A generalization of the Gram determinant of type A

Rhea Palak Bakshi; Dionne Ibarra; Sujoy Mukherjee; Józef H. Przytycki

doi:10.1016/j.topol.2021.107663

A generalization of the Gram determinant of type A

Rhea Palak Bakshi, Dionne Ibarra, Sujoy Mukherjee, Józef H. Przytycki

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

1 Citation (Scopus)

Abstract

The Gram determinant of type A was introduced by Lickorish in his work on invariants of 3 - manifolds. We generalize the theory of the Gram determinant of type A by evaluating, in the annulus, a bilinear form of non-intersecting connections in the disc. The main result provides a closed formula for this Gram determinant. We conclude the paper by discussing Chen's conjecture about the Gram determinant of type Mb evaluated in the Klein bottle.

Original language	English
Article number	107663
Number of pages	15
Journal	Topology and its Applications
Volume	295
DOIs	https://doi.org/10.1016/j.topol.2021.107663
Publication status	Published - 15 May 2021
Externally published	Yes

Keywords

Catalan numbers
Gram determinants
Klein bottle
Knot theory
Möbius band
Temperley - Lieb algebra
Witten - Reshetikhin - Turaev invariants

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10.1016/j.topol.2021.107663

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title = "A generalization of the Gram determinant of type A",

abstract = "The Gram determinant of type A was introduced by Lickorish in his work on invariants of 3 - manifolds. We generalize the theory of the Gram determinant of type A by evaluating, in the annulus, a bilinear form of non-intersecting connections in the disc. The main result provides a closed formula for this Gram determinant. We conclude the paper by discussing Chen's conjecture about the Gram determinant of type Mb evaluated in the Klein bottle.",

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AU - Bakshi, Rhea Palak

AU - Ibarra, Dionne

AU - Mukherjee, Sujoy

AU - Przytycki, Józef H.

N1 - Funding Information: The third author was partially supported by the CCAS Dean's Dissertation Completion Fellowship. The fourth author was partially supported by the Simons Foundation Collaboration Grant for Mathematicians - 316446 and the CCAS Dean's Research Chair award. Publisher Copyright: © 2021 Elsevier B.V.

PY - 2021/5/15

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N2 - The Gram determinant of type A was introduced by Lickorish in his work on invariants of 3 - manifolds. We generalize the theory of the Gram determinant of type A by evaluating, in the annulus, a bilinear form of non-intersecting connections in the disc. The main result provides a closed formula for this Gram determinant. We conclude the paper by discussing Chen's conjecture about the Gram determinant of type Mb evaluated in the Klein bottle.

AB - The Gram determinant of type A was introduced by Lickorish in his work on invariants of 3 - manifolds. We generalize the theory of the Gram determinant of type A by evaluating, in the annulus, a bilinear form of non-intersecting connections in the disc. The main result provides a closed formula for this Gram determinant. We conclude the paper by discussing Chen's conjecture about the Gram determinant of type Mb evaluated in the Klein bottle.

KW - Catalan numbers

KW - Gram determinants

KW - Klein bottle

KW - Knot theory

KW - Möbius band

KW - Temperley - Lieb algebra

KW - Witten - Reshetikhin - Turaev invariants

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SN - 0166-8641

VL - 295

JO - Topology and its Applications

JF - Topology and its Applications

M1 - 107663

ER -

A generalization of the Gram determinant of type A

Abstract

Keywords

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