A fast solution to the conjugacy problem in the four-strand braid group

Calvez, Matthieu; Wiest, Bert

doi:10.1515/jgth-2014-0020

Cited by 4 publications

(9 citation statements)

References 22 publications

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“…Indeed, suppose that the left inequality in (7) Hence ϕ(s −1 Z)s δ by Lemma 2.1b. Since ϕ(s −1 Z) = ϕ(Z) by Corollary 2.4, this means that ϕ(Z)s δ which implies the right inequality in (7). Thus, (7) holds.…”

Section: Products Of Two Atoms Normal Forms In Case (Iii) Of Theoremmentioning

confidence: 81%

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Algorithmic recognition of quasipositive 4-braids of algebraic length three

Orevkov

2015

Groups Complexity Cryptology

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The paper provides an empirical analysis of the macroeconomic factors that enhance revenue gap in South Africa using the multivariate cointegration techniques for the period 1965 to 2012. The results from the cointegration analysis indicate that the revenue gap in South Africa is negatively associated with the level of imports while positively related to external debt and underground economy. The former finding is consistent with the notion that imports are subjected to more taxation than domestic activities because of certain features of international trade that tend to make tax evasion difficult. On the other hand, the positive relationship between external debt and tax gap shows that the South African government relies upon external debt to finance its budget deficit resulting from missing revenues. Furthermore, the observed negative effect of the post-apartheid dummy confirms that the tax policy reforms that South Africa introduced following the liberation in 1994 have led to a reduction in missing revenues. The results from the Granger causality test also show that there is a unidirectional causality running from imports and underground economy to revenue gap, while revenue gap on the other hand is found to Granger-cause national income and external debt in South Africa.

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Section: Products Of Two Atoms Normal Forms In Case (Iii) Of Theoremmentioning

confidence: 81%

“…Thus, (7) holds. By symmetry, without loss of generality we may assume that the right inequality in (7) holds. Then x 1 s = ϕ(Z)s δ by Lemma 2.1b.…”

Section: Products Of Two Atoms Normal Forms In Case (Iii) Of Theoremmentioning

confidence: 99%

Algorithmic recognition of quasipositive 4-braids of algebraic length three

Orevkov

2015

Groups Complexity Cryptology

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“…If we want to decide the Nielsen-Thurston type of a given 4-braid, the algorithm in [12] should rather be used instead of the present one because it is implementable and it finds explicitly the reducing curves whenever they exist (in polynomial time). Using the Birman-Ko-Lee structure of B 4 , the author together with Bert Wiest show in [11] the existence of a bound as in (iii) (which depends on Masur-Minsky's constant c(4), see Proposition 17). Unfortunately, they do not know yet how to make explicit the constant c(4) (nor C(4)), so that the cardinality of the ultra summit set of a pseudo-Anosov rigid 4-braid is not explicitly known.…”

Section: Proofs Of Our Resultsmentioning

confidence: 99%

“…Unfortunately, they do not know yet how to make explicit the constant c(4) (nor C(4)), so that the cardinality of the ultra summit set of a pseudo-Anosov rigid 4-braid is not explicitly known. Nevertheless [11] presents a polynomial-time algorithm for solving the conjugacy problem in B 4 .…”

Section: Proofs Of Our Resultsmentioning

confidence: 99%

Fast Nielsen–Thurston classification of braids

Calvez

2014

Algebr. Geom. Topol.

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“…In [20] the author exhibit seven different classes of braids in B 3 and proved that any element of B 3 is conjugate to a unique element of these seven classes. To classify conjugacy classes of elements in braid groups with several strings is not trivial, during the last years a lot of work has been done to create efficient algorithms to attack the problem of determine if two given braids are conjugate, see [5], [6], [7], [9] and [12].…”

Section: Introductionmentioning

confidence: 99%