Related papers: On Krebes' tangle

An important difference between high dimensional smooth manifolds and smooth 4-manifolds that in a 4-manifold it is not always possible to represent every middle dimensional homology class with a smoothly embedded sphere. This is true even…

Consider a manifold, of dimension greater than 3, equipped with a bracket-generating distribution. In this article we prove complete h-principles for embedded regular horizontal curves and for embedded transverse curves. These results…

Suppose F is a compact orientable surface, K is a knot in F x I, and N is the 3-manifold obtained by some non-trivial surgery on K. If F x {0} compresses in N, then there is an annulus in F x I with one end K and the other end an essential…

In this paper we study further when tangles embed into the unknot, the unlink or a split link. In particular, we study obstructions to these properties through geometric characterizations, tangle sums and colorings. As an application we…

This paper discusses some geometric ideas associated with knots in real projective 3-space $\mathbb{R}P^3$. These ideas are borrowed from classical knot theory. Since knots in $\mathbb{R}P^3$ are classified into three disjoint classes, -…

A knot K in 1-bridge position with respect to a genus-g Heegaard surface in a 3-manifold can be moved by isotopy through knots in 1-bridge position until it lies in a union of n parallel genus-g surfaces tubed together by n-1 straight…

In a previous paper we outlined how discrete torsion can be understood geometrically as an analogue of orbifold U(1) Wilson lines. In this paper we shall prove the remaining details. More precisely, in this paper we describe gerbes in terms…

We show that, for any prime p, a knot K in the 3-sphere is determined by its p-fold cyclic unbranched covering. We also investigate when the m-fold cyclic unbranched covering of a knot coincides with the n-fold cyclic unbranched covering of…

In a lens space X of order r a knot K representing an element of the fundamental group pi_1 X = Z/rZ of order s <= r contains a connected orientable surface S properly embedded in its exterior X-N(K) such that the boundary of S intersects…

A graph G is intrinsically S^1-linked if for every embedding of the vertices of G into S^1, vertices that form the endpoints of two disjoint edges in G form a non-split link in the embedding. We show that a graph is intrinsically S^1-linked…

Let G be a simple complex algebraic group. By using a notion of a G-category we define invariants of tangles with flat G-connections in their complements. We also show that quantized universal enveloping algebras at roots of unity provide…

Let $K$ be a knot with an unknotting tunnel $\gamma$ and suppose that $K$ is not a 2-bridge knot. There is an invariant $\rho = p/q \in \mathbb{Q}/2 \mathbb{Z}$, $p$ odd, defined for the pair $(K, \gamma)$. The invariant $\rho$ has…

The classical abelian invariants of a knot are the Alexander module, which is the first homology group of the the unique infinite cyclic covering space of S^3-K, considered as a module over the (commutative) Laurent polynomial ring, and the…

Whitney's theorem states that every 3-connected planar graph is uniquely embeddable on the sphere. On the other hand, it has many inequivalent embeddings on another surface. We shall characterize structures of a $3$-connected $3$-regular…

The algebraic genus of a knot is an invariant that arises when one considers upper bounds for the topological slice genus coming from Freedman's theorem that Alexander polynomial one knots are topologically slice. This paper develops…

Let K be a knot in the 3-sphere with 2-fold branched covering space M. If for some prime p congruent to 3 mod 4 the p-torsion in the first homology of M is cyclic with odd exponent, then K is of infinite order in the knot concordance group.…

We present a family of index 1 abelian current graphs whose derived embeddings can be modified into triangulations of $K_{12s}$ for $s \geq 4$. Our construction is significantly simpler than previous methods for finding genus embeddings of…

A knot type is exchange reducible if an arbitrary closed n-braid representative can be changed to a closed braid of minimum braid index by a finite sequence of braid isotopies, exchange moves and +/- destabilizations. In the manuscript [J…

In knot concordance three genera arise naturally, g(K), g_4(K), and g_c(K): these are the classical genus, the 4-ball genus, and the concordance genus, defined to be the minimum genus among all knots concordant to K. Clearly 0 <= g_4(K) <=…

Attaching a 2-handle to a genus two or greater boundary component of a 3-manifold is a natural generalization of Dehn filling a torus boundary component. We prove that there is an interesting relationship between an essential surface in a…