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Jonathan Spreer - One of the best experts on this subject based on the ideXlab platform.
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A Polynomial-Time Algorithm to Compute Turaev–Viro Invariants $$\mathrm {TV}_{4,q}$$ TV 4 , q of 3-Manifolds with Bounded First Betti Number
Foundations of Computational Mathematics, 2020Co-Authors: Clément Maria, Jonathan SpreerAbstract:In this article, we introduce a fixed-parameter tractable algorithm for computing the Turaev–Viro invariants $$\mathrm {TV}_{4,q}$$ TV 4 , q , using the first Betti Number, i.e. the dimension of the first homology group of the manifold with $$\mathbb {Z}_2$$ Z 2 -coefficients, as parameter. This is, to our knowledge, the first parameterised algorithm in computational 3-manifold topology using a topological parameter. The computation of $$\mathrm {TV}_{4,q}$$ TV 4 , q is known to be #P-hard in general; using a topological parameter provides an algorithm polynomial in the size of the input triangulation for the family of 3-manifolds with first $$\mathbb {Z}_2$$ Z 2 -homology group of bounded dimension. Our algorithm is easy to implement, and running times are comparable with running times to compute integral homology groups for standard libraries of triangulated 3-manifolds. The invariants we can compute this way are powerful: in combination with integral homology and using standard data sets, we are able to almost double the pairs of 3-manifolds we can distinguish. We hope this qualifies $$\mathrm {TV}_{4,q}$$ TV 4 , q to be added to the short list of standard properties (such as orientability, connectedness and Betti Numbers) that can be computed ad hoc when first investigating an unknown triangulation.
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a polynomial time algorithm to compute turaev viro invariants mathrm tv _ 4 q of 3 manifolds with bounded first Betti Number
Foundations of Computational Mathematics, 2020Co-Authors: Clément Maria, Jonathan SpreerAbstract:In this article, we introduce a fixed-parameter tractable algorithm for computing the Turaev–Viro invariants $$\mathrm {TV}_{4,q}$$, using the first Betti Number, i.e. the dimension of the first homology group of the manifold with $$\mathbb {Z}_2$$-coefficients, as parameter. This is, to our knowledge, the first parameterised algorithm in computational 3-manifold topology using a topological parameter. The computation of $$\mathrm {TV}_{4,q}$$ is known to be #P-hard in general; using a topological parameter provides an algorithm polynomial in the size of the input triangulation for the family of 3-manifolds with first $$\mathbb {Z}_2$$-homology group of bounded dimension. Our algorithm is easy to implement, and running times are comparable with running times to compute integral homology groups for standard libraries of triangulated 3-manifolds. The invariants we can compute this way are powerful: in combination with integral homology and using standard data sets, we are able to almost double the pairs of 3-manifolds we can distinguish. We hope this qualifies $$\mathrm {TV}_{4,q}$$ to be added to the short list of standard properties (such as orientability, connectedness and Betti Numbers) that can be computed ad hoc when first investigating an unknown triangulation.
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A Polynomial-Time Algorithm to Compute Turaev–Viro Invariants $$\mathrm {TV}_{4,q}$$TV4,q of 3-Manifolds with Bounded First Betti Number
Foundations of Computational Mathematics, 2019Co-Authors: Clément Maria, Jonathan SpreerAbstract:In this article, we introduce a fixed-parameter tractable algorithm for computing the Turaev–Viro invariants $$\mathrm {TV}_{4,q}$$ TV 4 , q , using the first Betti Number, i.e. the dimension of the first homology group of the manifold with $$\mathbb {Z}_2$$ Z 2 -coefficients, as parameter. This is, to our knowledge, the first parameterised algorithm in computational 3-manifold topology using a topological parameter. The computation of $$\mathrm {TV}_{4,q}$$ TV 4 , q is known to be #P-hard in general; using a topological parameter provides an algorithm polynomial in the size of the input triangulation for the family of 3-manifolds with first $$\mathbb {Z}_2$$ Z 2 -homology group of bounded dimension. Our algorithm is easy to implement, and running times are comparable with running times to compute integral homology groups for standard libraries of triangulated 3-manifolds. The invariants we can compute this way are powerful: in combination with integral homology and using standard data sets, we are able to almost double the pairs of 3-manifolds we can distinguish. We hope this qualifies $$\mathrm {TV}_{4,q}$$ TV 4 , q to be added to the short list of standard properties (such as orientability, connectedness and Betti Numbers) that can be computed ad hoc when first investigating an unknown triangulation.
