T. Ben-itzhak - Academia.edu (original) (raw)

Papers by T. Ben-itzhak

In this paper we present a topological algorithm for conjugating two powers of half-twists in the... more In this paper we present a topological algorithm for conjugating two powers of half-twists in the braid group. The algorithm allows us to compute the conjugation without knowing the algebraic structure of the half-twist’s dif-feomorphism. The motivation of using the graph structure is to ignore the longer algebraic structure, and more important, to allow us a trivial comparison of the results without using the solution of the word problem in the braid group. Using the topological properties of the half-twist, we present an efficient solution for the conjugation of high powers of half-twists. The algorithm is motivated by the Hurwitz equivalence problem which is based on conjugation and comparison of power of half-twists. The results will be applied to the classification of algebraic surfaces up to deformation. It is already known that there exist surfaces that are diffeo-morphic to each other but are not deformations of each other (First example by Manetti). We have constructed a ne...

ABSTRACT. In this paper we prove certain Hurwitz equivalence properties in Bn. Our main result is... more ABSTRACT. In this paper we prove certain Hurwitz equivalence properties in Bn. Our main result is that every two Artin’s factorizations of ∆ 2 n of the form Hi1 · · ·Hi n(n−1)

ABSTRACT. In this paper we prove certain Hurwitz equivalence properties in the braid group. Our m... more ABSTRACT. In this paper we prove certain Hurwitz equivalence properties in the braid group. Our main result is that every two factorizations of ∆ 2 n where the elements of the factorization are semi-frame are Hurwitz equivalent. The results of this paper are generalization of the results in [8]. We use a new presentation of the braid group, called the Birman-Ko-Lee presentation, to define the semi-frame structure. The main result of this paper can be applied to compute the BMT invariant of surfaces (presented in [6] or [7]). The BMT is the class of Hurwitz equivalent factorizations of the central element of the braid group. The BMT distinguish among diffeomorphic surfaces which are not deformation of each other. 1 Topological Background In this section we recall some basic definitions and statements from [5]: Let D be a closed disk on R 2, K ⊂ D finite set, u ∈ ∂D. Any diffeomorphism of D which fixes K and is the identity on ∂D acts naturally on Π1 = Π1(D − K, u). We say that two su...

arXiv: Algebraic Geometry, 2002

In this paper we give an algorithm for solving a main case of the conjugacy problem in the braid ... more In this paper we give an algorithm for solving a main case of the conjugacy problem in the braid groups. We also prove that half-twists satisfy a special root property which allows us to reduce the solution for the conjugacy problem in half-twists into the free group. Using this algorithm one is able to check conjugacy of a given braid to one of E. Artin's generators in any power, and compute its root. Moreover, the braid element which conjugates a given half-twist to one of E. Artin's generators in any power can be restored. The result is applicable to calculations of braid monodromy of branch curves and verification of Hurwitz equivalence of braid monodromy factorizations, which are essential in order to determine braid monodromy type of algebraic surfaces and symplectic 4-manifolds.

In this paper we prove certain Hurwitz equivalence properties of BnB_nBn. In particular we prove th... more In this paper we prove certain Hurwitz equivalence properties of BnB_nBn. In particular we prove that for n=3n=3n=3 every two Artin's factorizations of Delta32\Delta _3 ^2Delta32 of the form Hi1...Hi6,quadFj1...Fj6H_{i_1} ... H_{i_6}, \quad F_{j_1} ... F_{j_6}Hi1...Hi6,quadFj1...Fj6 (with ik,jkin1,2i_k, j_k \in \{1,2 \}ik,jkin1,2) where H1,H2,F1,F2\{H_1, H_2 \}, \{F_1, F_2 \}H_1,H_2,F_1,F_2 are frames, are Hurwitz equivalent. The proof provided here is geometric, based on a newly defined frame type. The results will be applied to the classification of algebraic surfaces up to deformation. It is already known that there exist surfaces that are diffeomorphic to each other but are not deformations of each other (Manetti example). We are constructing a new invariant based on Hurwitz equivalence class of factorization, to distinguish among diffeomorphic surfaces which are not deformation of each other.

