A proof of the Gordon Conjecture

广义Conway--Gordon--Sachs定理的一个简短证明

A short proof of the generalized Conway--Gordon--Sachs theorem

本文针对广义Conway--Gordon--Sachs定理，在整数环上给出了一个比现有证明更简短的证明，该定理将六顶点完全图的情形推广到任意n个顶点的完全图。

Alternative proofs of the Conway-Gordon-Sachs Theorems

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.

Whitney towers, Gropes and Casson-Gordon style invariants of links

In this paper, we prove a conjecture of Friedl and Powell that their Casson-Gordon type invariant of 2-component link with linking number one is actually an obstruction to being height 3.5 Whitney tower/grope concordant to the Hopf Link. The proof employs the notion of solvable cobordism of 3-manifolds with boundary, which was introduced by Cha. We also prove that the Blanchfield form and the Alexander polynomial of links in $S^3$ give obstructions to height 3 Whitney tower/grope concordance. This generalizes the results of Hillman and Kawauchi.

On lattice cohomology and left-orderability

It has been recently conjectured by Boyer-Gordon-Watson that a closed, orientable, irreducible $3$-manifold $M$ is a Heegaard Floer $L$-space if and only if $π_1(M)$ is not left-orderable. In this article, we study this conjecture from the point of view of lattice cohomology, an invariant introduced by Némethi which is conjecturally isomorphic to the $HF^+$ version of Heegaard Floer homology. Using the invariant's combinatorial tractability as a stepping stone, we produce some interesting quite general families of negative-definite graph manifolds against which to test the Boyer-Gordon-Watson …

Algebraic proofs of linear versions of the Conway--Gordon--Sachs theorem and the van Kampen--Flores theorem

In this paper we present short algebraic proofs of the Linear Conway--Gordon--Sachs and the Linear van Kampen--Flores theorems in the spirit of the Radon theorem on convex hulls. {\bf Theorem.} {\it Take any $n+3$ general position points in $\mathbb{R}^n$. If $n$ is odd, then there are two linked $(n+1)/2$-simplices with the vertices at these points. If $n$ is even, then one can choose two disjoint $(n+2)/2$-tuples such that the interiors $(n/2)$-simplices with the vertices at these $(n+2)/2$-tuples intersect each other.} This theorem is interesting even in case of small dimensions.

A refinement of the Conway-Gordon theorems

In 1983, Conway-Gordon showed that for every spatial complete graph on 6 vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on 7 vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruent to 1 modulo 2. In this article, we give integral lifts of the Conway-Gordon theorems above in terms of the square of the linking number and the second coefficient of the Conway polynomial. As applications, we give alternative topological proofs of theorems of Brown-Ramirez Alfo…