. Arrow-moves. Arrow moves consist of the virtual Reidemeister moves VR1, VR2, VR3, where they may contain W-arrows, and the local moves AR1, ..., AR10 in [PITH_FULL_IMAGE:figures/full_fig_…
Figure 12
Figure 12. Figure 12: Arrow moves AR1–AR10 We say that LT is obtained from L by a Wk-move if T is a Wk-tree. In particular, we say that LT is obtained from L by a self Wk-move if all ends of T are contained in the same component of L. Using the move (I) in [PITH_FULL_IMAGE:figures/full_fi…
Figure 13
Figure 13. Figure 13: Expansion (E) of a tree (E) (E) [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]
Figure 14
Figure 14. Figure 14: Expansions of a W3-tree 8 It is shown that if l > k, then a (self) Wl-move is realized by (self) Wk-moves [4] [PITH_FULL_IMAGE:figures/full_fig_p014_14.png]
Figure 15
Figure 15. Figure 15: W-tree moves 3.3. Welded-concordance. Two n-component diagrams L and L ′ are welded-concordant if one can be deformed into the other by a sequence of welded equivalence and the birth/death and saddle moves of [PITH_FULL_IMAGE:figures/full_fig_p015_15.png]
Figure 16
Figure 16. Figure 16: Birth/death and saddle moves Remark 3.5. If L and L ′ are classical links, then the usual link concordance implies the welded-concordance. In the definition of the welded-concordance, the condition on the numbers of birth/death and saddle moves corresponds to the fact…
Figure 17
Figure 17. Figure 17: Inverse of expansion (where [x, y] := xy−1x −1 y) 3.5. Self Wk-concordance and Minor invariants. The content of this section is due to [4]. The self Wk-concordance is an equivalence relation on diagrams obtained by combining self Wk-equivalence and welded-concordance.…
Figure 18
Figure 18. Figure 18: A surface-link (surface-knot) diagram [PITH_FULL_IMAGE:figures/full_fig_p020_18.png]
Figure 19
Figure 19. Figure 19: But in general, as in the center of Figure 19, an intersection containing a point [PITH_FULL_IMAGE:figures/full_fig_p021_19.png]
Figure 20
Figure 20. Figure 20: Relation zxz−1 y −1 (x, y, z ∈ {aij}ij ) 4.3. Nets of diagrams. In this section, we intuitively explain cut-diagrams of classical links and of surface-links. As we see below, cut-diagrams are ‘nets’ of diagrams. 4.3.1. Cut-diagrams of classical link diagrams. First, w…
Figure 21
Figure 21. Figure 21: A link (knot) diagram and its cut-diagram 4.3.2. Cut-diagrams of surface-link diagrams. By applying the idea of ‘net’ to surface-link diagrams, we obtain cut-diagrams of surface-link diagrams. The figure on the right side of [PITH_FULL_IMAGE:figures/full_fig_p023_21.png]
Figure 22
Figure 22. Figure 22: A surface-link diagram (spherical-link diagram) and its cut￾diagram In general, surface-link diagrams may contain branch points and triple points. Then cut-diagrams have ‘local nets’ as illustrated in Figures 23 (resp. 24), which contain end points corresponding to th…
Figure 23
Figure 23. Figure 23: Cut-diagrams (local nets) around branch points C A B net −→ C C A B A B C C A B net −→ C C A B A B C [PITH_FULL_IMAGE:figures/full_fig_p024_23.png]
Figure 24
Figure 24. Figure 24: Cut-diagrams (local nets) around triple points 4.4.1. 1-dimensional cut-diagrams. A cut diagram of an n-component link diagram can be seen as a set S of n circles (1-dim) with signed points P (0-dim) arranged on S, where each point in P is labeled by an arc of S − P. …
Figure 25
Figure 25. Figure 25: Relation zxz−1 y −1 (x, y, z ∈ {aij}ij ) 13This is almost a link diagram but it may contain end points that correspond to branch points 14For a detailed definition, see [3, Subsection 1.2.1] [PITH_FULL_IMAGE:figures/full_fig_p025_25.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [3]

    Audoux, J.B

    B. Audoux, J.B. Meilhan and A. Yasuhara, Milnor concordance invariants for knotted surfaces and beyond, preprint

    work page

  2. [1]

    Audoux, P

    B. Audoux, P. Bellingeri, J.-B. Meilhan and E. Wagner, Homotopy classification of ribbon tubes and welded string links , Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), no. 2, 713–761

    work page 2017

  3. [2]

    Audoux and J.B

    B. Audoux and J.B. Meilhan, Characterization of the reduced peripheral system of links , to appear in J. Inst. Math. Jussieu

    work page

  4. [4]

    Audoux, J.B

    B. Audoux, J.B. Meilhan and A. Yasuhara, k-reduced groups and Milnor invariants , to appear in Math. Research Letters

    work page

  5. [5]

