scholarly journals Multivariable connected sums and multiple polylogarithms

2021 ◽  
Vol 9 (1) ◽  
Author(s):  
Hanamichi Kawamura ◽  
Takumi Maesaka ◽  
Shin-ichiro Seki
Keyword(s):  
Connected Sum ◽  
Fundamental Identity ◽  
Connected Sums ◽  
Special Values ◽  
Multiple Polylogarithms

AbstractWe introduce the multivariable connected sum which is a generalization of Seki–Yamamoto’s connected sum and prove the fundamental identity for these sums by series manipulation. This identity yields explicit procedures for evaluating multivariable connected sums and for giving relations among special values of multiple polylogarithms. In particular, our class of relations contains Ohno’s relations for multiple polylogarithms.

scholarly journals The SL(2, C) Casson Invariant for Knots and the Â-polynomial

2016 ◽  
Vol 68 (1) ◽  
pp. 3-23 ◽  
Author(s):  
Hans U. Boden ◽  
Cynthia L. Curtis
Keyword(s):  
Large Class ◽  
Product Formula ◽  
Integral Homology ◽  
Connected Sum ◽  
Connected Sums ◽  
Knot Invariant ◽  
Definition Of ◽  
Arbitrary Knots

AbstractIn this paper, we extend the definition of the SL(2,ℂ) Casson invariant to arbitrary knots K in integral homology 3-spheres and relate it to the m-degree of the Â-polynomial of K. We prove a product formula for the Â-polynomial of the connected sum K1#K2 of two knots in S3 and deduce additivity of the SL(2,ℂ) Casson knot invariant under connected sums for a large class of knots in S3. We also present an example of a nontrivial knot K in S3 with trivial Â-polynomial and trivial SL(2,ℂ) Casson knot invariant, showing that neither of these invariants detect the unknot.

Connected sum of virtual knots and F-polynomials

2021 ◽  
Author(s):  
Maxim Ivanov
Keyword(s):  
Infinite Family ◽  
Connected Sum ◽  
Connected Sums ◽  
Virtual Knots ◽  
Knot Diagrams

It is known that connected sum of two virtual knots is not uniquely determined and depends on knot diagrams and choosing the points to be connected. But different connected sums of the same virtual knots cannot be distinguished by Kauffman’s affine index polynomial. For any pair of virtual knots [Formula: see text] and [Formula: see text] with [Formula: see text]-dwrithe [Formula: see text] we construct an infinite family of different connected sums of [Formula: see text] and [Formula: see text] which can be distinguished by [Formula: see text]-polynomials.

scholarly journals Variations of Mixed Hodge Structures of Multiple Polylogarithms

2004 ◽  
Vol 56 (6) ◽  
pp. 1308-1338 ◽  
Author(s):  
Jianqiang Zhao
Keyword(s):  
Number Field ◽  
Zeta Functions ◽  
Explicit Construction ◽  
Hodge Structures ◽  
Special Values ◽  
Multiple Polylogarithms ◽  
Real Analytic ◽  
Mixed Hodge Structures

AbstractIt is well known that multiple polylogarithms give rise to good unipotent variations of mixed Hodge-Tate structures. In this paper we shall explicitly determine these structures related to multiple logarithms and some other multiple polylogarithms of lower weights. The purpose of this explicit construction is to give some important applications. First we study the limit of mixed Hodge-Tate structures and make a conjecture relating the variations of mixed Hodge-Tate structures of multiple logarithms to those of general multiple polylogarithms. Then following Deligne and Beilinson we describe an approach to defining the single-valued real analytic version of the multiple polylogarithms which generalizes the well-known result of Zagier on classical polylogarithms. In the process we find some interesting identities relating single-valued multiple polylogarithms of the same weight k when k = 2 and 3. At the end of this paper, motivated by Zagier's conjecture we pose a problem which relates the special values of multiple Dedekind zeta functions of a number field to the single-valued version of multiple polylogarithms.

