Related papers: Spindle configurations of skew lines
We define an invariant of tangles and framed tangles given a finite crossed module and a pair of functions, called a Reidemeister pair, satisfying natural properties. We give several examples of Reidemeister pairs derived from racks,…
We define an integer valued invariant for two-component links in S^3 by counting projective SU(2) representations of the link group having non-trivial second Stiefel-Whitney class. We show that our invariant is, up to sign, the linking…
For certain classes of knots we define geometric invariants called higher-order genera. Each of these invariants is a refinement of the slice genus of a knot. We find lower bounds for the higher-order genera in terms of certain von Neumann…
We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extension-of-scalars. We deduce that, given a group $G$, the derived and the stable categories…
This is an application of the theory of tilting objects to the geometric setting of perverse sheaves. We show that this theory is a natural framework for Beilinson's gluing of perverse sheaves construction. In the special case of Schubert…
We define a skew-closed structure for Gray-categories extending the mapping space construction of the author.
This paper gives new and elementary combinatorial topological proofs of the classification of unoriented and oriented rational knots and links. These proofs are based on the known classification of alternating knots through flyping, and the…
We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, {\sc…
We introduce and study configuration schemes, which are obtained by ``glueing'' usual schemes along closed embeddings. The category of coherent sheaves on a configuration scheme is investigated. Smooth configuration schemes provide…
We prove necessary and sufficient conditions on a family of (generalised) gridding matrices to determine when the corresponding permutation classes are partially well-ordered. One direction requires an application of Higman's Theorem and…
The class of skew lattices can be seen as an algebraic category. It models an algebraic theory in the category of Sets where the Green's relation D is a congruence describing an adjunction to the category of Lattices. In this paper we will…
We construct smooth manifolds with order two $\pi_1$ and even intersection forms which are irreducible, meaning they do not decompose into non-trivial connected sums. Their intersection forms being even implies that their universal covers…
In this article it is proven that if a knot, K, bounds an imbedded grope of class n, then the knot is n/2-trivial in the sense of Gusarov and Stanford. That is, all type n/2 invariants vanish on K. We also give a simple way to construct all…
A branched twist spin is a generalization of twist spun knots, which appeared in the study of locally smooth circle actions on the $4$-sphere due to Montgomery, Yang, Fintushel and Pao. In this paper, we give a sufficient condition to…
Tilings of the plane resemble the simplicial and other complexes from algebraic topology, but have not been studied from this perspective. We construct finite categories corresponding to polygons with labeled directed edges, and introduce…
The simple permutations in two permutation classes --- the 321-avoiding permutations and the skew-merged permutations --- are enumerated using a uniform method. In both cases, these enumerations were known implicitly, by working backwards…
To each rail knotoid we associate two unoriented knots along with their oriented counterparts, thus deriving invariants for rail knotoids based on these associations. We then translate them to invariants of rail isotopy for rail arcs.
We establish a number of results about smooth and topological concordance of knots in $S^1\times S^2$. The winding number of a knot in $S^1\times S^2$ is defined to be its class in $H_1(S^1\times S^2;\mathbb{Z})\cong \mathbb{Z}$. We show…
One of the major problems in the structural theory of skew braces consists in the classification of skew braces of finite order up to isomorphism. In this light, the open question of the existence of a Cauchy theorem for finite skew braces…
We give a necessary and suffcient condition for almost-flat manifolds with cyclic holonomy to admit a Spin structure. Using this condition we find all 4-dimensional orientable almost- flat manifolds with cyclic holonomy that do not admit a…