insight - ScientificComputing - # Rogers-Ramanujan Identities

Multiple Rogers-Ramanujan Type Identities for the Quot and Motivic Cohen-Lenstra Zeta Functions of (2, 2k) Torus Links

Q: Can the methods used in this paper be extended to derive similar results for higher-dimensional torus knots or links?

Extending the methods in the paper to higher-dimensional torus knots and links is a challenging but potentially fruitful avenue for future research. Here's a breakdown of the challenges and possibilities: Challenges: Increased Complexity: Higher-dimensional torus knots and links have more complicated algebraic descriptions compared to their planar counterparts. This translates to a significant increase in the complexity of the associated Quot and motivic Cohen-Lenstra zeta functions. Lack of Explicit Formulas: The paper heavily relies on explicit formulas for the zeta functions associated with (2,n) torus knots and links. Deriving analogous formulas for higher-dimensional cases is a substantial hurdle. Hall-Littlewood Polynomial Generalizations: The paper utilizes Hall-Littlewood polynomials, which are inherently linked to partitions and symmetric functions. Generalizing these polynomials and their combinatorial interpretations to higher dimensions is non-trivial. Possibilities: Geometric Insights: Higher-dimensional torus knots and links are fundamental objects in knot theory and topology. If successful, extending these methods could provide new geometric insights into these objects through the lens of algebraic geometry and q-series identities. New Families of Identities: Just as the paper connects (2,n) torus links to Rogers-Ramanujan type identities, exploring higher-dimensional cases might lead to the discovery of entirely new families of q-series identities with rich combinatorial and representation-theoretic interpretations. Computational Exploration: While finding general formulas might be difficult, computational exploration using computer algebra systems could provide valuable data and suggest potential patterns for specific families of higher-dimensional torus knots and links.

Q: Are there alternative representations of the Quot and motivic Cohen-Lenstra zeta functions that could lead to different families of Rogers-Ramanujan type identities?

Yes, exploring alternative representations of the Quot and motivic Cohen-Lenstra zeta functions is a promising direction for uncovering new connections with Rogers-Ramanujan type identities. Here are some potential avenues: Combinatorial Representations: Seeking combinatorial interpretations of these zeta functions, perhaps in terms of weighted partitions or other combinatorial objects, could provide a direct link to the combinatorial structure underlying Rogers-Ramanujan type identities. Representation Theory: The appearance of Hall-Littlewood polynomials hints at a possible connection with representation theory. Investigating representations of suitable algebras or groups where these zeta functions appear as characters or graded dimensions could be insightful. Different Specializations: The paper focuses on specializations of the parameters (e.g., t = ±1) that lead to Rogers-Ramanujan type identities. Exploring different specializations or limits of these parameters might reveal connections to other families of q-series identities. Duality: Investigating potential duality relations satisfied by these zeta functions could provide new perspectives and lead to alternative representations that are more amenable to q-series manipulations.

Q: What are the implications of these findings for the study of quantum invariants of knots and 3-manifolds?

While not directly addressed in the paper, the findings have intriguing potential implications for the study of quantum invariants of knots and 3-manifolds: New Computational Tools: The explicit formulas and identities derived in the paper could potentially lead to new methods for computing quantum invariants, particularly for those invariants that are related to the colored Jones polynomial or its generalizations. Categorification: Rogers-Ramanujan type identities often have deep connections to representation theory and categorification. The findings in the paper might suggest ways to categorify certain quantum invariants, potentially leading to new knot homologies or other categorical structures. Geometric Interpretations: Quantum invariants are known to capture subtle geometric and topological information about knots and 3-manifolds. The connection between these invariants and the algebraic geometry of torus links, as hinted at in the paper, could provide new geometric interpretations of these invariants. Connections to Topological Quantum Field Theory: The motivic nature of the Cohen-Lenstra zeta function suggests a possible link to topological quantum field theory (TQFT). Exploring this connection further could provide a deeper understanding of the relationship between quantum invariants and TQFTs.

Core Concepts

This paper derives simplified k-fold summation expressions for the Quot and motivic Cohen-Lenstra zeta functions associated with (2, 2k) torus links, leading to multiple Rogers-Ramanujan type identities and confirming a conjecture by Huang and Jiang.