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A Polynomial-Time Algorithm to Compute Turaev–Viro Invariants $$\mathrm {TV}_{4,q}$$ of 3-Manifolds with Bounded First Betti Number
Foundations of Computational Mathematics, 2019Co-Authors: Clément Maria, Jonathan SpreerAbstract:In this article, we introduce a fixed-parameter tractable algorithm for computing the Turaev–Viro invariants $$\mathrm {TV}_{4,q}$$, using the first Betti Number, i.e. the dimension of the first homology group of the manifold with $$\mathbb {Z}_2$$-coefficients, as parameter. This is, to our knowledge, the first parameterised algorithm in computational 3-manifold topology using a topological parameter. The computation of $$\mathrm {TV}_{4,q}$$ is known to be #P-hard in general; using a topological parameter provides an algorithm polynomial in the size of the input triangulation for the family of 3-manifolds with first $$\mathbb {Z}_2$$-homology group of bounded dimension. Our algorithm is easy to implement, and running times are comparable with running times to compute integral homology groups for standard libraries of triangulated 3-manifolds. The invariants we can compute this way are powerful: in combination with integral homology and using standard data sets, we are able to almost double the pairs of 3-manifolds we can distinguish. We hope this qualifies $$\mathrm {TV}_{4,q}$$ to be added to the short list of standard properties (such as orientability, connectedness and Betti Numbers) that can be computed ad hoc when first investigating an unknown triangulation.
Clément Maria - One of the best experts on this subject based on the ideXlab platform.
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A Polynomial-Time Algorithm to Compute Turaev–Viro Invariants $$\mathrm {TV}_{4,q}$$ TV 4 , q of 3-Manifolds with Bounded First Betti Number
Foundations of Computational Mathematics, 2020Co-Authors: Clément Maria, Jonathan SpreerAbstract:In this article, we introduce a fixed-parameter tractable algorithm for computing the Turaev–Viro invariants $$\mathrm {TV}_{4,q}$$ TV 4 , q , using the first Betti Number, i.e. the dimension of the first homology group of the manifold with $$\mathbb {Z}_2$$ Z 2 -coefficients, as parameter. This is, to our knowledge, the first parameterised algorithm in computational 3-manifold topology using a topological parameter. The computation of $$\mathrm {TV}_{4,q}$$ TV 4 , q is known to be #P-hard in general; using a topological parameter provides an algorithm polynomial in the size of the input triangulation for the family of 3-manifolds with first $$\mathbb {Z}_2$$ Z 2 -homology group of bounded dimension. Our algorithm is easy to implement, and running times are comparable with running times to compute integral homology groups for standard libraries of triangulated 3-manifolds. The invariants we can compute this way are powerful: in combination with integral homology and using standard data sets, we are able to almost double the pairs of 3-manifolds we can distinguish. We hope this qualifies $$\mathrm {TV}_{4,q}$$ TV 4 , q to be added to the short list of standard properties (such as orientability, connectedness and Betti Numbers) that can be computed ad hoc when first investigating an unknown triangulation.