Motivated by the problem of Hurwitz equivalence of Delta2\Delta ^2Delta2 factorization in the braid group, ... more Motivated by the problem of Hurwitz equivalence of Delta2\Delta ^2Delta2 factorization in the braid group, we address the problem of Hurwitz equivalence in the symmetric group, obtained by projecting the Delta2\Delta ^2Delta2 factorizations into SnS_nSn. We get 1Sn1_{S_n}1_S_n factorizations with transposition factors. Looking at the transpositions as the edges in a graph, we show that two factorizations are Hurwitz equivalent if and only if their graphs have the same weighted connected components. The main result of this paper will help us to compute the "Braid Monodromy Type" invariant. The graph structure gives a weaker but very easy to compute invariant to distinguish between diffeomorphic surfaces which are not deformation of each other.

In this paper we prove certain Hurwitz equivalence properties in the braid group. Our main result... more In this paper we prove certain Hurwitz equivalence properties in the braid group. Our main result is that every two factorizations of Deltan2\Delta_n ^2Deltan2 where the elements of the factorization are semi-frame are Hurwitz equivalent. The results of this paper are generalization of the results in \cite{B4}. We use a new presentation of the braid group, called the Birman-Ko-Lee presentation, to define the semi-frame structure. The main result of this paper can be applied to compute the BMT invariant of surfaces. The BMT is the class of Hurwitz equivalent factorizations of the central element of the braid group. The BMT distinguish among diffeomorphic surfaces which are not deformation of each other.

In this paper we prove certain Hurwitz equivalence properties of BnB_nBn. In particular we prove th... more In this paper we prove certain Hurwitz equivalence properties of BnB_nBn. In particular we prove that for n=3n=3n=3 every two Artin's factorizations of Delta32\Delta _3 ^2Delta32 of the form Hi1...Hi6,quadFj1...Fj6H_{i_1} ... H_{i_6}, \quad F_{j_1} ... F_{j_6}Hi1...Hi6,quadFj1...Fj6 (with ik,jkin1,2i_k, j_k \in \{1,2 \}ik,jkin1,2) where H1,H2,F1,F2\{H_1, H_2 \}, \{F_1, F_2 \}H_1,H_2,F_1,F_2 are frames, are Hurwitz equivalent. The proof provided here is geometric, based on a newly defined frame type. The results will be applied to the classification of algebraic surfaces up to deformation. It is already known that there exist surfaces that are diffeomorphic to each other but are not deformations of each other (Manetti example). We are constructing a new invariant based on Hurwitz equivalence class of factorization, to distinguish among diffeomorphic surfaces which are not deformation of each other.

Israel Journal of Mathematics, 2003

Motivated by the problem of Hurwitz equivalence of A s factorization in the braid group, we addre... more Motivated by the problem of Hurwitz equivalence of A s factorization in the braid group, we address the problem of Hurwitz equivalence in the symmetric group, of ls~ factorizations with transposition factors. Looking at the transpositions as the edges in a graph, we show that two factorizations are Hurwitz equivalent if and only if their graphs have the same weighted connected components. The graph structure allows us to compute Hurwitz equivalence in the symmetric group. Using this result, one can compute non-Hurwitz equivalence in the braid group.

Journal of Algebra, 2003

In this paper we prove certain Hurwitz equivalence properties in B n. Our main result is that eve... more In this paper we prove certain Hurwitz equivalence properties in B n. Our main result is that every two Artin's factorizations of ∆ 2 n of the form H i 1 • • • H i n(n−1) , F j 1 • • • F j n(n−1) (with i k , j k ∈ {1, ..., n − 1}), where {H 1 , ..., H n−1 }, {F 1 , ..., F n−1 } are frames, are Hurwitz equivalent. This theorem is a generalization of the theorem we have proved in [6], using an algebraic approach unlike the proof in [6] which is geometric. The application of the result is applied to the classification of algebraic surfaces up to deformation. It is already known that there exist surfaces that are diffeomorphic to each other but are not deformations of each other (Manetti example). We construct a new invariant based on a Hurwitz equivalence class of factorization, to distinguish among diffeomorphic surfaces which are not deformations of each other. The precise definition of the new invariant can be found in [4] or [5]. The main result of this paper will help us to compute the new invariant.

