[Paper Review] Alternative proofs of the Conway-Gordon-Sachs Theorems

Arseny Zimin|arXiv (Cornell University)|Nov 12, 2013

Computability, Logic, AI Algorithms4 references3 citations

TL;DR

This paper presents alternative, planar-based proofs of the Conway-Gordon-Sachs Theorems, reducing 3D linking theorems for $K_6$ and $K_{4,4}$ to invariants in the plane. By leveraging the van Kampen invariant modulo 2 and planar general position arguments, it establishes that any piecewise-linear embedding of $K_6$ or $K_{4,4}$ in $bR^3$ must contain at least one pair of linked triangles or quadrilaterals, respectively, via topological invariance and parity arguments.

ABSTRACT

In this paper we present new proofs of the Conway-Gordon-Sachs and Sachs Theorems on the linked cycles in graphs embedded in $\R^3$. We reduce these theorems to certain property of graphs mapped to the plane.

Motivation & Objective

• To provide new, planar-reduction-based proofs of the Conway-Gordon-Sachs Theorems, avoiding traditional isotopy or sphere intersection arguments.
• To establish that any piecewise-linear embedding of $K_6$ in $bR^3$ contains at least one pair of linked 3-cycles, using planar invariants.
• To extend the same approach to $K_{4,4}$, proving the existence of linked 4-cycles in any such embedding via a planar $K_{3,3}$ projection.
• To demonstrate that the linking parity is invariant under continuous deformation, using the invariance of the van Kampen invariant modulo 2.
• To simplify the original proofs by reducing 3D topological complexity to planar combinatorial topology and intersection parity.

Proposed method

• Reduces the 3D linking problem to a planar setting by projecting embeddings of $K_6$ and $K_{4,4}$ onto a generic plane.
• Uses the van Kampen invariant $v(f) mod 2$ for configurations of five or six points in general position in the plane.
• Proves that $v(f) = 1$ for any five-point configuration in general position, establishing a topological invariant.
• Applies the invariance of $v(f)$ under continuous deformation of point positions (with fixed four-point base), using broken-line homotopy and parity of intersections.
• Relies on planar intersection lemmas: two triangles in general position in the plane intersect at an even number of points.
• Translates the 3D linking condition (via convex hull intersection) into a parity condition on planar segment intersections, using the 'higher than' relation from a viewpoint.

Experimental results

Research questions

• RQ1 Can the Conway-Gordon-Sachs Theorems be reproven using planar invariants instead of 3D isotopy or sphere intersection theorems?
• RQ2 Is the van Kampen invariant modulo 2 invariant under continuous deformation of five points in the plane with four fixed?
• RQ3 Does the linking of cycles in $K_6$ and $K_{4,4}$ in $bR^3$ follow from a planar topological invariant of $K_5$ and $K_{3,3}$ respectively?
• RQ4 Can the parity of linking number in 3D be reduced to the parity of segment intersections in a planar projection?
• RQ5 Is the linking of two 4-cycles in $K_{4,4}$ in $bR^3$ detectable via the planar van Kampen invariant of a $K_{3,3}$ subgraph?

Key findings

• The van Kampen invariant $v(f)$ for any five-point configuration in general position in the plane is always 1 modulo 2.
• The invariant $v(f)$ is independent of the position of the fifth point, provided the other four are fixed, proving its topological invariance.
• For any piecewise-linear embedding of $K_6$ in $bR^3$, there exists at least one pair of linked 3-cycles, as shown by reduction to planar $K_5$ and invariance of $v(f)$.
• For any piecewise-linear embedding of $K_{4,4}$ in $bR^3$, there exists at least one pair of linked 4-cycles, established via projection to $K_{3,3}$ and $v(f) = 1$ in the plane.
• The linking condition in 3D (intersection of one triangle with the convex hull of another at a single point) is equivalent to an odd number of planar segment intersections under the 'higher than' relation.
• The proof avoids isotopy and sphere intersection theorems, instead relying on planar combinatorics and parity, offering a simpler alternative to the original Conway-Gordon-Sachs proof.