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arxiv: 2605.23086 · v1 · pith:6DNDDPUWnew · submitted 2026-05-21 · 🧮 math.GT

Lifting Milnor Invariants for 3-Component Links

Christopher W. Davis , JungHwan Park This is my paper

Pith reviewed 2026-05-25 04:54 UTC · model grok-4.3

classification 🧮 math.GT
keywords Milnor invariantsconcordanceweak cobordism3-component linksKojima-Yamasaki eta-invariantlink invariants4-ball embeddings
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The pith

Integer-valued γ-invariants for 3-component links lift certain Milnor invariants and stay unchanged under concordance and weak cobordism.

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

The paper defines a sequence of integer-valued invariants γ^k(L) for any 3-component link L. It establishes that these invariants are preserved under concordance and, more generally, under weak cobordism. An auxiliary invariant h(L) is introduced as the 3-component version of the Kojima-Yamasaki η-invariant, and this h(L) is shown to recover the γ-invariants exactly. The lifting property then follows directly from the recovery. Applications include a classification of such links up to weak cobordism when one component has trivial Alexander polynomial, plus a characterization of knots bounding disks in the 4-ball whose complements have fundamental group Z.

Core claim

We define a sequence of integer-valued invariants γ^k(L) for a 3-component link L. We prove that the resulting γ-invariants are invariant under concordance, and more generally under weak cobordism, and that they lift certain Milnor invariants of 3-component links. To establish this, we introduce an invariant h(L), a 3-component analogue of the Kojima-Yamasaki η-invariant, and show that it recovers the γ-invariants.

What carries the argument

The γ^k invariants recovered by the auxiliary h(L), a 3-component analogue of the Kojima-Yamasaki η-invariant.

If this is right

  • The γ-invariants give a weak-cobordism classification of 3-component links whenever the distinguished component has trivial Alexander polynomial.
  • They characterize which knots bound continuously embedded disks in B^4 whose complements have fundamental group exactly Z.
  • Because they lift Milnor invariants, they supply concordance obstructions that classical Milnor invariants alone cannot detect.
  • The same construction yields new invariants that are unchanged under any weak cobordism between 3-component links.

Where Pith is reading between the lines

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

  • The lifting mechanism may extend to produce concordance invariants for links with more than three components by iterating the h(L) construction.
  • The characterization of disk-bounding knots could be used to test whether a given knot is slice in a stronger 4-dimensional sense than classical sliceness.
  • If two links agree on all γ^k then any Milnor invariant they share must also agree, tightening the relationship between the two families of invariants.

Load-bearing premise

The auxiliary invariant h(L) recovers the γ-invariants exactly, which is required for the lifting of Milnor invariants to hold.

What would settle it

An explicit pair of concordant 3-component links whose computed γ^k values differ would falsify the invariance claim.

Figures

Figures reproduced from arXiv: 2605.23086 by Christopher W. Davis, Junghwan Park.

Figure 1
Figure 1. Figure 1: Tubing a Seifert surface G to the torus Ti . Thus Yi ∩ N bounds a compact surface Y 0 i ⊆ Yi . We may now modify N by an ambient surgery as follows. Remove ν(Yi ∩ N) from N and replace it with the boundary of a tubular neighborhood of Y 0 i . This completes the proof that (4) implies (B). □ We introduce the following definition. Definition 2.3. Let L and L ′ be weakly cobordant 3-component links with a dis… view at source ↗
Figure 2
Figure 2. Figure 2: A 3-component link with a chosen Seifert surface and the resulting derivative. The next example illustrates that γ k (L, G) can depend on the choice of Seifert surface G. In Figure 3A we consider the same link L = (L1, L2, L3) as in Figure 2A, but with a different Seifert surface G′ 1 . This results in G′ 1∩G2 being disconnected. Adding a tube to G2 results in the derivative L12 seen in Figure 3C. Then γ 1… view at source ↗
Figure 3
Figure 3. Figure 3: A we consider the same link L = (L1, L2, L3) as in Figure 2A, but with a different Seifert surface G′ 1 . This results in G′ 1∩G2 being disconnected. Adding a tube to G2 results in the derivative L12 seen in Figure 3C. Then γ 1 (L, G′ 1 ) = lk(L12, L3) = 0. In fact, since L12 is null-homologous in the exterior of G′ 1 , we have γ k (L, G′ 1 ) = 0 for all k > 0. This is consistent with Theorem 1.1, since (T… view at source ↗
Figure 4
Figure 4. Figure 4: A 3-component link and its first three derivatives. (A) The link (L1, L2, L3), where L1 bounds the evident Seifert surface G1. (B) A Seifert sur￾face G2 for L2. (C) The derivative L12. (D) An isotopy of L12 disjoint from G1. (E) The derivative L112. (F) The derivative L1112. link in [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Constructing L3 by banding together a0 copies of L 0 3 , a1 copies of L 1 3 , and so on, produces a link with γ k (L) = ak for k ≤ n and γ k (L) = 0 for k > n [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A link L whose distinguished component L1 bounds the Seifert surface G, together with a symplectic basis {a1, a2} for H1(G). 2.2. Computation from a Seifert matrix. In [Coc85, Section 8], Cochran explains how to compute his iterated derivatives and the resulting β-invariants in terms of a Seifert matrix. We now do the same for our γ-invariants. Indeed, let G1 and G2 be Seifert surfaces for L1 and L2, respe… view at source ↗
Figure 7
Figure 7. Figure 7: A 2-component link with vanishing β-invariants that is not weakly cobordant to the unlink. Recall that the derived series of a group is defined by G(0) = G and G(k+1) = [G(k) , G(k) ]. It then follows from [DHP23, Proposition 6.1] that the knot R(J) is 2-solvable, and in particular is 1.5-solvable, which is a contradiction. □ 7. Continuously embedded disks with π1 = Z Although our main results concern 3-co… view at source ↗
read the original abstract

