Generating Infinitely Many Sequences of Distinct Prime Hyperbolic Knots Using Plat Closures of PseudoAnosov Braids
Core Concepts
This paper presents a method for constructing infinitely many sequences of distinct prime hyperbolic knots by analyzing the plat closures of pseudoAnosov braids and their relationship to the Hempel distance of Heegaard splittings.
Abstract

Bibliographic Information: Engelhardt, C., & Hovland, S. (2024). Generating Infinitely Many Hyperbolic Knots with Plats. arXiv preprint arXiv:2410.17443.

Research Objective: To develop a method for constructing infinitely many sequences of distinct prime hyperbolic knots, addressing the challenge posed by the nongenericity of hyperbolic links as crossing number increases.

Methodology: The authors leverage the relationship between links in plat position, braid group dynamics, and Heegaard splittings of double branched covers of S^3 over a link. They utilize the Hempel distance as a measure of complexity for both Heegaard splittings and links in bridge position. By analyzing the plat closures of pseudoAnosov braids and their corresponding Heegaard splittings, they establish a lower bound for the Hempel distance of the knots.

Key Findings:
 The Hempel distance of the Heegaard splitting of the double branched cover obtained from a plat is a lower bound for the Hempel distance of that plat.
 For n ≥ 3, plat closures of certain powers of a generic pseudoAnosov braid with a 1component plat closure generate a sequence of infinitely many distinct prime hyperbolic knots with strictly increasing volumes.
 Plat closures of braids in B6 (braid group on 6 strands) are generically hyperbolic.

Main Conclusions: The study provides a novel construction method for generating infinite families of distinct prime hyperbolic knots with increasing complexity. This method offers insights into the relationship between knot theory, braid groups, and Heegaard splittings.

Significance: This research significantly contributes to knot theory by providing a new tool for constructing and analyzing hyperbolic knots, particularly those with high crossing numbers. It also sheds light on the relationship between the Hempel distance of Heegaard splittings and the geometry of knots.

