arXiv subjects
Publications and source records attributed to Simon A. King.
We complete the calculation of the Ext algebra of the principal 2-block of the Mathieu group M_11, extending work of Benson-Carlson and of Pawloski - and duplicating work of Generalov.
We provide a non-commutative version of the F5 algorithm, namely for right-modules over path algebra quotients. It terminates, if the path algebra quotient is a basic algebra. In addition, we use the F5 algorithm in negative degree monomial orderings to compute Loewy layers.
By explicit machine computation we obtain the mod-2 cohomology ring of the third Conway group Co_3. It is Cohen-Macaulay, has dimension 4, and is detected on the maximal elementary abelian 2-subgroups.
We describe the computation of the mod-2 cohomology rings of all 2328 groups of order 128. One consequence is that all groups of order less than 256 satisfy the strong form of Benson's Regularity Conjecture.
We introduce and investigate notions of persistent homology for p-groups and for coclass trees of p-groups. Using computer techniques we show that persistent homology provides fairly strong homological invariants for p-groups of order at most 81. The strength of these invariants, and some elementary theoretical properties, suggest that persistent homology may be a useful tool in the study of prime-power groups.
A very classical subject in Commutative Algebra is the Invariant Theory of finite groups. In our work on 3-dimensional topology (S. King, Ideal Turaev-Viro invariants. To appear in Top. Appl.), we found certain examples of group actions on polynomial rings. When we tried to compute the invariant ring using Singular or Magma, it turned out that the existing algorithms did not suffice. We present here a new algorithm for the computation of secondary invariants, if primary invariants are given. Our benchmarks show that the implementation of our algorithm in the library "finvar" of Singular marks a dramatic improvement in the manageable problem size. A particular benefit of our algorithm is that the computation of irreducible secondary invariants does not involve the explicit computation of reducible secondary invariants, which may save resources. The implementation of our algorithm in Singular is for the non-modular case; however, the key theorem of our algorithm holds in the modular case as well and might be useful also there.
A Turaev-Viro invariant is a state sum, i.e., a polynomial that can be read off from a special spine or a triangulation of a compact 3-manifold. If the polynomial is evaluated at the solution of a certain system of polynomial equations (Biedenharn-Elliott equations) then the result is a homeomorphism invariant of the manifold (``numerical Turaev-Viro invariant''). The equation system defines an ideal, and actually the coset of the polynomial with respect to that ideal is a homeomorphism invariant as well (``ideal Turaev-Viro invariant''). It is clear that ideal Turaev-Viro invariants are at least as strong as numerical Turaev-Viro invariants, and we show that there is reason to expect that they are strictly stronger. They offer a more unified approach, since many numerical Turaev-Viro invariants can be captured in a singly ideal Turaev-Viro invariant. Using computer algebra, we obtain computational results on some examples of ideal Turaev-Viro invariants.
We introduce a numerical isomorphism invariant p(T) for any triangulation T of S^3. Although its definition is purely topological (inspired by the bridge number of knots), p(T) reflects the geometric properties of T. Specifically, if T is polytopal or shellable then p(T) is `small' in the sense that we obtain a linear upper bound for p(T) in the number n=n(T) of tetrahedra of T. Conversely, if p(T) is `small' then T is `almost' polytopal, since we show how to transform T into a polytopal triangulation by O((p(T))^2) local subdivisions. The minimal number of local subdivisions needed to transform T into a polytopal triangulation is at least $\frac{p(T)}{3n}-n-2$. Using our previous results [math.GT/0007032], we obtain a general upper bound for p(T) exponential in n^2. We prove here by explicit constructions that there is no general subexponential upper bound for p(T) in n. Thus, we obtain triangulations that are `very far' from being polytopal. Our results yield a recognition algorithm for S^3 that is conceptually simpler, though somewhat slower, as the famous Rubinstein-Thompson algorithm.
We present a new and shorter proof of Stocking's result that any strongly irreducible Heegaard surface of a closed orientable triangulated 3-manifold is isotopic to an almost normal surface. We also re-prove a result of Jaco and Rubinstein on normal spheres. Both proofs are based on the "reduction" technique introduced by the author.
It is known that any two triangulations of a compact 3-manifold are related by finite sequences of certain local transformations. We prove here an upper bound for the length of a shortest transformation sequence relating any two triangulations of the 3-dimensional projective space, in terms of the number of tetrahedra.
Any two triangulations of a closed surface with the same number of vertices can be transformed into each other by a sequence of regular flips, provided the number of vertices exceeds a number N depending on the surface. Examples show that in general N is bigger than the minimal number of vertices of a triangulation. The existence of N was known, but no estimate. This paper provides an estimate for N that is linear in the Euler characteristic of the surface.
We study the crossing number of links that are formed by edges of a triangulation T of the 3-sphere with n tetrahedra. We show that the crossing number is bounded from above by an exponential function of n^2. In general, this bound can not be replaced by a subexponential bound. However, if T is polytopal (resp. shellable) then there is a quadratic (resp. biquadratic) upper bound in n for the crossing number. In our proof, we use a numerical invariant p(T), called polytopality, that we have introduced in math.GT/0009216.