In this paper we prove certain Hurwitz equivalence properties in the braid group. Our main result... more In this paper we prove certain Hurwitz equivalence properties in the braid group. Our main result is that every two factorizations of Deltan2\Delta_n ^2Deltan2 where the elements of the factorization are semi-frame are Hurwitz equivalent. The results of this paper are generalization of the results in \cite{B4}. We use a new presentation of the braid group, called the Birman-Ko-Lee presentation,

In this paper we present a topological algorithm for conjugating two powers of half-twists in the... more In this paper we present a topological algorithm for conjugating two powers of half-twists in the braid group. The algorithm allows us to compute the conjugation without knowing the algebraic structure of the half-twist's dif- feomorphism. The motivation of using the graph structure is to ignore the longer algebraic structure, and more important, to allow us a trivial comparison of

In this paper we give an algorithm for solving a main case of the conjugacy problem in the braid ... more In this paper we give an algorithm for solving a main case of the conjugacy problem in the braid groups. We also prove that half-twists satisfy a special root property which allows us to reduce the solution for the conjugacy problem in half-twists into the free group. Using this algorithm one is able to check conjugacy of a given braid to one of E. Artin's generators in any power, and compute its root. Moreover, the braid element which conjugates a given half-twist to one of E. Artin's generators in any power can be restored. The result is applicable to calculations of braid monodromy of branch curves and verification of Hurwitz equivalence of braid monodromy factorizations, which are essential in order to determine braid monodromy type of algebraic surfaces and symplectic 4-manifolds.

Motivated by the problem of Hurwitz equivalence of Delta2\Delta ^2Delta2 factorization in the braid group, ... more Motivated by the problem of Hurwitz equivalence of Delta2\Delta ^2Delta2 factorization in the braid group, we address the problem of Hurwitz equivalence in the symmetric group, obtained by projecting the Delta2\Delta ^2Delta2 factorizations into SnS_nSn. We get 1Sn1_{S_n}1_S_n factorizations with transposition factors. Looking at the transpositions as the edges in a graph, we show that two factorizations are Hurwitz equivalent if and only if their graphs have the same weighted connected components. The main result of this paper will help us to compute the "Braid Monodromy Type" invariant. The graph structure gives a weaker but very easy to compute invariant to distinguish between diffeomorphic surfaces which are not deformation of each other.

Motivated by the problem of Hurwitz equivalence of A s factorization in the braid group, we addre... more Motivated by the problem of Hurwitz equivalence of A s factorization in the braid group, we address the problem of Hurwitz equivalence in the symmetric group, of ls~ factorizations with transposition factors.

International Journal of Algebra and Computation, 2003

In this paper we present a topological algorithm for conjugating two powers of half-twists in the... more In this paper we present a topological algorithm for conjugating two powers of half-twists in the braid group. The algorithm allows us to compute the conjugation without knowing the algebraic structure of the half-twist’s dif-feomorphism. The motivation of using the graph structure is to ignore the longer algebraic structure, and more important, to allow us a trivial comparison of the results without using the solution of the word problem in the braid group. Using the topological properties of the half-twist, we present an efficient solution for the conjugation of high powers of half-twists. The algorithm is motivated by the Hurwitz equivalence problem which is based on conjugation and comparison of power of half-twists. The results will be applied to the classification of algebraic surfaces up to deformation. It is already known that there exist surfaces that are diffeo-morphic to each other but are not deformations of each other (First example by Manetti). We have constructed a ne...

ABSTRACT. In this paper we prove certain Hurwitz equivalence properties in Bn. Our main result is... more ABSTRACT. In this paper we prove certain Hurwitz equivalence properties in Bn. Our main result is that every two Artin’s factorizations of ∆ 2 n of the form Hi1 · · ·Hi n(n−1)

ABSTRACT. In this paper we prove certain Hurwitz equivalence properties in the braid group. Our m... more ABSTRACT. In this paper we prove certain Hurwitz equivalence properties in the braid group. Our main result is that every two factorizations of ∆ 2 n where the elements of the factorization are semi-frame are Hurwitz equivalent. The results of this paper are generalization of the results in [8]. We use a new presentation of the braid group, called the Birman-Ko-Lee presentation, to define the semi-frame structure. The main result of this paper can be applied to compute the BMT invariant of surfaces (presented in [6] or [7]). The BMT is the class of Hurwitz equivalent factorizations of the central element of the braid group. The BMT distinguish among diffeomorphic surfaces which are not deformation of each other. 1 Topological Background In this section we recall some basic definitions and statements from [5]: Let D be a closed disk on R 2, K ⊂ D finite set, u ∈ ∂D. Any diffeomorphism of D which fixes K and is the identity on ∂D acts naturally on Π1 = Π1(D − K, u). We say that two su...

