On geometric bases for A-polynomials II: su3 and Kuperberg bracket.
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| Title: | On geometric bases for A-polynomials II: su3 and Kuperberg bracket. |
|---|---|
| Authors: | Galakhov, Dmitry1,2,3,4 (AUTHOR) d.galakhov.pion@gmail.com, Morozov, Alexei1,2,3,4 (AUTHOR) morozov@itep.ru |
| Source: | European Physical Journal C -- Particles & Fields. Aug2025, Vol. 85 Issue 8, p1-18. 18p. |
| Subjects: | Knot theory, Quantum groups, R-matrices, Gauge symmetries |
| Abstract: | We continue the study of quantum A-polynomials – equations for knot polynomials with respect to their coloring (representation-dependence) – as the relations between different links, obtained by hanging additional "simple" components on the original knot. Depending on the choice of this "decoration", the knot polynomial is either multiplied by a number or decomposes into a sum over "surrounding" representations by a cabling procedure. What happens is that these two of decorations, when complicated enough, become dependent – and this provides an equation. Remarkably it can be made independent of the representation. However, the equivalence of links is not a topological property – it follows from the properties of R-matrices, and strongly depends on the choice the gauge group and particular links. The relatively well studied part of the story concerns su 2 , where R-matrices can be chosen in an especially convenient Kauffman form, what makes the derivation of equations rather geometrical. To make these geometric methods somewhat simpler we suggest to use an arcade formalism/representation of the braid group to simplify decorating links universally. Here we attempt to extend this technique to the next case, su 3 , where the Kauffman rule is substituted by a more involved Kuperberg rule, still remains more geometric than generic analysis of MOY-diagrams, needed for higher ranks. Already in this case we encounter a classification problem for possible "decorations" and emergence of two-lined Young diagrams in enumeration of representations. [ABSTRACT FROM AUTHOR] |
| Copyright of European Physical Journal C -- Particles & Fields is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Header | DbId: egs DbLabel: Engineering Source An: 187993675 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: On geometric bases for A-polynomials II: su3 and Kuperberg bracket. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Galakhov%2C+Dmitry%22">Galakhov, Dmitry</searchLink><relatesTo>1,2,3,4</relatesTo> (AUTHOR)<i> d.galakhov.pion@gmail.com</i><br /><searchLink fieldCode="AR" term="%22Morozov%2C+Alexei%22">Morozov, Alexei</searchLink><relatesTo>1,2,3,4</relatesTo> (AUTHOR)<i> morozov@itep.ru</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22European+Physical+Journal+C+--+Particles+%26+Fields%22">European Physical Journal C -- Particles & Fields</searchLink>. Aug2025, Vol. 85 Issue 8, p1-18. 18p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Knot+theory%22">Knot theory</searchLink><br /><searchLink fieldCode="DE" term="%22Quantum+groups%22">Quantum groups</searchLink><br /><searchLink fieldCode="DE" term="%22R-matrices%22">R-matrices</searchLink><br /><searchLink fieldCode="DE" term="%22Gauge+symmetries%22">Gauge symmetries</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: We continue the study of quantum A-polynomials – equations for knot polynomials with respect to their coloring (representation-dependence) – as the relations between different links, obtained by hanging additional "simple" components on the original knot. Depending on the choice of this "decoration", the knot polynomial is either multiplied by a number or decomposes into a sum over "surrounding" representations by a cabling procedure. What happens is that these two of decorations, when complicated enough, become dependent – and this provides an equation. Remarkably it can be made independent of the representation. However, the equivalence of links is not a topological property – it follows from the properties of R-matrices, and strongly depends on the choice the gauge group and particular links. The relatively well studied part of the story concerns su 2 , where R-matrices can be chosen in an especially convenient Kauffman form, what makes the derivation of equations rather geometrical. To make these geometric methods somewhat simpler we suggest to use an arcade formalism/representation of the braid group to simplify decorating links universally. Here we attempt to extend this technique to the next case, su 3 , where the Kauffman rule is substituted by a more involved Kuperberg rule, still remains more geometric than generic analysis of MOY-diagrams, needed for higher ranks. Already in this case we encounter a classification problem for possible "decorations" and emergence of two-lined Young diagrams in enumeration of representations. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of European Physical Journal C -- Particles & Fields is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1140/epjc/s10052-025-14648-2 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 18 StartPage: 1 Subjects: – SubjectFull: Knot theory Type: general – SubjectFull: Quantum groups Type: general – SubjectFull: R-matrices Type: general – SubjectFull: Gauge symmetries Type: general Titles: – TitleFull: On geometric bases for A-polynomials II: su3 and Kuperberg bracket. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Galakhov, Dmitry – PersonEntity: Name: NameFull: Morozov, Alexei IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 08 Text: Aug2025 Type: published Y: 2025 Identifiers: – Type: issn-print Value: 14346044 Numbering: – Type: volume Value: 85 – Type: issue Value: 8 Titles: – TitleFull: European Physical Journal C -- Particles & Fields Type: main |
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