of Sciences, ul. Gubkina 8, Moscow, 119991 Russia'
author:
- 'Sergey A. Melikhov'
title: Combinatorics of embeddings
---

[^1]

Introduction {#intro}
============

This introduction attempts to motivate the notions we eventually arrive at. The fast reader may want to first look at the Main Theorem and its corollaries in §\[dichotomial spheres\]; these involve only the (very short) definitions of linkless and knotless embeddings (in §\[graphs\]), of a cell complex (in §\[notation\]) and of an $h$-minor (in §\[h-minors\]). However most of the examples and remarks in §\[section:main\] do depend on much of the preceding development.

The constructively minded reader might be best guided by §\[algorithmic\], which explains how non-algorithmic topological notions such as PL spheres and contractible polyhedra can be eliminated from our results and conjectures.

All posets, and in particular simplicial complexes, shall be implicitly assumed to be finite. By a [*graph*]{} we mean a $1$-dimensional simplicial complex (so no loops or multiple edges). By a [*circuit*]{} in a graph we mean a connected subgraph with all vertices of degree $2$. Following the terminology of PL topology (as opposed to that of convex geometry), by a [*polyhedron*]{} we mean a space triangulable by a simplicial complex, and moreover endowed with 