arXiv: Algebraic Geometry, 2002

In this paper we give an algorithm for solving a main case of the conjugacy problem in the braid ... more In this paper we give an algorithm for solving a main case of the conjugacy problem in the braid groups. We also prove that half-twists satisfy a special root property which allows us to reduce the solution for the conjugacy problem in half-twists into the free group. Using this algorithm one is able to check conjugacy of a given braid to one of E. Artin's generators in any power, and compute its root. Moreover, the braid element which conjugates a given half-twist to one of E. Artin's generators in any power can be restored. The result is applicable to calculations of braid monodromy of branch curves and verification of Hurwitz equivalence of braid monodromy factorizations, which are essential in order to determine braid monodromy type of algebraic surfaces and symplectic 4-manifolds.

In this paper we prove certain Hurwitz equivalence properties of BnB_nBn. In particular we prove th... more In this paper we prove certain Hurwitz equivalence properties of BnB_nBn. In particular we prove that for n=3n=3n=3 every two Artin's factorizations of Delta32\Delta _3 ^2Delta32 of the form Hi1...Hi6,quadFj1...Fj6H_{i_1} ... H_{i_6}, \quad F_{j_1} ... F_{j_6}Hi1...Hi6,quadFj1...Fj6 (with ik,jkin1,2i_k, j_k \in \{1,2 \}ik,jkin1,2) where H1,H2,F1,F2\{H_1, H_2 \}, \{F_1, F_2 \}H_1,H_2,F_1,F_2 are frames, are Hurwitz equivalent. The proof provided here is geometric, based on a newly defined frame type. The results will be applied to the classification of algebraic surfaces up to deformation. It is already known that there exist surfaces that are diffeomorphic to each other but are not deformations of each other (Manetti example). We are constructing a new invariant based on Hurwitz equivalence class of factorization, to distinguish among diffeomorphic surfaces which are not deformation of each other.

Motivated by the problem of Hurwitz equivalence of Delta2\Delta ^2Delta2 factorization in the braid group, ... more Motivated by the problem of Hurwitz equivalence of Delta2\Delta ^2Delta2 factorization in the braid group, we address the problem of Hurwitz equivalence in the symmetric group, obtained by projecting the Delta2\Delta ^2Delta2 factorizations into SnS_nSn. We get 1Sn1_{S_n}1_S_n factorizations with transposition factors. Looking at the transpositions as the edges in a graph, we show that two factorizations are Hurwitz equivalent if and only if their graphs have the same weighted connected components. The main result of this paper will help us to compute the "Braid Monodromy Type" invariant. The graph structure gives a weaker but very easy to compute invariant to distinguish between diffeomorphic surfaces which are not deformation of each other.

In this paper we prove certain Hurwitz equivalence properties in the braid group. Our main result... more In this paper we prove certain Hurwitz equivalence properties in the braid group. Our main result is that every two factorizations of Deltan2\Delta_n ^2Deltan2 where the elements of the factorization are semi-frame are Hurwitz equivalent. The results of this paper are generalization of the results in \cite{B4}. We use a new presentation of the braid group, called the Birman-Ko-Lee presentation, to define the semi-frame structure. The main result of this paper can be applied to compute the BMT invariant of surfaces. The BMT is the class of Hurwitz equivalent factorizations of the central element of the braid group. The BMT distinguish among diffeomorphic surfaces which are not deformation of each other.

In this paper we prove certain Hurwitz equivalence properties of BnB_nBn. In particular we prove th... more In this paper we prove certain Hurwitz equivalence properties of BnB_nBn. In particular we prove that for n=3n=3n=3 every two Artin's factorizations of Delta32\Delta _3 ^2Delta32 of the form Hi1...Hi6,quadFj1...Fj6H_{i_1} ... H_{i_6}, \quad F_{j_1} ... F_{j_6}Hi1...Hi6,quadFj1...Fj6 (with ik,jkin1,2i_k, j_k \in \{1,2 \}ik,jkin1,2) where H1,H2,F1,F2\{H_1, H_2 \}, \{F_1, F_2 \}H_1,H_2,F_1,F_2 are frames, are Hurwitz equivalent. The proof provided here is geometric, based on a newly defined frame type. The results will be applied to the classification of algebraic surfaces up to deformation. It is already known that there exist surfaces that are diffeomorphic to each other but are not deformations of each other (Manetti example). We are constructing a new invariant based on Hurwitz equivalence class of factorization, to distinguish among diffeomorphic surfaces which are not deformation of each other.

