Invariants of closed braids via counting surfaces

Michael Brandenbursky

doi:10.1142/S0218216513500119

Invariants of closed braids via counting surfaces

Research output: Contribution to journal › Article › peer-review

Abstract

A Gauss diagram is a simple, combinatorial way to present a link. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting subdiagrams of certain combinatorial types. In this paper we present simple formulas for an infinite family of invariants in terms of counting surfaces of a certain genus and number of boundary components in a Gauss diagram associated with a closed braid. We then identify the resulting invariants with partial derivatives of the HOMFLY-PT polynomial.

Original language	American English
Article number	13500119
Journal	Journal of Knot Theory and its Ramifications
Volume	22
Issue number	3
DOIs	https://doi.org/10.1142/S0218216513500119
State	Published - 1 Mar 2013
Externally published	Yes

Keywords

Braid groups
Gauss diagram formulas
finite type invariants
knot polynomials

All Science Journal Classification (ASJC) codes

Algebra and Number Theory

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10.1142/S0218216513500119

Cite this

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abstract = "A Gauss diagram is a simple, combinatorial way to present a link. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting subdiagrams of certain combinatorial types. In this paper we present simple formulas for an infinite family of invariants in terms of counting surfaces of a certain genus and number of boundary components in a Gauss diagram associated with a closed braid. We then identify the resulting invariants with partial derivatives of the HOMFLY-PT polynomial.",

keywords = "Braid groups, Gauss diagram formulas, finite type invariants, knot polynomials",

author = "Michael Brandenbursky",

note = "Funding Information: The author would like to thank Michael Polyak and Hao Wu for helpful conversations. We would like to thank the anonymous referee for careful reading of our paper and for his/her helpful comments and remarks. Part of this work has been done during the author{\textquoteright}s stay in Mathematisches Forschungsinstitut Oberwolfach. The author wishes to express his gratitude to the institute. He was supported by the Oberwolfach Leibniz fellowship.",

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N2 - A Gauss diagram is a simple, combinatorial way to present a link. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting subdiagrams of certain combinatorial types. In this paper we present simple formulas for an infinite family of invariants in terms of counting surfaces of a certain genus and number of boundary components in a Gauss diagram associated with a closed braid. We then identify the resulting invariants with partial derivatives of the HOMFLY-PT polynomial.

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KW - knot polynomials

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ER -

Invariants of closed braids via counting surfaces

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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