Surgery on links with unknotted components and three-manifolds

It is shown that any closed three-manifold M obtained by integral surgery on a knot in the three-sphere can always be constructed from integral surgeries on a 3-component link L with each component being an unknot in the three-sphere. It is also interesting to notice that infinitely many different integral surgeries on the same link L could give the same three-manifold M.Comment: 10 pages, 8 figure

What is NP? - Interpretation of a Chinese paradox "white horse is not horse"

The notion of nondeterminism has disappeared from the current definition of NP, which has led to ambiguities in understanding NP, and caused fundamental difficulties in studying the relation P versus NP. In this paper, we question the equivalence of the two definitions of NP, the one defining NP as the class of problems solvable by a nondeterministic Turing machine in polynomial time, and the other defining NP as the class of problems verifiable by a deterministic Turing machine in polynomial time, and reveal cognitive biases in this equivalence. Inspired from a famous Chinese paradox white horse is not horse, we further analyze these cognitive biases. The work shows that these cognitive biases arise from the confusion between different levels of nondeterminism and determinism, due to the lack of understanding about the essence of nondeterminism. Therefore, we argue that fundamental difficulties in understanding P versus NP lie firstly at cognition level, then logic level

On free ZpZ_p-torus actions in dimension two and three

We confirm the Halperin-Carlsson Conjecture for free $Z_p$-torus actions (p is a prime) on 2-dimensional finite CW-complexes and free $Z_2$-torus actions on compact 3-manifolds.Comment: 26 pages, no figure. The contents of the paper are reorganized and some proofs are simplifie

On lower bounds of the sum of multigraded Betti numbers of simplicial complexes

We find some general lower bounds of the sum of certain families of multigraded Betti numbers of any simplicial complex with a vertex coloring.Comment: 15 pages, 2 figures. Minor revisions are made (two pictures and some new references are added

On the renormalization of Coulomb interactions in two-dimensional tilted Dirac fermions

We investigate the effects of long-ranged Coulomb interactions in a tilted Dirac semimetal in two dimensions by using the perturbative renormalization-group method. Depending on the magnitude of the tilting parameter, the undoped system can have either Fermi points (type-I) or Fermi lines (type-II). Previous studies usually performed the renormalization-group transformations by integrating out the modes with large momenta. This is problematic when the Fermi surface is open, like type-II Dirac fermions. In this work, we study the effects of Coulomb interactions, following the spirit of Shankar\cite{Shankar}, by introducing a cutoff in the energy scale around the Fermi surface and integrating out the high-energy modes. For type-I Dirac fermions, our result is consistent with that of the previous work. On the other hand, we find that for type-II Dirac fermions, the magnitude of the tilting parameter increases monotonically with lowering energies. This implies the stability of type-II Dirac fermions in the presence of Coulomb interactions, in contrast with previous results. Furthermore, for type-II Dirac fermions, the velocities in different directions acquire different renormalization even if they have the same bare values. By taking into account the renormalization of the tilting parameter and the velocities due to the Coulomb interactions, we show that while the presence of a charged impurity leads only to charge redistribution around the impurity for type-I Dirac fermions, for type-II Dirac fermions, the impurity charge is completely screened, albeit with a very long screening length. The latter indicates that the temperature dependence of physical observables are essentially determined by the RG equations we derived. We illustrate this by calculating the temperature dependence of the compressibility and specific heat of the interacting tilted Dirac fermions.Comment: 15 pages, 4 figure