    Boden and M

    H.U. Boden and M. Chrisman, Virtual concordance and the generalized Alexander polynomial , J. Knot Theory Ramifications 30, (2021) Paper No. 2150030, 35 pp

    work page 2021

  6. [6]

    Casson, Link cobordism and Milnor’s invariant , Bull

    A.J. Casson, Link cobordism and Milnor’s invariant , Bull. London Math. Soc. 7 (1975), 39–40

    work page 1975

  7. [7]

    Carter, S

    J.S. Carter, S. Kamada and M. Saito, Stable equivalence of knots on surfaces and virtual knot cobordisms, J. Knot Theory Ramifications 11(2002), 311–322

    work page 2002

  8. [8]

    J. S. Carter and M. Saito, Reidemeister moves for surface isotopies and their interpretation as moves to movies , J. Knot Theory Ramifications 2 (1993), 251–284

    work page 1993

Show all 45 references
  1. [9]

    J. S. Carter and M. Saito, Knotted surfaces and their diagrams , Mathematical Surveys and Mono- graphs, 55 Providence, RI: American Mathematical Society, 1998

    1998

  2. [10]

    Chen, it Commutator calculus and link invariants, Proc

    K.T. Chen, it Commutator calculus and link invariants, Proc. Amer. Math. Soc., 3 (1952), 44—45

    1952

  3. [11]

    Chrisman, Milnor’s concordance invariants for knots on surfaces, Algebr

    M. Chrisman, Milnor’s concordance invariants for knots on surfaces, Algebr. Geom. Topol.22 (2022), 2293–2353

    2022

  4. [12]

    Cochran, Derivatives of links: Milnor’s concordance invariants and Massey’s products , Mem

    T. Cochran, Derivatives of links: Milnor’s concordance invariants and Massey’s products , Mem. Amer. Math. Soc. 84 (1990), no. 427, x+73 pp

    1990

  5. [13]

    Colombari, A diagrammatic characterization of Milnor invariants , preprint

    B. Colombari, A diagrammatic characterization of Milnor invariants , preprint

  6. [14]

    Conant, R

    J. Conant, R. Schneiderman and P. Teichner, Whitney tower concordance of classical links , Geom. Topol. 16 2012, 1419–1479

    2012

  7. [15]

    Fenn, Techniques of geometric topology , London Mathematical Society Lecture Note Series, 57, Cambridge University Press, Cambridge, 1983

    R. Fenn, Techniques of geometric topology , London Mathematical Society Lecture Note Series, 57, Cambridge University Press, Cambridge, 1983

    1983

  8. [16]

    R. Fenn, R. Rim´ anyi and C. Rourke, The braid-permutation group , Topology 36 (1997), no. 1, 123–135

    1997

  9. [17]

    Fleming and A

    T. Fleming and A. Yasuhara, Milnor’s invariants and self Ck-equivalence, Proc. Amer. Math. Soc. 137 (2009) 761–770

    2009

  10. [18]

    R.Gaudreau, Classification of virtual string links up to cobordism , Ars Math. Contemp. 19(2020) 37–49

    2020

  11. [19]

    Goussarov, M

    M. Goussarov, M. Polyak and O. Viro, Finite-type invariants of classical and virtual knots , Topology 39 (2000), 1045–1068

    2000

  12. [20]

    Habegger and X.S

    N. Habegger and X.S. Lin, The classification of links up to link-homotopy , J. Amer. Math. Soc. 3 (1990), 389–419

    1990

  13. [21]

    Habiro, Claspers and finite type invariants of links , Geom

    K. Habiro, Claspers and finite type invariants of links , Geom. Topol. 4 (2000), 1–83

    2000

  14. [22]

    L. H. Kauffman, Virtual knot theory , European J. Combin. 20 (1999), no. 7, 663–690. MILNOR INV ARIANTS 30

    1999

  15. [23]

    Kawauchi, A survey of Knot Theory , Translated and revised from the 1990 Japanese original by the author

    A. Kawauchi, A survey of Knot Theory , Translated and revised from the 1990 Japanese original by the author. Birkh¨ auser Verlag, Basel, 1996

    1990

  16. [24]

    Kotorii, The Milnor µ invariants and nanophrases , J

    Y. Kotorii, The Milnor µ invariants and nanophrases , J. Knot Theory Ramifications 22 (2013), no. 2, 1250142, 28 pp

    2013

  17. [25]

    Kravchenko and M

    O. Kravchenko and M. Polyak, Diassociative algebras and Milnor’s invariants for tangles , Lett. Math. Phys. 95 (2011), no. 3, 297–316

    2011

  18. [26]

    Karrass and D

    W.Magnus, A. Karrass and D. Solitar, Combinatorial group theory : presentations of groups in terms of generators and relations , Dover books on advanced mathematics, Dover, New York, 1976

    1976

  19. [27]

    Meilhan, Linking number and Milnor invariants Encyclopedia of Knot Theory, Taylor & Francis, (2021), Chapter 83

    J.B. Meilhan, Linking number and Milnor invariants Encyclopedia of Knot Theory, Taylor & Francis, (2021), Chapter 83