scholarly journals Log-Sine Evaluations of Mahler Measures, II

Integers ◽  
2012 ◽  
Vol 12 (6) ◽  
Author(s):  
David Borwein ◽  
Jonathan M. Borwein ◽  
Armin Straub ◽  
James Wan
Keyword(s):  
Worked Examples ◽  
Special Values ◽  
Easy Proof ◽  
Multiple Polylogarithms

Abstract.We continue the analysis of higher and multiple Mahler measures using log-sine integrals as started in “Log-sine evaluations of Mahler measures” and “Special values of generalized log-sine integrals” by two of the authors. This motivates a detailed study of various multiple polylogarithms and worked examples are given. Our techniques enable the reduction of several multiple Mahler measures, and supply an easy proof of two conjectures by Boyd.

scholarly journals Simple crystallizations of 4-manifolds

2016 ◽  
Vol 16 (1) ◽  
Author(s):  
Biplab Basak ◽  
Jonathan Spreer
Keyword(s):  
Combinatorial Structure ◽  
K3 Surface ◽  
Standard Type ◽  
Connected Sum ◽  
Connected Sums ◽  
Topological Features ◽  
Natural Objects ◽  
Simply Connected

AbstractMinimal crystallizations of simply connected PL 4-manifolds are very natural objects. Many of their topological features are reflected in their combinatorial structure which, in addition, is preserved under the connected sum operation. We present a minimal crystallization of the standard PL K3 surface. In combination with known results this yields minimal crystallizations of all simply connected PL 4-manifolds of “standard” type, that is, all connected sums of ℂℙ

scholarly journals Haken spheres for genus two Heegaard splittings

2017 ◽  
Vol 165 (3) ◽  
pp. 563-572 ◽  
Author(s):  
SANGBUM CHO ◽  
YUYA KODA
Keyword(s):  
Heegaard Splitting ◽  
Combinatorial Structure ◽  
Lens Spaces ◽  
Heegaard Splittings ◽  
Precise Description ◽  
Connected Sum ◽  
Connected Sums ◽  
Genus 2 ◽  
Genus Two

AbstractA manifold which admits a reducible genus-2 Heegaard splitting is one of the 3-sphere, S2 × S1, lens spaces or their connected sums. For each of those splittings, the complex of Haken spheres is defined. When the manifold is the 3-sphere, S2 × S1 or a connected sum whose summands are lens spaces or S2 × S1, the combinatorial structure of the complex has been studied by several authors. In particular, it was shown that those complexes are all contractible. In this work, we study the remaining cases, that is, when the manifolds are lens spaces. We give a precise description of each of the complexes for the genus-2 Heegaard splittings of lens spaces. A remarkable fact is that the complexes for most lens spaces are not contractible and even not connected.

CONIFOLD TRANSITIONS FOR COMPLETE INTERSECTION CALABI–YAU THREEFOLDS IN PRODUCTS OF PROJECTIVE SPACES

2013 ◽  
Vol 15 (04) ◽  
pp. 1350009
Author(s):  
JINXING XU
Keyword(s):  
Complete Intersection ◽  
Projective Spaces ◽  
Line Bundles ◽  
Complex Structures ◽  
Connected Sum ◽  
Connected Sums

We prove that a generic complete intersection Calabi–Yau threefold defined by sections of ample line bundles on a product of projective spaces admits a conifold transition to a connected sum of S3 × S3. In this manner, we obtain complex structures with trivial canonical bundles on some connected sums of S3 × S3. This construction is an analogue of that made by Friedman, Lu and Tian who used quintics in ℙ4.

λ-unknotting-number-one knots need not be prime

1999 ◽  
Vol 08 (07) ◽  
pp. 831-834 ◽  
Author(s):  
N. ASKITAS
Keyword(s):  
Connected Sum ◽  
Connected Sums

We show that for certain unknotting operations λ their unknotting number uλ is not additive with respect to connected sums. In general the unknotting number of the connected sum of two knots is smaller than or equal to the sum of the unknotting numbers of the summands; we provide examples where the inequality is strict.

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