Kirkman's first mathematical publication was in the
Cambridge and Dublin Mathematical Journal in 1846, on a problem involving
Steiner triple systems that had been published two years earlier in ''
The Lady's and Gentleman's Diary'' by
Wesley S. B. Woolhouse. Generalizing the
quaternions and
octonions, Kirkman called a
pluquaternion Qa a representative of a system with
a imaginary units,
a > 3. Kirkman's paper was dedicated to confirming Cayley's assertions concerning two equations among triple-products of units as sufficient to determine the system in case
a = 3 but not
a = 4. By 1900 these number systems were called
hypercomplex numbers, and later treated as part of the theory of
associative algebras.
Polyhedral combinatorics Beginning in 1853, Kirkman began working on
combinatorial enumeration problems concerning
polyhedra, beginning with a proof of
Euler's formula and concentrating on simple polyhedra (the polyhedra in which each vertex has three incident edges). He also studied
Hamiltonian cycles in polyhedra, and provided an example of a polyhedron with no Hamiltonian cycle, prior to the work of
William Rowan Hamilton on the
Icosian game. He
enumerated cubic Halin graphs, over a century before the work of Halin on these graphs. He showed that every polyhedron can be generated from a pyramid by face-splitting and vertex-splitting operations, and he studied
self-dual polyhedra.
Late work Kirkman was inspired to work in
group theory by a prize offered beginning in 1858 (but in the end never awarded) by the
French Academy of Sciences. His contributions in this area include an enumeration of the transitive
group actions on sets of up to ten elements. However, as with much of his work on polyhedra, Kirkman's work in this area was weighed down by newly invented terminology and, perhaps because of this, did not significantly influence later researchers. In the early 1860s, Kirkman fell out with the mathematical establishment and in particular with
Arthur Cayley and
James Joseph Sylvester, over the poor reception of his works on polyhedra and groups and over issues of priority. Much of his later mathematical work was published (often in
doggerel) in the problem section of the
Educational Times and in the obscure
Proceedings of the Literary and Philosophical Society of Liverpool. However, in 1884 he began serious work on
knot theory, and with
Peter Guthrie Tait published an enumeration of the knots with up to ten crossings. He remained active in mathematics even after retirement, until his death in 1895. ==Awards and honours==