Arrow ribbon graphs

Robert Bradford; Clark Butler; Sergei Chmutov

doi:10.1142/S0218216512400020

Arrow ribbon graphs

Robert Bradford, Clark Butler, Sergei Chmutov

Mathematics

Research output: Contribution to journal › Review article › peer-review

7 Scopus citations

Abstract

We introduce an additional arrow structure on ribbon graphs. We extend the dichromatic polynomial to ribbon graphs with this structure. This extended polynomial satisfies the contractiondeletion relations and behaves naturally with respect to the partial duality of ribbon graphs. From a virtual link, we construct an arrow ribbon graph whose extended dichromatic polynomial specializes to the arrow polynomial of the virtual link recently introduced by H. Dye and L. Kauffman. This result generalizes the classical Thistlethwaite theorem to the arrow polynomial of virtual links.

Original language	English (US)
Article number	1240002
Journal	Journal of Knot Theory and its Ramifications
Volume	21
Issue number	13
DOIs	https://doi.org/10.1142/S0218216512400020
State	Published - Nov 2012

All Science Journal Classification (ASJC) codes

Algebra and Number Theory

Keywords

BollobásRiordan polynomial
Graphs on surfaces
Tutte polynomial
arrow polynomial
dichromatic polynomial
duality
ribbon graphs
virtual links

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10.1142/S0218216512400020

Cite this

@article{12965f16368a49b89671108e2864fcc7,

title = "Arrow ribbon graphs",

abstract = "We introduce an additional arrow structure on ribbon graphs. We extend the dichromatic polynomial to ribbon graphs with this structure. This extended polynomial satisfies the contractiondeletion relations and behaves naturally with respect to the partial duality of ribbon graphs. From a virtual link, we construct an arrow ribbon graph whose extended dichromatic polynomial specializes to the arrow polynomial of the virtual link recently introduced by H. Dye and L. Kauffman. This result generalizes the classical Thistlethwaite theorem to the arrow polynomial of virtual links.",

keywords = "Bollob{\'a}sRiordan polynomial, Graphs on surfaces, Tutte polynomial, arrow polynomial, dichromatic polynomial, duality, ribbon graphs, virtual links",

author = "Robert Bradford and Clark Butler and Sergei Chmutov",

note = "Funding Information: This work was done as part of the Summer 2010 undergraduate research working group http://www.math.ohio-state.edu/∼chmutov/wor-gr-su10/wor-gr.htm “Knots and Graphs” at the Ohio State University. We are grateful to all participants of the group for valuable discussions, to the OSU Honors Program Research Fund for the student financial support, to Ilya Kofman and Iain Moffatt for useful comments, and to an anonymous referee for various suggestions which drastically improved the exposition of the paper.",

year = "2012",

month = nov,

doi = "10.1142/S0218216512400020",

language = "English (US)",

volume = "21",

journal = "Journal of Knot Theory and its Ramifications",

issn = "0218-2165",

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number = "13",

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TY - JOUR

T1 - Arrow ribbon graphs

AU - Bradford, Robert

AU - Butler, Clark

AU - Chmutov, Sergei

N1 - Funding Information: This work was done as part of the Summer 2010 undergraduate research working group http://www.math.ohio-state.edu/∼chmutov/wor-gr-su10/wor-gr.htm “Knots and Graphs” at the Ohio State University. We are grateful to all participants of the group for valuable discussions, to the OSU Honors Program Research Fund for the student financial support, to Ilya Kofman and Iain Moffatt for useful comments, and to an anonymous referee for various suggestions which drastically improved the exposition of the paper.

PY - 2012/11

Y1 - 2012/11

N2 - We introduce an additional arrow structure on ribbon graphs. We extend the dichromatic polynomial to ribbon graphs with this structure. This extended polynomial satisfies the contractiondeletion relations and behaves naturally with respect to the partial duality of ribbon graphs. From a virtual link, we construct an arrow ribbon graph whose extended dichromatic polynomial specializes to the arrow polynomial of the virtual link recently introduced by H. Dye and L. Kauffman. This result generalizes the classical Thistlethwaite theorem to the arrow polynomial of virtual links.

AB - We introduce an additional arrow structure on ribbon graphs. We extend the dichromatic polynomial to ribbon graphs with this structure. This extended polynomial satisfies the contractiondeletion relations and behaves naturally with respect to the partial duality of ribbon graphs. From a virtual link, we construct an arrow ribbon graph whose extended dichromatic polynomial specializes to the arrow polynomial of the virtual link recently introduced by H. Dye and L. Kauffman. This result generalizes the classical Thistlethwaite theorem to the arrow polynomial of virtual links.

KW - BollobásRiordan polynomial

KW - Graphs on surfaces

KW - Tutte polynomial

KW - arrow polynomial

KW - dichromatic polynomial

KW - duality

KW - ribbon graphs

KW - virtual links

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U2 - 10.1142/S0218216512400020

DO - 10.1142/S0218216512400020

M3 - Review article

AN - SCOPUS:84868155822

SN - 0218-2165

VL - 21

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

IS - 13

M1 - 1240002

ER -

Arrow ribbon graphs

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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