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Quasipositivity test via unitary representations of braid groups and its applications to real algebraic curves. (English) Zbl 1030.20026
From the introduction: Let \(B_m=\langle\sigma_1,\dots,\sigma_{m-1}\mid\sigma_j\sigma_{j+1}\sigma_j=\sigma_{j+1}\sigma_j\sigma_{j+1}\), \(\sigma_j\sigma_k=\sigma_k\sigma_j\) for \(|j-k|>1\rangle\). We call the elements of \(B_m\) \(m\)-braids or braids. An \(m\)-braid \(b\) is called quasipositive if \(b=\prod^k_{j=1}a_j\sigma_1a_j^{-1}\) for some \(a_j\in B_m\).
In a series of previous papers we exploited the observation that the quasipositivity of a certain braid provides a necessary condition for the realizability of a given isotopy type by a plane real algebraic curve of a given degree. As a test for quasipositivity we used the Murasugi-Tristram signature inequality, elementary arguments based on linking numbers, or the Garside normal form for braids with three strings. Here we propose a new simple test for quasipositivity and give an example when it gives some new restrictions for real algebraic curves of 7th degree.
The test is based on the following elementary observation. Suppose we are interested in whether a given braid \(b\) is quasipositive. If it is, then the number \(k\) of the factors in any quasipositive presentation is just the image of \(b\) under the Abelianization \(B_m\to\mathbb{Z}\). Let \(\rho\colon B_m\to\text{SU}(n)\) be any unitary representation. Then the matrix \(\rho(b)\) is a product of \(k\) matrices, each of which is conjugated to \(\rho(\sigma_1)\). A necessary and sufficient condition for a given matrix to be presented as a product of matrices from given conjugacy classes was obtained by S. Agnihotri and C. Woodward [Math. Res. Lett. 5, No. 6, 817-836 (1998; Zbl 1004.14013)].

MSC:
20F36 Braid groups; Artin groups
14P05 Real algebraic sets
57M25 Knots and links in the \(3\)-sphere (MSC2010)
20C15 Ordinary representations and characters
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