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a polynomial time algorithm to compute turaev viro invariants mathrm tv _ 4 q of 3 manifolds with bounded first Betti Number
Foundations of Computational Mathematics, 2020Co-Authors: Clément Maria, Jonathan SpreerAbstract:In this article, we introduce a fixed-parameter tractable algorithm for computing the Turaev–Viro invariants $$\mathrm {TV}_{4,q}$$, using the first Betti Number, i.e. the dimension of the first homology group of the manifold with $$\mathbb {Z}_2$$-coefficients, as parameter. This is, to our knowledge, the first parameterised algorithm in computational 3-manifold topology using a topological parameter. The computation of $$\mathrm {TV}_{4,q}$$ is known to be #P-hard in general; using a topological parameter provides an algorithm polynomial in the size of the input triangulation for the family of 3-manifolds with first $$\mathbb {Z}_2$$-homology group of bounded dimension. Our algorithm is easy to implement, and running times are comparable with running times to compute integral homology groups for standard libraries of triangulated 3-manifolds. The invariants we can compute this way are powerful: in combination with integral homology and using standard data sets, we are able to almost double the pairs of 3-manifolds we can distinguish. We hope this qualifies $$\mathrm {TV}_{4,q}$$ to be added to the short list of standard properties (such as orientability, connectedness and Betti Numbers) that can be computed ad hoc when first investigating an unknown triangulation.
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A Polynomial-Time Algorithm to Compute Turaev–Viro Invariants $$\mathrm {TV}_{4,q}$$TV4,q of 3-Manifolds with Bounded First Betti Number
Foundations of Computational Mathematics, 2019Co-Authors: Clément Maria, Jonathan SpreerAbstract:In this article, we introduce a fixed-parameter tractable algorithm for computing the Turaev–Viro invariants $$\mathrm {TV}_{4,q}$$ TV 4 , q , using the first Betti Number, i.e. the dimension of the first homology group of the manifold with $$\mathbb {Z}_2$$ Z 2 -coefficients, as parameter. This is, to our knowledge, the first parameterised algorithm in computational 3-manifold topology using a topological parameter. The computation of $$\mathrm {TV}_{4,q}$$ TV 4 , q is known to be #P-hard in general; using a topological parameter provides an algorithm polynomial in the size of the input triangulation for the family of 3-manifolds with first $$\mathbb {Z}_2$$ Z 2 -homology group of bounded dimension. Our algorithm is easy to implement, and running times are comparable with running times to compute integral homology groups for standard libraries of triangulated 3-manifolds. The invariants we can compute this way are powerful: in combination with integral homology and using standard data sets, we are able to almost double the pairs of 3-manifolds we can distinguish. We hope this qualifies $$\mathrm {TV}_{4,q}$$ TV 4 , q to be added to the short list of standard properties (such as orientability, connectedness and Betti Numbers) that can be computed ad hoc when first investigating an unknown triangulation.
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A Polynomial-Time Algorithm to Compute Turaev–Viro Invariants $$\mathrm {TV}_{4,q}$$ of 3-Manifolds with Bounded First Betti Number
Foundations of Computational Mathematics, 2019Co-Authors: Clément Maria, Jonathan SpreerAbstract:In this article, we introduce a fixed-parameter tractable algorithm for computing the Turaev–Viro invariants $$\mathrm {TV}_{4,q}$$, using the first Betti Number, i.e. the dimension of the first homology group of the manifold with $$\mathbb {Z}_2$$-coefficients, as parameter. This is, to our knowledge, the first parameterised algorithm in computational 3-manifold topology using a topological parameter. The computation of $$\mathrm {TV}_{4,q}$$ is known to be #P-hard in general; using a topological parameter provides an algorithm polynomial in the size of the input triangulation for the family of 3-manifolds with first $$\mathbb {Z}_2$$-homology group of bounded dimension. Our algorithm is easy to implement, and running times are comparable with running times to compute integral homology groups for standard libraries of triangulated 3-manifolds. The invariants we can compute this way are powerful: in combination with integral homology and using standard data sets, we are able to almost double the pairs of 3-manifolds we can distinguish. We hope this qualifies $$\mathrm {TV}_{4,q}$$ to be added to the short list of standard properties (such as orientability, connectedness and Betti Numbers) that can be computed ad hoc when first investigating an unknown triangulation.
Marie-françoise Roy - One of the best experts on this subject based on the ideXlab platform.