Abstract

Bibliographic Information: Chern, S. (2024). Multiple Rogers-Ramanujan type identities for torus links. arXiv:2411.07198v1 [math.NT].
Research Objective: The paper aims to find simplified expressions for the Quot and motivic Cohen-Lenstra zeta functions associated with (2, 2k) torus links and explore their connections to Rogers-Ramanujan type identities. This work is motivated by a conjecture by Huang and Jiang regarding the evaluation of the motivic Cohen-Lenstra zeta function at t = -1.
Methodology: The author employs techniques from q-series theory, including the use of q-Pochhammer symbols, q-binomial coefficients, Jacobi's triple product identity, q-hypergeometric functions, and Heine's transformations. The author derives a finite version of a multiple Rogers-Ramanujan type identity and then uses it to prove the desired results for the zeta functions.
Key Findings:
- The paper establishes a k-fold summation formula for the motivic Cohen-Lenstra zeta function associated with (2, 2k) torus links, analogous to a known formula for (2, 2k+1) torus knots.
- The paper proves a conjecture by Huang and Jiang concerning the evaluation of the motivic Cohen-Lenstra zeta function at t = -1.
- The author provides a finite analog of the main theorem, leading to a simplified expression for the Quot zeta function.
- The paper confirms a reflection formula for the Quot zeta function using q-theoretic methods.
- The nonnegativity conjecture proposed by Huang and Jiang for the zeta functions is proven.
Main Conclusions: The paper successfully connects the theory of motivic Cohen-Lenstra zeta functions and Quot zeta functions to Rogers-Ramanujan type identities. The simplified expressions derived for these zeta functions and the proof of Huang and Jiang's conjecture contribute significantly to the understanding of these mathematical objects and their applications in algebraic geometry and number theory.
Significance: This research provides a deeper understanding of the relationship between algebraic geometry, particularly the study of torus links, and q-series identities. The results have implications for the study of knot theory, modular forms, and partition theory.
Limitations and Future Research: The paper focuses specifically on (2, 2k) torus links. Exploring similar connections and identities for other types of knots and links could be a potential direction for future research. Additionally, investigating the combinatorial interpretations of the obtained identities in terms of partitions or other combinatorial objects could be of interest.

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Multiple Rogers--Ramanujan type identities for torus links

by Shane Chern at arxiv.org 11-12-2024

https://arxiv.org/pdf/2411.07198.pdf

Multiple Rogers--Ramanujan type identities for torus links

Deeper Inquiries

Can the methods used in this paper be extended to derive similar results for higher-dimensional torus knots or links?

Extending the methods in the paper to higher-dimensional torus knots and links is a challenging but potentially fruitful avenue for future research. Here's a breakdown of the challenges and possibilities:
Challenges:

Increased Complexity:  Higher-dimensional torus knots and links have more complicated algebraic descriptions compared to their planar counterparts. This translates to a significant increase in the complexity of the associated Quot and motivic Cohen-Lenstra zeta functions.
Lack of Explicit Formulas:  The paper heavily relies on explicit formulas for the zeta functions associated with (2,n) torus knots and links.  Deriving analogous formulas for higher-dimensional cases is a substantial hurdle.
Hall-Littlewood Polynomial Generalizations: The paper utilizes Hall-Littlewood polynomials, which are inherently linked to partitions and symmetric functions.  Generalizing these polynomials and their combinatorial interpretations to higher dimensions is non-trivial.
Possibilities:

Geometric Insights: Higher-dimensional torus knots and links are fundamental objects in knot theory and topology.  If successful, extending these methods could provide new geometric insights into these objects through the lens of algebraic geometry and q-series identities.
New Families of Identities:  Just as the paper connects (2,n) torus links to Rogers-Ramanujan type identities, exploring higher-dimensional cases might lead to the discovery of entirely new families of q-series identities with rich combinatorial and representation-theoretic interpretations.
Computational Exploration:  While finding general formulas might be difficult, computational exploration using computer algebra systems could provide valuable data and suggest potential patterns for specific families of higher-dimensional torus knots and links.

Are there alternative representations of the Quot and motivic Cohen-Lenstra zeta functions that could lead to different families of Rogers-Ramanujan type identities?

Yes, exploring alternative representations of the Quot and motivic Cohen-Lenstra zeta functions is a promising direction for uncovering new connections with Rogers-Ramanujan type identities. Here are some potential avenues:

Combinatorial Representations:  Seeking combinatorial interpretations of these zeta functions, perhaps in terms of weighted partitions or other combinatorial objects, could provide a direct link to the combinatorial structure underlying Rogers-Ramanujan type identities.
Representation Theory:  The appearance of Hall-Littlewood polynomials hints at a possible connection with representation theory.  Investigating representations of suitable algebras or groups where these zeta functions appear as characters or graded dimensions could be insightful.
Different Specializations: The paper focuses on specializations of the parameters (e.g., t = ±1) that lead to Rogers-Ramanujan type identities. Exploring different specializations or limits of these parameters might reveal connections to other families of q-series identities.
Duality:  Investigating potential duality relations satisfied by these zeta functions could provide new perspectives and lead to alternative representations that are more amenable to q-series manipulations.

What are the implications of these findings for the study of quantum invariants of knots and 3-manifolds?

While not directly addressed in the paper, the findings have intriguing potential implications for the study of quantum invariants of knots and 3-manifolds:

New Computational Tools: The explicit formulas and identities derived in the paper could potentially lead to new methods for computing quantum invariants, particularly for those invariants that are related to the colored Jones polynomial or its generalizations.
Categorification:  Rogers-Ramanujan type identities often have deep connections to representation theory and categorification.  The findings in the paper might suggest ways to categorify certain quantum invariants, potentially leading to new knot homologies or other categorical structures.
Geometric Interpretations:  Quantum invariants are known to capture subtle geometric and topological information about knots and 3-manifolds.  The connection between these invariants and the algebraic geometry of torus links, as hinted at in the paper, could provide new geometric interpretations of these invariants.
Connections to Topological Quantum Field Theory:  The motivic nature of the Cohen-Lenstra zeta function suggests a possible link to topological quantum field theory (TQFT).  Exploring this connection further could provide a deeper understanding of the relationship between quantum invariants and TQFTs.