    2021

  20. [28]

    Meilhan and A

    J.B. Meilhan and A. Yasuhara, Characterization of finite type string link invariants of degree < 5, Math. Proc. Cambridge Philos. Soc. 148 (2010) 439–472

    2010

  21. [29]

    Meilhan and A

    J.B. Meilhan and A. Yasuhara, Arrow calculus for welded and classical links , Algebr. Geom. Topol. 19 (2019), 397–456

    2019

  22. [30]

    Meilhan and A

    J.B. Meilhan and A. Yasuhara, Link concordances as surfaces in 4-space and the 4-dimensional Milnor invariants , Indiana Univ. Math. J. 71 (2022), 2647–2669

    2022

  23. [31]

    Milnor, Link groups, Ann

    J. Milnor, Link groups, Ann. of Math. (2) 59 (1954), 177—195

    1954

  24. [32]

    Milnor, Isotopy of links , Algebraic geometry and topology, A symposium in honor of S

    J. Milnor, Isotopy of links , Algebraic geometry and topology, A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, N. J., (1957), 280–306

    1957

  25. [33]

    Miyazawa, K

    H.A. Miyazawa, K. Wada and A. Yasuhara, Combinatorial Approach to Milnor Invariants of Welded Links, Michigan Math. J. 73 (2023), 141–170

    2023

  26. [34]

    Murasugi, Nilpotent coverings of links and Milnor’s invariant , Low dimensional topology, 3rd Topology Semin

    K. Murasugi, Nilpotent coverings of links and Milnor’s invariant , Low dimensional topology, 3rd Topology Semin. Univ. Sussex 1982, Lond. Math. Soc. Lect. Note Ser. 95 (1985), 106–142

    1985

  27. [35]

    Turaev, The Milnor invariants and Massey products , Zap

    V.G. Turaev, The Milnor invariants and Massey products , Zap. Nauchn. Sem. LOMI, 66 (1976), 189–203

    1976

  28. [36]

    Porter, The Milnor’s µ-invariants and Massey products , Trans

    R. Porter, The Milnor’s µ-invariants and Massey products , Trans. Amer. Math. Soc. 284(1984), 40–424

    1984

  29. [37]

    Przytycki and W

    J.H. Przytycki and W. Rosicki, Cocycle invariants of codimension 2 embeddings of manifolds , Knots in Poland III. Proceedings of the 3rd conference, Warsaw, Poland, July 18–25, 2010 and B¸ edlewo, Poland, July 25 – August 4, 2010, (2014) 251–289

    2014

  30. [38]

    Roseman, Reidemeister-type moves for surfaces in four-dimensional space , Knot theory, Proceed- ings of the mini-semester, Warsaw, Poland, 1995, (1998), 347–380

    D. Roseman, Reidemeister-type moves for surfaces in four-dimensional space , Knot theory, Proceed- ings of the mini-semester, Warsaw, Poland, 1995, (1998), 347–380

    1998

  31. [39]

    Roseman, Projections of codimension two embeddings , Knots in Hellas ’98

    D. Roseman, Projections of codimension two embeddings , Knots in Hellas ’98. Proceedings of the international conference on knot theory an its ramifications, European Cultural Centre of Delphi, Greece, August 7–15, 1998,World Scientific, (2000) 380–410

    2000

  32. [40]

    Roseman, Elementary moves for higher dimensional knots , Fundam

    D. Roseman, Elementary moves for higher dimensional knots , Fundam. Math. bf 184, (2004) 291– 310

    2004

  33. [41]

    Stallings, Homology and central series of groups , J

    J. Stallings, Homology and central series of groups , J. Algebra, 2 (1965), 170–181

    1965

  34. [42]

    Tucker, Boundary-reducible 3-manifolds and Waldhausen ’s theorem , Michigan Math.J

    T.W. Tucker, Boundary-reducible 3-manifolds and Waldhausen ’s theorem , Michigan Math.J. 20 (1973), 321 – 327

    1973

  35. [43]

    Takabatake, T

    Y. Takabatake, T. Kuboyama, H. Sakamoto, stringcmp:Faster calculation for Milnor invariant, avail- able at https://code.google.com/archive/p/stringcmp/

  36. [44]

    Yasuhara, Classification of string links up to self delta-moves and concordance , Alg

    A. Yasuhara, Classification of string links up to self delta-moves and concordance , Alg. Geom. Topol. 9 (2009), 265–275

    2009

  37. [45]

    Yasuhara, Self Delta-equivalence for Links Whose Milnor’s Isotopy Invariants Vanish , Trans

    A. Yasuhara, Self Delta-equivalence for Links Whose Milnor’s Isotopy Invariants Vanish , Trans. Amer. Math. Soc. 361 (2009), 4721–4749. F aculty of Commerce, W aseda University, 1-6-1 Nishi-W aseda, Shinjuku-ku, Tokyo 169- 8050, Japan Email address : yasuhara@waseda.jp

    2009

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.