Narrow Results

We investigate algebraic and computability-theoretic properties of orderable magmas. A magma is an algebraic structure with a single binary operation. A right order on a magma is a linear ordering of its domain, which is right-invariant with respect to the magma operation. We use tools of computability theory to investigate Turing complexity of orders on computable orderable magmas. A magma is computable if it is finite, or if its domain can be identified with the set of natural numbers and the magma operation is computable. Interesting orderable magmas that are not even associative come from knot theory.

A ribbon is a smooth mapping (possibly self-intersecting) of an annulus $S^1\times I$ in 3-space having constant width $R$. Given a regular parametrization $x(s)$, and a smooth unit vector field $u(s)$ based along $x$, for a knot $K$, we may define a ribbon of width $R$ associated to $x$ and $u$ as the set of all points $x(s)+ru(s)$, $r\in [0,R]$. For large $R$, ribbons, and their outer edge curves, may have self-intersections. In this paper, we analyze how the knot type of the outer ribbon edge $x(s)+Ru(s)$ relates to that of the original knot $K$. Generically, as $R\to \infty$, there is an eventual constant knot type. This eventual knot type is one of only finitely many possibilities which depend just on the vector field $u$. The particular knot type within the finite set depends on the parametrized curves $x(s)$, $u(s)$, and their interactions. We demonstrate a way to control the curves and their parametrizations so that given two knot types $K_1$ and $K_2$, we can find a smooth ribbon of constant width connecting curves of these two knot types.

We directly connect topological changes that can occur in mathematical three-space via surgery, with black hole formation, the formation of wormholes and new generalizations of these phenomena. This work widens the bridge between topology and natural sciences and creates a new platform for exploring geometrical physics.

The non-orientable 4-genus of a knot $K$ in $S^3$ is defined to be the minimum first Betti number of a non-orientable surface $F$ smoothly embedded in $B^4$ so that $K$ bounds $F$. We will survey the tools used to compute the non-orientable 4-genus, and use various techniques to calculate this invariant for non-alternating 11 crossing knots. We will also view obstructions to a knot bounding a Möbius band given by the double branched cover of $S^3$ branched over $K$.

The role of topology in the statistical mechanics of surfaces in the lattice is considered. The possibility that a phase transition driven by the number of boundary components occurs is investigated. Bounds on the limiting free energy is derived and conditions for the existence of a critical point in the phase diagram are presented.