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.

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 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 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 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.

scholarly journals Digital k-Contractibility of an n-Times Iterated Connected Sum of Simple Closed k-Surfaces and Almost Fixed Point Property

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 345 ◽  
Author(s):  
Sang-Eon Han
Keyword(s):  
Fixed Point ◽  
Digital Image ◽  
Fixed Point Property ◽  
Point Property ◽  
Connected Sum ◽  
Connected Sums

The paper firstly establishes the so-called n-times iterated connected sum of a simple closed k-surface in Z 3 , denoted by C k n , k ∈ { 6 , 18 , 26 } . Secondly, for a simple closed 18-surface M S S 18 , we prove that there are only two types of connected sums of it up to 18-isomorphism. Besides, given a simple closed 6-surface M S S 6 , we prove that only one type of M S S 6 ♯ M S S 6 exists up to 6-isomorphism, where ♯ means the digital connected sum operator. Thirdly, we prove the digital k-contractibility of C k n : = M S S k ♯ ⋯ ♯ M S S k ︷ n - times , k ∈ { 18 , 26 } , which leads to the simply k-connectedness of C k n , k ∈ { 18 , 26 } , n ∈ N . Fourthly, we prove that C 6 2 and C k n do not have the almost fixed point property (AFPP, for short), k ∈ { 18 , 26 } . Finally, assume a closed k-surface S k ( ⊂ Z 3 ) which is ( k , k ¯ ) -isomorphic to ( X , k ) in the picture ( Z 3 , k , k ¯ , X ) and the set X is symmetric according to each of x y -, y z -, and x z -planes of R 3 . Then we prove that S k does not have the AFPP. In this paper given a digital image ( X , k ) is assumed to be k-connected and its cardinality | X | ≥ 2 .

Homology of groups of surfaces in the 4-sphere

1981 ◽  
Vol 89 (1) ◽  
pp. 113-117 ◽  
Author(s):  
C. McA. Gordon
Keyword(s):  
Positive Integer ◽  
Fundamental Group ◽  
Orientable Surface ◽  
Connected Sum ◽  
Connected Sums ◽  
Ambient Isotopy

1. Let F be a closed, connected, orientable surface of genus g ≥ 0 smoothly embedded in S4, and let π denote the fundamental group π1(S4 − F). Then H2(π) is a quotient of H2(S4 − F) ≅ H1(F) ≅ Z2g. If F is unknotted, that is, if there is an ambient isotopy taking F to the standardly embedded surface of genus g in S3 ⊂ S4, then π ≅ Z, so H2(π) = 0. More generally, if F is the connected sum of an unknotted surface and some 2-sphere S, then π ≅ π1 (S4 − S), so again H2(π) = 0. The question of whether H2(π) could ever be non-zero was raised in (5), Problem 4.29, and (10), Conjecture 4.13, and answered in (7) and (1). There, surfaces are constructed with H2(π)≅ Z/2, and hence, by forming connected sums, with H2(π) ≅ (Z/2)n for any positive integer n. In fact, (1) produces tori T in S4 with H2(π) ≅ Z/2, and hence surfaces of genus g with H2(π) ≅ (Z/2)g.

Quasipositive Links and Connected Sums

2020 ◽  
Vol 54 (1) ◽  
pp. 64-67
Author(s):  
S. Yu. Orevkov
Keyword(s):  
Connected Sums

Constructing symplectic manifolds

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Final Section ◽  
Symplectic Manifolds ◽  
Homotopy Equivalence ◽  
Codimension Two ◽  
Open Manifolds ◽  
Connected Sums ◽  
Symplectic Forms ◽  
Ambient Manifold ◽  
Blowing Up ◽  
Four Manifolds

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.

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