that sends each transposition , which interchanges the th element with its neighbour and leaves the other elements fixed, to the nontrivial element .
Permutations in the kernel of are called even permutations, and the rest are called odd permutations.
The signature is well-defined.
One way of seeing this is invoking a standard group presentation of where generators for to (representing ) are subject to relations
and checking that the sign applied to both sides of a relation equation gives the same result.
Another is by invoking a tautological representation of on a polynomial algebra ,
(where core refers to the groupoid of invertible morphisms) and recognizing that for the special polynomial
we have, for each permutation , either or . (The polynomial , which is invariant under the action, is called the discriminant.)
There are various means for computing the signature (also called sign) of a permutation.
Each crossing corresponds to a pair of elements such that , called an inversion.
Another method which does not depend on choosing a total order is to exhibit a permutation through its cycle decomposition. Each cycle of period contributes a sign , and the overall sign is the product of these contributions taken over all the cycles. Thus the signature is given by the parity of the number of cycles of even length.
This cycle description can actually be used to give an independent definition of the signature. It is manifestly well-defined and invariant on conjugacy classes. To check that it defines a homomorphism to , it suffices to check that multiplication by a transposition changes the parity of the number of even-length cycles by one. This is easy if we note that transposing two elements belonging to different cycles merges two cycles into one, whereas transposing two elements belonging to the same cycle splits one cycle into two.