Machine Kinematics
solved, then drawn
Concept

Cospectral — where it appears

Having the same characteristic polynomial while not being the same graph. Among the 230 ten-link chains there are two such pairs, and on one of them the two chains do not even share a link assortment — so the cheapest invariant in the subject separates them and the spectrum does not.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Different numbers of ternary links, and the same spectrum. Two of the 230 ten-link chains whose adjacency matrices have identical characteristic polynomials — identical in every one of the eleven coefficients — and which are not the same chain. They do not even share their assortment — 6×2 + 2×3 + 2×4 on the left and 4×2 + 6×3 on the right. Counting the ternary links tells them apart and the spectrum does not. That is worth pausing on: the spectrum is the more sophisticated invariant, it is the one that got written into the literature as a test, and here it is beaten by the first thing anybody would try. The polynomial both of them have is λ^10 − 13λ^8 + 52λ^6 − 4λ^5 − 76λ^4 + 8λ^3 + 32λ^2.

Right until the size nobody checked

The characteristic polynomial of a chain's adjacency matrix is a fingerprint that costs nothing and separates every six-link chain and every eight-link one. At ten links it fails on two pairs — and on one of them, counting the ternary links tells the two chains apart while the polynomial does not.

topology · Topology One, two, sixteen, two hundred and thirty. Every planar chain of mobility one, up to ten links, counted by enumeration rather than quoted. The pins column is forced: a chain of 10 links has one degree of freedom only if it has exactly (3n − 4)/2 pins, which is why no odd link count appears. Pass the count is how many graphs satisfy Grübler's rule, are connected, are simple and give every link at least two pins. Are chains is how many of those survive the fourth condition, that no proper subchain is already a structure — and the gap between the two columns is the whole of this field's first argument: at ten links 1,878 graphs pass a rule that 230 of them deserve. Mechanisms is larger again, because a chain is not a mechanism until a link is held still, and how many different mechanisms that gives is a question about the chain's own symmetry.

Six things a chain is not

A count read as a verdict, a rank trusted where it is blind, a fingerprint used as a proof, a list of five taken for a complete one, a solver treated as a convenience, and a census read as a catalogue of machines. Six claims, each with the number that kills it.

wrong · Misconception

Named alongside it

The objects these essays reach for when they reach for this one.

Canonical formGraph isomorphismInversionKinematic chainType synthesisAssur groupAutomorphismCharacteristic polynomialDegenerate chainDesign ruleEigenvalueGrübler's criterion

All concepts