Mod 2 Pairwise Linking in Topology

• Mod 2 pairwise linking is defined by reducing classical linking numbers modulo 2 to capture binary link behavior in knots, graphs, and algebraic structures.
• The methodology converts combinatorial problems, such as the Independent Set Problem, into a symmetric n×n linking matrix over F2 that encodes pairwise relations.
• This approach implies that NP-hardness in trivial sublink decision problems originates from binary linking, eliminating the need for higher-order invariants.

Mod 2 pairwise linking refers to the paper and use of binary (modulo 2) pairwise linking invariants in various mathematical domains, particularly in low-dimensional topology, combinatorial group theory, algebraic geometry, and arithmetic. This concept appears in multiple forms depending on context: as a mod 2 reduction of classical linking numbers in knot theory, as binary adjacency encoding in computational topology and graph theory, and as the core of specific invariants in algebraic and arithmetic settings. The central idea is that, for a given collection of objects (such as knots, graph vertices, or algebraic generators), the information about "linkage" between any two elements is captured not by full numeric invariants, but by the parity (0 or 1) of links, thus fitting naturally into mod 2 linear algebra and combinatorics. This leads to profound implications for both the structure of invariants and for the computational complexity of related decision problems.

1. Mod 2 Pairwise Linking: Definitions and Topological Foundations

Mod 2 pairwise linking arises most transparently in classical knot theory, where the linking number $\operatorname{lk}(K_1, K_2)$ of two oriented link components $K_1$ and $K_2$ in $S^3$ is computed as half the sum of signed crossings between the two components. Its value is an integer and encodes how many times the components are intertwined. The reduction modulo 2,

$$\ell_{i, j} \equiv \operatorname{lk}(K_i, K_j) \pmod{2}$$

distills this to a binary invariant that takes value 1 if the components are "oddly" linked and 0 otherwise. This mod 2 version completely characterizes, for certain algorithmic problems, whether components are nontrivially linked in the minimal sense.

In computational settings, specifically in the context of the Trivial Sublink Problem, mod 2 pairwise linking refers to constructing the $n \times n$ symmetric linking matrix $L$ of an $n$-component link diagram, with entries in $\mathbb{F}_2$, representing the mod 2 linking numbers. The entry $L_{i,j}=1$ if components $i$ and $j$ are mod 2 linked, and zero otherwise. This matrix then serves as the adjacency matrix for the encoding of set systems, allowing combinatorial algorithms to be expressed directly in terms of mod 2 linear algebra.

2. Reduction from Combinatorial Problems to Mod 2 Pairwise Linking

A key technical contribution of (Cheng et al., 16 Sep 2025) is a reduction from the Independent Set Problem to the Trivial Sublink Problem via mod 2 pairwise linking. Given a graph with adjacency matrix $A$, the reduction constructs a link $L_A$ with $n$ components such that $L_A$'s linking matrix (modulo 2) equals $A$. This is achieved explicitly by building a pure braid on $n$ strands whose trace closure yields $L_A$, with the property that off-diagonal entries of the linking matrix (interpreted mod 2) exactly capture the original graph's adjacency.

The correspondence is exact: a collection $\{K_{i_1},...,K_{i_k}\}$ forms a trivial $k$-component sublink (i.e., all pairwise mod 2 linking numbers vanish) if and only if the corresponding set of vertices in the graph is an independent set. Thus, the mod 2 pairwise linking data is both necessary and sufficient for the computational encoding. There is no need for integral linking numbers or higher-order Milnor invariants: the complexity is already present in the parity structure.

3. Contrasts with Higher-Order and Integral Linking

Earlier NP-hardness results for related problems in 3-manifold topology (e.g., those by Koenig–Tsvietkova, de Mesmay–Rieck–Sedgwick–Tancer) leveraged Brunnian link constructions, such as the Borromean rings, where every proper sublink is trivial yet the whole link is nontrivially linked due to higher-order (Milnor) invariants. Such examples require analysis beyond pairwise linking—specifically, triple and higher-order linking numbers, which are sensitive to more intricate topological entanglements.

In contrast, the present reduction in (Cheng et al., 16 Sep 2025) is conceptually and technically simpler, relying only on binary (mod 2) pairwise data. The conclusion is that, for decision problems such as finding a trivial $k$-component sublink, mod 2 pairwise linking accounts for all the computational difficulty, and higher-order invariants or integral data are irrelevant for hardness. The related classical formula,

$$\operatorname{lk}(K_i, K_j) = \frac{1}{2} \sum_{\text{crossings } c \text{ between } K_i \text{ and } K_j} \operatorname{sign}(c)$$

is invoked, but the computational reduction uses only the parity of the sum.

4. Algorithmic and Pedagogical Implications

The proof strategy is fully elementary—employing only linking numbers (mod 2), braid group operations, the Reidemeister moves, and a reduction from the well-known NP-complete Independent Set Problem. The construction of the link from a graph is explicit and can be visualized step-by-step. Each component of the link realized as the closure of a strand in the braid corresponds to a graph vertex; the presence of a pairwise mod 2 linking between components encodes an edge.

This pedagogical clarity allows the theory to be presented without reference to deep 3-manifold topology or advanced algebraic topology, making it well suited for both undergraduate instruction and computational visualization (as facilitated by the accompanying web application).

5. Broader Computational and Theoretical Consequences

The reduction demonstrates that NP-hardness of the Trivial Sublink Problem is an inherent feature already at the mod 2 pairwise level, not requiring the complexity of invariants arising from the full link group or higher Massey products. Consequently, the computational intractability of seemingly topologically deep problems can be ascribed to parity-based pairwise structures.

This observation has direct implications for the computational and algorithmic analysis of topology, suggesting that mod 2 invariants are surprisingly "expressive" for encoding combinatorial complexity. It further implies that the problem of finding the largest unlinked sublink, or approximating its size, inherits the NP-hardness of the Maximum Independent Set Problem in graphs.

6. Connections to Related Mathematical Frameworks

This binary perspective on linking has analogues in other areas:

• In bipartite network analysis, the removal of "internal links" leaving the 1-mode projection unchanged is tantamount to preserving the mod 2 pairwise connectivity structure (Allali et al., 2011).
• In arithmetic topology, mod 2 Milnor invariants measure linkage among arithmetic cycles or primes via quadratic (mod 2) relations in Galois groups (Efrat, 12 Mar 2024), again encoding linkages at the mod 2 level.
• In algebraic geometry, reductions modulo 2 establish correspondences between roots of even lattices and elements satisfying $q(v) = 1$ in finite quadratic spaces, giving rise to a mod 2 pairing structure (Beauville, 2022).

These analogies underscore a unifying theme: mod 2 pairwise linking, despite its apparent simplicity, underpins rich and computationally deep phenomena across topology, algebra, and arithmetic.

7. Summary Table: Mod 2 Pairwise Linking in Link Decision Problems

Feature	Mod 2 Pairwise Linking	Higher-order/Integral Linking
Captures NP-hardness of sublink problem	Yes	Not necessary for hardness
Computational encoding	Adjacency matrix in $\mathbb{F}_2$	Requires full linking/integral data
Construction via pure braids	Sufficient	Not required
Preserves all combinatorial complexity	Yes	Redundant for these decision problems
Example	Reduction from Independent Set	Brunnian links in previous reductions

This encapsulates the principle that the elementary mod 2 pairwise linking structure of a link suffices to encode—and is responsible for—the computational complexity of key link invariants and decision problems in low-dimensional topology.