某些内在链接定理的逆定理

四维空间中2复形的内在链接

Intrinsic Linking of 2-complexes in $\mathbb{R}^4$

研究构造了无限多个四维空间中内在链接的2复形，证明含K6为子图的悬垂图嵌入四维空间时包含非平凡链接的1和2循环。

Intrinsic knotting and linking of complete graphs

We show that for every m in N, there exists an n in N such that every embedding of the complete graph K_n in R^3 contains a link of two components whose linking number is at least m. Furthermore, there exists an r in N such that every embedding of K_r in R^3 contains a knot Q with |a_2(Q)| > m-1, where a_2(Q) denotes the second coefficient of the Conway polynomial of Q.

单纯 $n$-复形在 $\mathbb{R}^{2n}$ 中的内在链环：一个额外的最小 $n$-复形

Intrinsic linking of simplicial $n$-complexes in $\mathbb{R}^{2n}$: An additional minimal $n$-complex

本文构造了一个额外的单纯 $n$-复形，该复形在任意嵌入 $\mathbb{R}^{2n}$ 时必然包含一个不可拆分的两分支链环（由 $(n-1)$-球面和 $n$-球面组成），扩展了作者之前的最小 $n$-复形系列。

Generalization of the Conway-Gordon theorem and intrinsic linking on complete graphs

Conway and Gordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent two-component links is odd, and Kazakov and Korablev proved that for every spatial complete graph with arbitrary number of vertices greater than six, the sum of the linking numbers over all of the constituent two-component Hamiltonian links is even. In this paper, we show that for every spatial complete graph whose number of vertices is greater than six, the sum of the square of the linking numbers over all of the two-component Hamiltonian links is determin…

Intrinsic linking and knotting are arbitrarily complex in directed graphs

Fleming and Foisy recently proved the existence of a digraph whose every embedding contains a $4$-component link, and left open the possibility that a directed graph with an intrinsic $n$-component link might exist. We show that, indeed, this is the case. In fact, much as Flapan, Mellor, and Naimi show for graphs, knotting and linking are arbitrarily complex in directed graphs. Specifically, we prove the analog for digraphs of the main theorem of their paper: for any $n$ and $α$, every embedding of a sufficiently large complete digraph in $\mathbb{R}^3$ contains an oriented link with component…

Intrinsic linking with linking numbers of specified divisibility

Let $n$, $q$ and $r$ be positive integers, and let $K_N^n$ be the $n$-skeleton of an $(N-1)$-simplex. We show that for $N$ sufficiently large every embedding of $K_N^n$ in $\mathbb{R}^{2n+1}$ contains a link $L_1\cup\cdots\cup L_r$ consisting of $r$ disjoint $n$-spheres, such that the linking number $link(L_i,L_j)$ is a nonzero multiple of $q$ for all $i\neq j$. This result is new in the classical case $n=1$ (graphs embedded in $\mathbb{R}^3$) as well as the higher dimensional cases $n\geq 2$; and since it implies the existence of a link $L_1\cup\cdots\cup L_r$ such that $|link(L_i,L_j)|\geq q…

Intrinsic linking and knotting are arbitrarily complex

We show that, given any $n$ and $α$, every embedding of any sufficiently large complete graph in $\mathbb{R}^3$ contains an oriented link with components $Q_1$, ..., $Q_n$ such that for every $i\not =j$, $|\lk(Q_i,Q_j)|\geqα$ and $|a_2(Q_i)|\geqα$, where $a_{2}(Q_i)$ denotes the second coefficient of the Conway polynomial of $Q_i$.

嵌入 $\mathbb{R}^{2n}$ 中的 simplicial $n$-复形的内在链接

Intrinsic linking of a simplicial $n$-complex embedded in $\mathbb{R}^{2n}$

研究证明在任何嵌入 $\mathbb{R}^{2n}$ 中，最小的 simplicial $n$-复形必然包含一个不可分割的两组件链接，由一个 $(n-1)$-球面和一个 $n$-球面组成，扩展了非分离平面图的高维泛化。

Some Ramsey-type results on intrinsic linking of n-complexes

Define the complete n-complex on N vertices to be the n-skeleton of an (N-1)-simplex. We show that embeddings of sufficiently large complete n-complexes in R^{2n+1} necessarily exhibit complicated linking behaviour, thereby extending known results on embeddings of large complete graphs in R^3 (the case n=1) to higher dimensions. In particular, we prove the existence of links of the following types: r-component links, with the linking pattern of a chain, necklace or keyring; 2-component links with linking number at least lambda in absolute value; and 2-component links with linking number a non-…

空间图的不变量关联

Linking invariants of spatial graphs

该研究回顾空间图的环绕数与Wu–Simon数定义，揭示Conway–Gordon–Sachs定理的逆命题，提出其向多重环绕推广的猜想，论述基于平面图表，无需空间几何知识。

Generalizations of the Conway-Gordon theorems and intrinsic knotting on complete graphs

In 1983, Conway and Gordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent two-component links is odd, and that for every spatial complete graph on seven vertices, the sum of the Arf invariants over all of the Hamiltonian knots is odd. In 2009, the second author gave integral lifts of the Conway-Gordon theorems in terms of the square of the linking number and the second coefficient of the Conway polynomial. In this paper, we generalize the integral Conway-Gordon theorems to complete graphs with arbitrary number of vertice…

超图的可实现性与内在链接理论

Realizability of hypergraphs and intrinsic link theory

本文通过降维方法给出了三维空间内在链接和四维空间内在交集定理的简洁证明，并探讨了超图可实现性问题。