Invariants for Homology 3-Spheres

Saveliev, Nikolai

doi:10.1007/978-3-662-04705-7

Cited by 122 publications

(98 citation statements)

References 180 publications

(439 reference statements)

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“…With the aim of obtain the coupling constant hierarchy we consider tree plumbing graphs p Γ of more general type [5] [13], whose basic structure blocks are Seifert fibered Brieskorn homology spheres [7]. Let 1 2 3 , , a a a be pairwise relatively prime positive numbers, the Brieskorn homology sphere (Bh-sphere) is defined as the link of Brieskorn singularity ( ) ( ) { } 5 3 1 2 1 2 3 1 2 3 : , , : , , a a a Σ is given in Figure 2 [1].…”

Section: Continued Fractions and Graph Manifoldsmentioning

confidence: 99%

Algorithm for Fast Calculation of Hirzebruch-Jung Continued Fraction Expansions to Coding of Graph Manifolds

Lopez¹,

Ефремов²,

Magdaleno³

2015

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We present a new algorithm for the fast expansion of rational numbers into continued fractions. This algorithm permits to compute the complete set of integer Euler numbers of the sophisticate tree graph manifolds, which we used to simulate the coupling constant hierarchy for the universe with five fundamental interactions. Moreover, we can explicitly compute the integer Laplacian block matrix associated with any tree plumbing graph. This matrix coincides up to sign with the integer linking matrix (the main topological invariant) of the graph manifold corresponding to the plumbing graph. The need for a special algorithm appeared during computations of these topological invariants of complicated graph manifolds since there emerged a set of special rational numbers (fractions) with huge numerators and denominators; for these rational numbers, the ordinary methods of expansion in continued fraction became unusable.

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Section: Continued Fractions and Graph Manifoldsmentioning

confidence: 99%

Algorithm for Fast Calculation of Hirzebruch-Jung Continued Fraction Expansions to Coding of Graph Manifolds

Lopez¹,

Ефремов²,

Magdaleno³

2015

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“…a a a I R Σ =  , the resulting graph three-manifold will be integer homology sphere [4] [9] (-homology sphere), which in general case does not have the global Seifert fibration. But we can construct the JSJ-covering { } : 1, ,…”

Section: Block Matrix Representation For a Graph P γ Of Tree Typementioning

confidence: 99%

“…We constructed various models [2] [3] where the role of internal spaces is played by a specific family of 3-dimensional graph manifolds, whose rational linking matrices describe the hierarchy of gauge coupling constants of the real universe. The basic structure blocks of these graph manifolds are Seifert fibered Brieskorn homology spheres, defined as the link of Brieskorn singularity ( ) ( ) { } 1 2 3 , , a a a pairwise relatively prime positive numbers [4]. Bh-spheres belong to the class of Seifert fibered homology spheres (Sfh-spheres).…”

Section: Introductionmentioning

confidence: 99%

Block Matrix Representation of a Graph Manifold Linking Matrix Using Continued Fractions

Lopez¹,

Ефремов²,

Magdaleno³

2014

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We consider the block matrices and 3-dimensional graph manifolds associated with a special type of tree graphs. We demonstrate that the linking matrices of these graph manifolds coincide with the reduced matrices obtained from the Laplacian block matrices by means of Gauss partial diagonalization procedure described explicitly by W. Neumann. The linking matrix is an important topological invariant of a graph manifold which is possible to interpret as a matrix of coupling constants of gauge interaction in Kaluza-Klein approach, where 3-dimensional graph manifold plays the role of internal space in topological 7-dimensional BF theory. The Gauss-Neumann method gives us a simple algorithm to calculate the linking matrices of graph manifolds and thus the coupling constants matrices.

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“…[13], and the second fact is due to Whitehead [19]. For a more recent source, see for instance [15,16]. In all cases, the minus sign corresponds to a reversal of the orientation.…”

Section: Theorem 53 the U (1) Homotopy Dw Invariants Distinguish Homentioning

confidence: 99%

Abelian homotopy Dijkgraaf-Witten theory

Hansen¹,

Slingerland²,

Turner³

2005

Advances in Theoretical and Mathematical Physics

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Invariants for Homology 3-Spheres

Cited by 122 publications

References 180 publications

Algorithm for Fast Calculation of Hirzebruch-Jung Continued Fraction Expansions to Coding of Graph Manifolds

Algorithm for Fast Calculation of Hirzebruch-Jung Continued Fraction Expansions to Coding of Graph Manifolds

Block Matrix Representation of a Graph Manifold Linking Matrix Using Continued Fractions

Abelian homotopy Dijkgraaf-Witten theory

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