Israel Journal of Mathematics, 2003

Motivated by the problem of Hurwitz equivalence of A s factorization in the braid group, we addre... more Motivated by the problem of Hurwitz equivalence of A s factorization in the braid group, we address the problem of Hurwitz equivalence in the symmetric group, of ls~ factorizations with transposition factors. Looking at the transpositions as the edges in a graph, we show that two factorizations are Hurwitz equivalent if and only if their graphs have the same weighted connected components. The graph structure allows us to compute Hurwitz equivalence in the symmetric group. Using this result, one can compute non-Hurwitz equivalence in the braid group.

Journal of Algebra, 2003

In this paper we prove certain Hurwitz equivalence properties in B n. Our main result is that eve... more In this paper we prove certain Hurwitz equivalence properties in B n. Our main result is that every two Artin's factorizations of ∆ 2 n of the form H i 1 • • • H i n(n−1) , F j 1 • • • F j n(n−1) (with i k , j k ∈ {1, ..., n − 1}), where {H 1 , ..., H n−1 }, {F 1 , ..., F n−1 } are frames, are Hurwitz equivalent. This theorem is a generalization of the theorem we have proved in [6], using an algebraic approach unlike the proof in [6] which is geometric. The application of the result is applied to the classification of algebraic surfaces up to deformation. It is already known that there exist surfaces that are diffeomorphic to each other but are not deformations of each other (Manetti example). We construct a new invariant based on a Hurwitz equivalence class of factorization, to distinguish among diffeomorphic surfaces which are not deformations of each other. The precise definition of the new invariant can be found in [4] or [5]. The main result of this paper will help us to compute the new invariant.

In this paper we prove certain Hurwitz equivalence properties in the braid group. Our main result... more In this paper we prove certain Hurwitz equivalence properties in the braid group. Our main result is that every two factorizations of Deltan2\Delta_n ^2Deltan2 where the elements of the factorization are semi-frame are Hurwitz equivalent. The results of this paper are generalization of the results in \cite{B4}. We use a new presentation of the braid group, called the Birman-Ko-Lee presentation,

In this paper we present a topological algorithm for conjugating two powers of half-twists in the... more In this paper we present a topological algorithm for conjugating two powers of half-twists in the braid group. The algorithm allows us to compute the conjugation without knowing the algebraic structure of the half-twist's dif- feomorphism. The motivation of using the graph structure is to ignore the longer algebraic structure, and more important, to allow us a trivial comparison of

In this paper we give an algorithm for solving a main case of the conjugacy problem in the braid ... more In this paper we give an algorithm for solving a main case of the conjugacy problem in the braid groups. We also prove that half-twists satisfy a special root property which allows us to reduce the solution for the conjugacy problem in half-twists into the free group. Using this algorithm one is able to check conjugacy of a given braid to one of E. Artin's generators in any power, and compute its root. Moreover, the braid element which conjugates a given half-twist to one of E. Artin's generators in any power can be restored. The result is applicable to calculations of braid monodromy of branch curves and verification of Hurwitz equivalence of braid monodromy factorizations, which are essential in order to determine braid monodromy type of algebraic surfaces and symplectic 4-manifolds.

Motivated by the problem of Hurwitz equivalence of Delta2\Delta ^2Delta2 factorization in the braid group, ... more Motivated by the problem of Hurwitz equivalence of Delta2\Delta ^2Delta2 factorization in the braid group, we address the problem of Hurwitz equivalence in the symmetric group, obtained by projecting the Delta2\Delta ^2Delta2 factorizations into SnS_nSn. We get 1Sn1_{S_n}1_S_n factorizations with transposition factors. Looking at the transpositions as the edges in a graph, we show that two factorizations are Hurwitz equivalent if and only if their graphs have the same weighted connected components. The main result of this paper will help us to compute the "Braid Monodromy Type" invariant. The graph structure gives a weaker but very easy to compute invariant to distinguish between diffeomorphic surfaces which are not deformation of each other.

Motivated by the problem of Hurwitz equivalence of A s factorization in the braid group, we addre... more Motivated by the problem of Hurwitz equivalence of A s factorization in the braid group, we address the problem of Hurwitz equivalence in the symmetric group, of ls~ factorizations with transposition factors.

International Journal of Algebra and Computation, 2003