Related papers: Bridge number and tangle products
Let $F$ be an incompressible, meridionally incompressible and not boundary-parallel surface with boundary in the complement of an algebraic tangle $(B,T)$. Then $F$ separates the strings of $T$ in $B$ and the boundary slope of $F$ is…
We prove the Categorified Wrapping Number Conjecture for large classes of annular links, including alternating annular links and tangle closures exhibiting plumbed link phenomena. We do so by characterizing when a resolution is sufficient…
We determine which three-manifolds are dominated by products. The result is that a closed, oriented, connected three-manifold is dominated by a product if and only if it is finitely covered either by a product or by a connected sum of…
Consider three qubits A, B, and C which may be entangled with each other. We show that there is a trade-off between A's entanglement with B and its entanglement with C. This relation is expressed in terms of a measure of entanglement called…
We describe, for a few small examples, the Kauffman bracket skein algebra of a surface crossed with an interval. If the surface is a punctured torus the result is a quantization of the symmetric algebra in three variables (and an algebra…
The triple linking number of an oriented surface link was defined as an analogical notion of the linking number of a classical link. We consider a certain $m$-component $T^2$-link ($m \geq 3$) determined from two commutative pure $m$-braids…
Two Seifert surfaces of links in $S^3$ are said to be twist equivalent if one can be obtained from the other, up to isotopy, by repeatedly performing operations consisting of cutting along an embedded arc, applying a full twist near one…
We consider the mixed three-qubit bound entangled state defined as the normalized projector on the subspace that is complementary to an Unextendible Product Basis [C. H. Bennett et. al., Phys. Rev. Lett. 82, 5385 (1999)]. Using the fact…
The entanglement of curves within a 3-periodic box provides a model for complicated space-filling entangled structures occurring in biological, chemical and physical systems. Quantifying the complexity of the entanglement within these…
We report new results and generalizations of our work on unextendible product bases (UPB), uncompletable product bases and bound entanglement. We present a new construction for bound entangled states based on product bases which are only…
The product of two complex-valued harmonic function is not in general complex-valued harmonic function. In this paper we show that if a complex-valued harmonic function is the product of two complex-valued harmonic functions, then it is the…
In this paper, we characterize closed incompressible surfaces of genus two in the complements of 3-bridge knots and links. This characterization includes that of essential 2-string tangle decompositions for 3-bridge knots and links.
A graph is an instrument which is extensively utilized to model various problems in different fields. Up to date, many graphs have been developed to represent algebraic structures, particularly rings in order to study their properties. In…
Every link is shown to be presentable as a boundary of an unknotted flat banded surface. A (flat) banded link is defined as a boundary of an unknotted (flat) banded surface. A link's (flat) band index is defined as the minimum number of…
Many identities written by $P=S=C$ are obtained, where $P$ infinite products, $S$ infinite series, and $C$ continued fractions. Such equality is called {\it triplicity}, and it can be used to compute the values of infinite series. It is…
We define the twisted tensor product of two enriched categories, which generalizes various sorts of `products' of algebraic structures, including the bicrossed product of groups, the twisted tensor product of (co)algebras and the double…
Using the combinatorial description of shuffle product, we prove or reformulate several shuffle product formulas of multiple zeta values, including a general formula of the shuffle product of two multiple zeta values, some restricted…
We use the combinatorial techniques of graphs of intersection to study reducible Dehn surgeries on knots in the three-sphere. In particular, in the event that a reducible surgery on a knot K in the three-sphere of slope r produces a…
We enumerate rooted triangulations of a sphere with multiple holes by the total number of edges and the length of each boundary component. The proof relies on a combinatorial identity due to W.T. Tutte.
We prove that a generic complete intersection Calabi-Yau 3-fold defined by sections of ample line bundles on a product of projective spaces admits a conifold transition to a connected sum of S^{3} \times S^{3}. In this manner, we obtain…