A quantum homomorphic encryption scheme for polynomial-sized circuits

Quantum homomorphic encryption (QHE) is an encryption method that allows quantum computation to be performed on one party's private data with the program provided by another party, without revealing much information about the data nor about the program to the opposite party. It is known that information-theoretically-secure QHE for circuits of unrestricted size would require exponential resources, and efficient computationally-secure QHE schemes for polynomial-sized quantum circuits have been constructed. In this paper we first propose a QHE scheme for a type of circuits of polynomial depth, based on the rebit quantum computation formalism. The scheme keeps the restricted type of data perfectly secure. We then propose a QHE scheme for a larger class of polynomial-depth quantum circuits, which has partial data privacy. Both schemes have good circuit privacy. We also propose an interactive QHE scheme with asymptotic data privacy, however, the circuit privacy is not good, in the sense that the party who provides the data could cheat and learn about the circuit. We show that such cheating would generally affect the correctness of the evaluation or cause deviation from the protocol. Hence the cheating can be caught by the opposite party in an interactive scheme with embedded verifications. Such scheme with verification has a minor drawback in data privacy. Finally, we show some methods which achieve some nontrivial level of data privacy and circuit privacy without resorting to allowing early terminations, in both the QHE problem and in secure evaluation of classical functions. The entanglement and classical communication costs in these schemes are polynomial in the circuit size and the security parameter (if any).Comment: 29 pages, 2 figures. Revised Section VIII, among other minor fixe

On the constructions of free and locally standard Z_2-torus actions on Manifolds

We introduce an elementary way of constructing principal (Z_2)^m-bundles over compact smooth manifolds. In addition, we will define a general notion of locally standard (Z_2)^m-actions on closed manifolds for all m>0, and then give a general way to construct all such (Z_2)^m-actions from the orbit space. Some related topology problems are also studied.Comment: 28 pages, 12 figures, some minor revisions are made, one picture and one reference are added

Quantum Boson Algebra and Poisson Geometry of the Flag Variety

In his work on crystal bases \cite{Kas}, Kashiwara introduced a certain degeneration of the quantized universal enveloping algebra of a semi-simple Lie algebra $\mathfrak g$, which he called a quantum boson algebra. In this paper, we construct Kashiwara operators associated to all positive roots and use them to define a variant of Kashiwara's quantum boson algebra. We show that a quasi-classical limit of the positive half of our variant is a Poisson algebra of the form $(P \simeq \mathbb C[\mathfrak n^{\ast}], \, \{~~,~~\}_P)$, where $\mathfrak n$ is the positive part of $\mathfrak g$ and $\{~~,~~\}_P$ is a Poisson bracket that has the same rank as, but is different from, the Kirillov-Kostant bracket $\{~~,~~\}_{KK}$ on $\mathfrak n^{\ast}$. Furthermore, we prove that, in the special case of type $A$, any linear combination $a_1 \{~~,~~\}_P + a_2 \{~~,~~\}_{KK}$, $a_1, a_2 \in \mathbb C$, is again a Poisson bracket. In the general case, we establish an isomorphism of $P$ and the Poisson algebra of regular functions on the open Bruhat cell in the flag variety. In type $A$, we also construct a Casimir function on the open Bruhat cell, together with its quantization, which may be thought of as an analog of the linear function on $\mathfrak n^{\ast}$ defined by a root vector for the highest root

Cubes and Generalized Real Bott Manifolds

We define a notion of facets-pairing structure and its seal space on a nice manifold with corners. We will study facets-pairing structures on any cube in detail and investigate when the seal space of a facets-pairing structure on a cube is a closed manifold. In particular, for any binary square matrix $A$ with zero diagonal in dimension n, there is a canonical facets-pairing structure $F_A$ on the n-dimensional cube. We will show that all the closed manifolds that we can obtain from the seal spaces of such $F_A$'s are neither more nor less than all the generalized real Bott manifolds --- a special class of real toric manifolds introduced by Choi, Masuda and Suh.Comment: Some small changes were made to the previous version. The introduction part was expanded and a new reference was adde

Gaiotto's Lagrangian subvarieties via loop groups

The purpose of this note is to give a simple proof of the fact that a certain substack, defined in [2], of the moduli stack $T^{\ast}Bun_G(\Sigma)$ of Higgs bundles over a curve $\Sigma$, for a connected, simply connected semisimple group $G$, possesses a Lagrangian structure. The substack, roughly speaking, consists of images under the moment map of global sections of principal $G$-bundles over $\Sigma$ twisted by a smooth symplectic variety with a Hamiltonian $G$-action