Limitations and Future Research: The study primarily focuses on knots obtained from plat closures of pseudoAnosov braids. Further research could explore the applicability of this method to other types of braids or develop alternative constructions for hyperbolic knots. Additionally, investigating the precise relationship between the Hempel distance and other knot invariants, such as genus and volume, could yield further insights.
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Generating Infinitely Many Hyperbolic Knots with Plats
Stats
For n ≥ 3, the genus of a knot K in bridge position is at least 1/2(M1), where M is the Hempel distance of K with respect to a splitting sphere.
The proportion of pseudoAnosov braids in the set of all braid words in B_n of length at most ℓ increases exponentially as ℓ increases.
Quotes
"Until recently, it was widely believed that generic prime links are hyperbolic."
"Malyutin proved [...] that, counterintuitively, the proportion of hyperbolic links as crossing number increases does not approach 1."
"This leads to a question: how can one construct prime hyperbolic knots of high crossing number?"
"The plat closures of braids in B6 are generically hyperbolic."
Deeper Inquiries
Can this method be extended to construct other classes of knots, such as those with specific geometric or topological properties?
This method, which leverages the properties of pseudoAnosov braids and the Hempel distance of their plat closures, holds significant potential for extension to construct knots with specific characteristics. Here's how:
Targetting Specific Genus: The paper establishes a lower bound for the genus of the constructed knots using Hempel distance. By carefully selecting pseudoAnosov braids and controlling the growth rate of Hempel distance with increasing powers, one could potentially target the construction of knots with a specific genus or genus bounds.
Exploring other Geometric Invariants: Beyond genus, other geometric invariants like hyperbolic volume, bridge number, and tunnel number are deeply connected to the structure of knots. Investigating how the choice of pseudoAnosov braid and the properties of its stable/unstable laminations influence these invariants could pave the way for constructing knots with desired geometric properties.
Exploiting Braid Group Representations: Different representations of braid groups, such as those in linear groups or mapping class groups, can shed light on knot invariants. Analyzing these representations for the chosen pseudoAnosov braids might reveal hidden structures in the resulting knots, enabling the construction of knots with specific topological properties.
However, challenges exist:
Complexity of Relationships: The relationship between braid properties and knot invariants is intricate. Finding braids that precisely yield desired properties might require a deeper understanding of these relationships and potentially new theoretical tools.
Computational Challenges: As the complexity of braids increases, computing invariants and verifying properties can become computationally demanding. Efficient algorithms and computational tools would be crucial for exploring a wider range of possibilities.
Could there be alternative measures of complexity, other than Hempel distance, that could be used to construct infinite families of hyperbolic knots?
Yes, alternative complexity measures beyond Hempel distance could potentially be used to construct infinite families of hyperbolic knots. Here are a few promising avenues:
Distance in the Curve Complex: While Hempel distance focuses on disks, one could consider distances between other types of essential subsurfaces in the complement of the knot. For example, the distance between annuli in the curve complex can provide information about the knot's tunnel number and could be used to distinguish hyperbolic knots.
Geometric Complexity Measures: Hyperbolic knots admit a hyperbolic metric, and geometric invariants like injectivity radius, volume entropy, and Cheeger constant capture different aspects of this geometry. Exploring how these invariants behave under specific braid operations could lead to new constructions of hyperbolic knots.
Complexity of Heegaard Splittings: Beyond Hempel distance, other complexity measures for Heegaard splittings, such as distance in the handlebody complex or Heegaard genus, could be relevant. Investigating how these measures relate to knot properties might offer new ways to construct hyperbolic knots.
Combinatorial Complexity of Knot Diagrams: Knot invariants derived from knot diagrams, such as crossing number, braid index, and knot polynomials, could also be used. Finding ways to systematically construct diagrams with increasing complexity while preserving hyperbolicity could lead to new families of hyperbolic knots.
The key challenge lies in finding measures that:
Distinguish Hyperbolic Knots: The measure should effectively differentiate between hyperbolic and nonhyperbolic knots.
Exhibit Growth: The measure should grow systematically within the constructed families, ensuring the creation of infinitely many distinct knots.
Be Amenable to Computation: Calculating the measure should be feasible, allowing for practical construction and analysis of knot families.
How does the understanding of knots and their properties translate to other areas of mathematics or even physics, considering the intricate connections between seemingly disparate fields?
The study of knots, seemingly abstract mathematical objects, has surprisingly deep connections to diverse areas within mathematics and physics, demonstrating the interconnected nature of scientific disciplines. Here are some key examples:
Within Mathematics:
Topology and Geometry: Knot theory is fundamentally intertwined with 3manifold topology. Understanding knots provides insights into the structure and classification of 3manifolds. For instance, knot invariants like the Jones polynomial have revolutionized our understanding of 3manifolds.
Algebra and Representation Theory: Knot invariants often arise from algebraic structures like quantum groups and braid groups. Representations of these groups provide powerful tools for studying knots and their properties.
Dynamical Systems: The study of pseudoAnosov maps, as seen in the paper, connects knot theory to dynamical systems on surfaces. Properties of these maps, such as entropy and stable/unstable foliations, have implications for knot invariants.
In Physics:
Statistical Mechanics: Knot invariants, particularly polynomial invariants, have found applications in statistical mechanics models, such as those describing polymers and DNA. The knotting and linking of these molecules influence their physical properties.
Quantum Field Theory: Knot theory plays a role in topological quantum field theories, where knots and links represent physical processes. Invariants like the Jones polynomial have interpretations in terms of quantum observables.
Condensed Matter Physics: Topological phases of matter, such as topological insulators and superconductors, exhibit exotic properties related to knotting and linking. Knot theory provides a mathematical framework for understanding these phases.
The intricate connections between knot theory and other fields highlight the power of abstract mathematical concepts to provide insights into realworld phenomena. As our understanding of knots deepens, we can expect further crossfertilization of ideas and applications in diverse areas of science and engineering.