Clasper Concordance, Whitney towers and repeating Milnor invariants

TL;DR

The paper establishes a precise equivalence between a link bounding a degree-$k$ Whitney tower in $B^4$ and being $C_k$-concordant to the unlink, with the associated tree data matching between Whitney towers and tree claspers. It develops a comprehensive obstruction theory tied to Milnor invariants, higher-order Sato–Levine and Arf invariants, and introduces $k$-repeating variants of Whitney towers and Milnor invariants to classify twisted self $C_k$-concordance. Central to the work is a detailed correspondence between Whitney towers, claspers, and Bing-doubling constructions, enabling a clean translation of geometric operations into combinatorial invariants. The results yield a robust algebraic classification of the $C_k$-concordance filtration in terms of Milnor invariants and higher-order Arf invariants, with a generalization to twisted and $k$-repeating settings that extends link-homotopy-type classifications to higher degrees. These findings provide a framework for understanding 4-dimensional link concordance through explicit, computable invariants and a unified tree-calculus perspective.

Abstract

We show that for each $k\in\mathbb{N}$, a link $L\subset S^3$ bounds a degree $k$ Whitney tower in the 4-ball if and only if it is \emph{$C_k$-concordant} to the unlink. This means that $L$ is obtained from the unlink by a finite sequence of concordances and degree $k$ clasper surgeries. In our construction the trees associated to the Whitney towers coincide with the trees associated to the claspers. As a corollary to our previous obstruction theory for Whitney towers in the 4-ball, it follows that the $C_k$-concordance filtration of links is classified in terms of Milnor invariants, higher-order Sato-Levine and Arf invariants. Using a new notion of $k$-repeating twisted Whitney towers, we also classify a natural generalization of the notion of link homotopy, called twisted \emph{self $C_k$-concordance}, in terms of $k$-repeating Milnor invariants and $k$-repeating Arf invariants.

Figures (7)

• Figure 1: Left: A tree $^3_1>\!\!\!-\!\!\!\!-\!\!\!\!\!-\!\!\!<^{\,2}_{\,1}\,=t(C)$ associated to a clasper $C$ on the $3$-component unlink. Center: The link $L$ resulting from surgery along $C$. Right: A Whitney tower $\mathcal{W}$ bounded by $L$ with $t(\mathcal{W})=\,^3_1>\!\!\!-\!\!\!\!-\!\!\!\!\!-\!\!\!<^{\,2}_{\,1}$ can be constructed from the track of a null-homotopy of $L$ which changes two red-green crossings.
• Figure 3:
• Figure 4: Left: $L=L_1\cup L_2$ bounds $\mathcal{W}$ with $t(\mathcal{W})=\pm\cdot t_p=\pm\langle (1,2),2\rangle$. Right: The link $L^N=\mathcal{W}\cap \partial N$, where $N$ is a $4$--ball neighborhood of $t_p$ containing all singularities of $\mathcal{W}$, and the planar surface cobordism $P=P_1\cup P_2\subset B^4\setminus \textrm{int}(N) \cong S^3\times[0,1]$ from $L$ to $L^N$. (The $s$-arrow indicates the interval factor of $S^3\times[0,1]$.)
• Figure 5: The cobordism $C^+\cup P^0\subset S^3\times[0,1]$ from $L$ to $L^N\amalg U^k$.
• Figure 6: The bottom section is the same as in Figure \ref{['fig:w-tower-to-clasper-2AB']}. The second from bottom section shows the concordance $C^0$ from $L^+$ to $L'$ formed by pushing the punctured maxima and saddles of $P^0$ (Figure \ref{['fig:w-tower-to-clasper-2AB']}) into $\partial N$ . The third section from bottom (purple) depicts clasper-surgery on $\Gamma$ as (the bottom stage of) a $3$-dimensional grope cobordism between $L'$ and $L"$ entirely contained in the $3$-dimensional slice $\partial N$. The top section shows the 'band-cutting' concordance $C^-$ from $L"$ (an internal band sum of an $(n+2+k)$-component unlink) to an $m$-component unlink $U^m$.

Theorems & Definitions (54)

• Theorem 1
• Corollary 2
• Corollary 3
• Corollary 4
• Theorem 5
• Definition 1.1
• Definition 1.2
• Definition 2.1
• Definition 2.2
• Definition 2.3