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Computing the First Betti Number of a Semi-Algebraic Set
Foundations of Computational Mathematics, 2008Co-Authors: Saugata Basu, Richard Pollack, Marie-françoise RoyAbstract:In this paper we describe a singly exponential algorithm for computing the first Betti Number of a given semi-algebraic set. Singly exponential algorithms for computing the zeroth Betti Number, and the Euler–Poincaré characteristic, were known before. No singly exponential algorithm was known for computing any of the individual Betti Numbers other than the zeroth one. As a consequence we also obtain algorithms for computing semi-algebraic descriptions of the semi-algebraically connected components of any given real algebraic or semi-algebraic set in singly exponential time, which improves on the complexity of the previously published algorithms for this problem.
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Computing the First Betti Number of a Semi-Algebraic Set
Foundations of Computational Mathematics, 2007Co-Authors: Saugata Basu, Richard Pollack, Marie-françoise RoyAbstract:In this paper we describe a singly exponential algorithm for computing the first Betti Number of a given semi-algebraic set. Singly exponential algorithms for computing the zeroth Betti Number, and the Euler–Poincare characteristic, were known before. No singly exponential algorithm was known for computing any of the individual Betti Numbers other than the zeroth one. As a consequence we also obtain algorithms for computing semi-algebraic descriptions of the semi-algebraically connected components of any given real algebraic or semi-algebraic set in singly exponential time, which improves on the complexity of the previously published algorithms for this problem.
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computing the first Betti Number and the connected components of semi algebraic sets
Symposium on the Theory of Computing, 2005Co-Authors: Saugata Basu, Richard Pollack, Marie-françoise RoyAbstract:In this paper we describe the first singly exponential algorithm for computing the first Betti Number of a given semi-algebraic set. We also describe algorithms for obtaining semi-algebraic descriptions of the semi-algebraically connected components of any given real algebraic or semi-algebraic set. Singly exponential algorithms for computing the zero-th Betti Number, and the Euler-Poincare characteristic, were known before. No singly exponential algorithm was known for computing any of the individual Betti Numbers other than the zero-th one.
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STOC - Computing the first Betti Number and the connected components of semi-algebraic sets
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing - STOC '05, 2005Co-Authors: Saugata Basu, Richard Pollack, Marie-françoise RoyAbstract:In this paper we describe the first singly exponential algorithm for computing the first Betti Number of a given semi-algebraic set. We also describe algorithms for obtaining semi-algebraic descriptions of the semi-algebraically connected components of any given real algebraic or semi-algebraic set. Singly exponential algorithms for computing the zero-th Betti Number, and the Euler-Poincare characteristic, were known before. No singly exponential algorithm was known for computing any of the individual Betti Numbers other than the zero-th one.
Saugata Basu - One of the best experts on this subject based on the ideXlab platform.
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Computing the First Betti Number of a Semi-Algebraic Set
Foundations of Computational Mathematics, 2008Co-Authors: Saugata Basu, Richard Pollack, Marie-françoise RoyAbstract:In this paper we describe a singly exponential algorithm for computing the first Betti Number of a given semi-algebraic set. Singly exponential algorithms for computing the zeroth Betti Number, and the Euler–Poincaré characteristic, were known before. No singly exponential algorithm was known for computing any of the individual Betti Numbers other than the zeroth one. As a consequence we also obtain algorithms for computing semi-algebraic descriptions of the semi-algebraically connected components of any given real algebraic or semi-algebraic set in singly exponential time, which improves on the complexity of the previously published algorithms for this problem.
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Computing the First Betti Number of a Semi-Algebraic Set
Foundations of Computational Mathematics, 2007Co-Authors: Saugata Basu, Richard Pollack, Marie-françoise RoyAbstract:In this paper we describe a singly exponential algorithm for computing the first Betti Number of a given semi-algebraic set. Singly exponential algorithms for computing the zeroth Betti Number, and the Euler–Poincare characteristic, were known before. No singly exponential algorithm was known for computing any of the individual Betti Numbers other than the zeroth one. As a consequence we also obtain algorithms for computing semi-algebraic descriptions of the semi-algebraically connected components of any given real algebraic or semi-algebraic set in singly exponential time, which improves on the complexity of the previously published algorithms for this problem.
