Legendrian DGA Representations and the Colored Kauffman Polynomial
For any Legendrian knot 𝛫 in standard contact ℝ³, we relate counts of ungraded (1-graded) representations of the Legendrian contact homology DG-algebra (A(𝛫), ∂) with the n-colored Kauffman polynomial. To do this, we introduce an ungraded n-colored ruling polynomial, R¹ₙ, 𝛫(q), as a linear combinati...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2020 |
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Інститут математики НАН України
2020
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Murray, Justin Rutherford, Dan 2025-12-12T10:34:31Z 2020 Legendrian DGA Representations and the Colored Kauffman Polynomial. Justin Murray and Dan Rutherford. SIGMA 16 (2020), 017, 33 pages 1815-0659 2020 Mathematics Subject Classification: 53D42; 57M27 arXiv:1908.08978 https://nasplib.isofts.kiev.ua/handle/123456789/210593 https://doi.org/10.3842/SIGMA.2020.017 For any Legendrian knot 𝛫 in standard contact ℝ³, we relate counts of ungraded (1-graded) representations of the Legendrian contact homology DG-algebra (A(𝛫), ∂) with the n-colored Kauffman polynomial. To do this, we introduce an ungraded n-colored ruling polynomial, R¹ₙ, 𝛫(q), as a linear combination of reduced ruling polynomials of positive permutation braids and show that (i) R¹ₙ, 𝛫(q) arises as a specialization 𝘍ₙ, 𝛫(a, q)∣ₐ⁻¹₌₀ of the n-colored Kauffman polynomial and (ii) when q is a power of two R¹ₙ, 𝛫(q) agrees with the total ungraded representation number, Rep₁(𝛫, 𝔽ⁿq), which is a normalized count of n-dimensional representations of (A(𝛫),∂) over the finite field 𝔽q. This complements results from [Leverson C., Rutherford D., Quantum Topol. 11 (2020), 55-118] concerning the colored HOMFLY-PT polynomial, m-graded representation numbers, and m-graded ruling polynomials with m≠1. This article is dedicated to Dmitry Fuchs, to whom the second author is grateful for his generosity and support through the years in grad school and beyond. Thank you, Dmitry! DR acknowledges support from the Simons Foundation grant #429536. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Legendrian DGA Representations and the Colored Kauffman Polynomial Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Legendrian DGA Representations and the Colored Kauffman Polynomial |
| spellingShingle |
Legendrian DGA Representations and the Colored Kauffman Polynomial Murray, Justin Rutherford, Dan |
| title_short |
Legendrian DGA Representations and the Colored Kauffman Polynomial |
| title_full |
Legendrian DGA Representations and the Colored Kauffman Polynomial |
| title_fullStr |
Legendrian DGA Representations and the Colored Kauffman Polynomial |
| title_full_unstemmed |
Legendrian DGA Representations and the Colored Kauffman Polynomial |
| title_sort |
legendrian dga representations and the colored kauffman polynomial |
| author |
Murray, Justin Rutherford, Dan |
| author_facet |
Murray, Justin Rutherford, Dan |
| publishDate |
2020 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
For any Legendrian knot 𝛫 in standard contact ℝ³, we relate counts of ungraded (1-graded) representations of the Legendrian contact homology DG-algebra (A(𝛫), ∂) with the n-colored Kauffman polynomial. To do this, we introduce an ungraded n-colored ruling polynomial, R¹ₙ, 𝛫(q), as a linear combination of reduced ruling polynomials of positive permutation braids and show that (i) R¹ₙ, 𝛫(q) arises as a specialization 𝘍ₙ, 𝛫(a, q)∣ₐ⁻¹₌₀ of the n-colored Kauffman polynomial and (ii) when q is a power of two R¹ₙ, 𝛫(q) agrees with the total ungraded representation number, Rep₁(𝛫, 𝔽ⁿq), which is a normalized count of n-dimensional representations of (A(𝛫),∂) over the finite field 𝔽q. This complements results from [Leverson C., Rutherford D., Quantum Topol. 11 (2020), 55-118] concerning the colored HOMFLY-PT polynomial, m-graded representation numbers, and m-graded ruling polynomials with m≠1.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/210593 |
| citation_txt |
Legendrian DGA Representations and the Colored Kauffman Polynomial. Justin Murray and Dan Rutherford. SIGMA 16 (2020), 017, 33 pages |
| work_keys_str_mv |
AT murrayjustin legendriandgarepresentationsandthecoloredkauffmanpolynomial AT rutherforddan legendriandgarepresentationsandthecoloredkauffmanpolynomial |
| first_indexed |
2025-12-17T12:04:17Z |
| last_indexed |
2025-12-17T12:04:17Z |
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1851756964648845312 |