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2013•St Petersburg Mathematical JournalOpen access

On independence of some pseudocharacters on braid groups

Ivan Alekseevich Dynnikov, Vladimir Shastin

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Abstract

It is proved that the pseudocharacter defined on the braid group by the signature of braid closures is linearly independent of all pseudocharacters obtained from the twist number via the Malyutin operators, provided that the number of strands is greater than 4. This pseudocharacter is shown to have a nontrivial kernel part. It is observed that the operators $I$ and $R$ defined by Malyutin on the space of pseudocharacters satisfy the Heisenberg relation, and that some of Malyutin’s results are standard consequences of this fact.

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It is proved that the pseudocharacter defined on the braid group by the signature of braid closures is linearly independent of all pseudocharacters obtained from the twist number via the Malyutin operators, provided that the number of strands is greater than 4. This pseudocharacter is shown to have a nontrivial kernel part. It is observed that the operators $I$ and $R$ defined by Malyutin on the space of pseudocharacters satisfy the Heisenberg relation, and that some of Malyutin’s results are standard consequences of this fact.

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Available abstract

It is proved that the pseudocharacter defined on the braid group by the signature of braid closures is linearly independent of all pseudocharacters obtained from the twist number via the Malyutin operators, provided that the number of strands is greater than 4. This pseudocharacter is shown to have a nontrivial kernel part. It is observed that the operators $I$ and $R$ defined by Malyutin on the space of pseudocharacters satisfy the Heisenberg relation, and that some of Malyutin’s results are standard consequences of this fact.

Key concepts: Braid, Mathematics, Braid group, Braid theory, Kernel (algebra), Pure mathematics, Independence (probability theory), Twist

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