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computing the first Betti Number and the connected components of semi algebraic sets
Symposium on the Theory of Computing, 2005Co-Authors: Saugata Basu, Richard Pollack, Marie-françoise RoyAbstract:In this paper we describe the first singly exponential algorithm for computing the first Betti Number of a given semi-algebraic set. We also describe algorithms for obtaining semi-algebraic descriptions of the semi-algebraically connected components of any given real algebraic or semi-algebraic set. Singly exponential algorithms for computing the zero-th Betti Number, and the Euler-Poincare characteristic, were known before. No singly exponential algorithm was known for computing any of the individual Betti Numbers other than the zero-th one.
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STOC - Computing the first Betti Number and the connected components of semi-algebraic sets
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing - STOC '05, 2005Co-Authors: Saugata Basu, Richard Pollack, Marie-françoise RoyAbstract:In this paper we describe the first singly exponential algorithm for computing the first Betti Number of a given semi-algebraic set. We also describe algorithms for obtaining semi-algebraic descriptions of the semi-algebraically connected components of any given real algebraic or semi-algebraic set. Singly exponential algorithms for computing the zero-th Betti Number, and the Euler-Poincare characteristic, were known before. No singly exponential algorithm was known for computing any of the individual Betti Numbers other than the zero-th one.
Hirofumi Sasahira - One of the best experts on this subject based on the ideXlab platform.
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Stable cohomotopy Seiberg–Witten invariants of connected sums of four-manifolds with positive first Betti Number, I: Non-vanishing theorem
International Journal of Mathematics, 2015Co-Authors: Masashi Ishida, Hirofumi SasahiraAbstract:We shall prove a new non-vanishing theorem for the stable cohomotopy Seiberg–Witten invariant [S. Bauer and M. Furuta, Stable cohomotopy refinement of Seiberg–Witten invariants: I, Invent. Math.155 (2004) 1–19; S. Bauer, Stable cohomotopy refinement of Seiberg–Witten invariants: II, Invent. Math.155 (2004) 21–40.] of connected sums of 4-manifolds with positive first Betti Number.
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stable cohomotopy seiberg witten invariants of connected sums of four manifolds with positive first Betti Number i non vanishing theorem
International Journal of Mathematics, 2015Co-Authors: Masashi Ishida, Hirofumi SasahiraAbstract:We shall prove a new non-vanishing theorem for the stable cohomotopy Seiberg–Witten invariant [S. Bauer and M. Furuta, Stable cohomotopy refinement of Seiberg–Witten invariants: I, Invent. Math.155 (2004) 1–19; S. Bauer, Stable cohomotopy refinement of Seiberg–Witten invariants: II, Invent. Math.155 (2004) 21–40.] of connected sums of 4-manifolds with positive first Betti Number.
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Stable Cohomotopy Seiberg-Witten Invariants of Connected Sums of Four-Manifolds with Positive First Betti Number
arXiv: Differential Geometry, 2008Co-Authors: Masashi Ishida, Hirofumi SasahiraAbstract:We shall prove a new non-vanishing theorem for the stable cohomotopy Seiberg-Witten invariant of connected sums of 4-manifolds with positive first Betti Number. The non-vanishing theorem enables us to find many new examples of 4-manifolds with non-trivial stable cohomotopy Seiberg-Witten invariants and it also gives a partial, but strong affirmative answer to a conjecture concerning non-vanishing of the invariant. Various new applications of the non-vanishing theorem are also given. For example, we shall introduce variants $\bar{\lambda}_k$ of Perelman's $\bar{\lambda}$ invariants for real Numbers $k$ and compute the values for a large class of 4-manifolds including connected sums of certain K{\"{a}}hler surfaces. The non-vanishing theorem is also used to construct the first examples of 4-manifolds with non-zero simplicial volume and satisfying the strict Gromov-Hitchin-Thorpe inequality, but admitting infinitely many distinct smooth structures for which no compatible Einstein metric exists. Moreover, we are able to prove a new result on the existence of exotic smooth structures.