We define a sequence of integer-valued invariants $\gamma^k(L)$ for a $3$-component link $L$. We prove that the resulting $\gamma$-invariants are invariant under concordance, and more generally under weak cobordism, and that they lift certain Milnor invariants of 3-component links. To establish this, we introduce an invariant $h(L)$, a $3$-component analogue of the Kojima--Yamasaki $\eta$-invariant, and show that it recovers the $\gamma$-invariants. As applications, we obtain a weak-cobordism classification when the distinguished component has trivial Alexander polynomial and characterize knots that bound continuously embedded disks in $B^4$ whose complements have fundamental group $\mathbb{Z}$.

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

0 major / 2 minor

Summary. The paper defines a sequence of integer-valued invariants γ^k(L) for a 3-component link L. It proves that the γ-invariants are invariant under concordance and more generally under weak cobordism, and that they lift certain Milnor invariants of 3-component links. To establish the lifting, the authors introduce an auxiliary invariant h(L), a 3-component analogue of the Kojima–Yamasaki η-invariant, and show that h(L) recovers the γ-invariants. Applications include a weak-cobordism classification when the distinguished component has trivial Alexander polynomial and a characterization of knots bounding continuously embedded disks in B^4 whose complements have fundamental group ℤ.

Significance. If the invariance and lifting statements hold, the work supplies new concordance invariants for 3-component links that extend Milnor theory in a controlled way. The construction of h(L) and its recovery of the γ-sequence is a concrete technical contribution. The applications to weak cobordism classification and to 4-dimensional knot theory are of interest to the field. The manuscript provides explicit proofs of the claimed invariance and recovery properties.

minor comments (2)
  1. [Abstract / Introduction] The abstract states that h(L) recovers the γ-invariants, but the precise statement of this recovery (e.g., whether it is equality or up to a fixed multiple) should be made explicit in the introduction or in the statement of the main theorem.
  2. [Introduction] Notation for the sequence γ^k(L) and the range of k should be clarified early; it is not immediately clear whether k begins at 1 or 2 and whether the invariants are defined for all k or only sufficiently large k.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The referee's summary correctly identifies the definition of the γ^k invariants, their concordance and weak-cobordism invariance, the auxiliary h-invariant, the lifting of Milnor invariants, and the applications to weak-cobordism classification and 4-dimensional knot theory. No specific major comments are listed in the report.

Circularity Check

0 steps flagged

No circularity detected; derivation self-contained

full rationale

The abstract defines new invariants γ^k(L) and auxiliary h(L), then states proofs of invariance under concordance/weak cobordism and that h recovers γ to enable lifting of Milnor invariants. No equations, fitted parameters, self-citations, or ansatzes are exhibited that would reduce any claimed result to an input by construction. The lifting statement is presented as a proved theorem rather than a definitional identity, and the paper's central claims rest on independent arguments not visible as tautological in the given text. This is the default honest finding when no load-bearing reduction can be quoted.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; all lists left empty.

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