Related papers: Classification of tangent and transverse knots in …

In this paper we show how to place Michael Berry's discovery of knotted zeros in the quantum states of hydrogen in the context of general knot theory and in the context of our formulations for quantum knots. Berry gave a time independent…

We establish second main theorems for holomorphic curves into a projective subvary $V \subset \mathbb{P}^n(\mathbb{C})$ of dimension $k$, intersecting hypersurfaces in $N$-subgeneral position with respect to $V$ $(N > k)$. Our results…

In view of the self-linking invariant, the number $|K|$ of framed knots in $S^3$ with given underlying knot $K$ is infinite. In fact, the second author previously defined affine self-linking invariants and used them to show that $|K|$ is…

Let D be a rank 2 bracket generating distribution on a 4 manifold, D is Engel if its growth vector is maximal. When this maximality fails the distribution is degenerate. We prove Sard's conjecture for the endpoint map in the case of…

For each $g>0$ we give infinitely many knots that are strongly negative amphichiral, hence rationally slice and representing 2-torsion in the smooth concordance group, yet which do not bound any locally flatly embedded surface in the 4-ball…

Many bundle gerbes constructed in practice are either infinite-dimensional, or finite-dimensional but built using submersions that are far from being fibre bundles. Murray and Stevenson proved that gerbes on simply-connected manifolds,…

Many three dimensional manifolds are two-fold branched covers of the three dimensional sphere. However, there are some that are not. This paper includes exposition about two-fold branched covers and many examples. It shows that there are…

We establish a full classification of degree $2$ codimension one distributions on $\mathbb{P}^3$ according to invariants of their tangent sheaves.

It is proved that the suspension of a closed n-dimensional manifold M, $n\ge1$, does not embed in a product of n+1 curves. In fact, the ultimate result will be proved in a much more general setting. This is a far-reaching generalization the…

We shall prove that there are totally real and real analytic embeddings of $S^k$ in $\cc^n$ which are not biholomorphically equivalent if $k\geq 5$ and $n=k+2[\frac{k-1}{4}]$. We also show that a smooth manifold $M$ admits a totally real…

We discuss differences between genera of smooth and locally-flat non-orientable surfaces in the 4-ball with boundary a given torus knot or 2-bridge knot. In particular, we establish that a result by Batson on the smooth non-orientable…

The three-dimensional topologies of the membrane of M-theory can be constructed by performing Dehn surgery along knot lines. We investigate membranes wrapped around a circle and the correponding subset of topologies (Seifert manifolds). The…

It is proved that the Wedderburn Theorem on finite division rings implies that all knots and links in the smooth 4-dimensional manifolds are trivial.

For a knot K in S^3, let T(K) be the characteristic toric sub-orbifold of the orbifold (S^3,K) as defined by Bonahon and Siebenmann. If K has unknotting number one, we show that an unknotting arc for K can always be found which is disjoint…

We classify nonnegatively curved simply connected 4-manifolds with circle symmetry up to equivariant diffeomorphisms. The main problem is rule out knotted curves in the singular set of the orbit space. As an extension of this work we…

We develop a purely combinatorial framework for the systematic enumeration of knot and link diagrams supported on the thickened torus $T^2\times I$. Using the theory of maps on surfaces, cellular $4$--regular torus projections are encoded…

Relative self-linking and linking "numbers" for pairs of knots in oriented 3-manifolds are defined in terms of intersection invariants of immersed surfaces in 4-manifolds. The resulting concordance invariants generalize the usual…

Associated to an embedded surface in the $3$-sphere, we construct a diagram of fundamental groups, and prove that it is a complete invariant, wherefrom we deduce complete invariants of handlebody links, tunnels of handlebody links, and…

We study pseudo-classical knots in the non-orientable thickening of a non-orientable surface, specifically knots that are orientation-preserving paths in a non-orientable $3$-manifold of the form (non-orientable surface) $\times$ $[0, 1]$.…

The total embedding distributions of a graph is consisted of the orientable embeddings and non- orientable embeddings and have been know for few classes of graphs. The genus distribution of Ringel ladders is determined in [Discrete…