An Algorithm for Proving Braids are NonOrderPreserving
Core Concepts
This paper introduces an algorithm that can definitively determine if a braid is nonorderpreserving, offering a novel approach to understanding braid properties and their connection to biorderability in free groups and link complements.
Abstract

Bibliographic Information: Johnson, J., Scherich, N., & Turner, H. (2024). SEARCHING FOR NONORDERPRESERVING BRAIDS ALGORITHMICALLY. arXiv preprint arXiv:2410.10595.

Research Objective: The paper aims to develop an algorithm that can determine if a given braid is nonorderpreserving, addressing the challenge of classifying braids based on their orderpreserving properties.

Methodology: The researchers developed an algorithm inspired by Calegari and Dunfield's work on leftorderability obstructions. The algorithm leverages the concept of kprecones, which are finite representations of positive cones in free groups. By iteratively searching for kprecones preserved by a given braid, the algorithm can determine if the braid is nonorderpreserving. The researchers implemented the algorithm in Python and made it available on Github.

Key Findings: The algorithm successfully proved that the braid σ1σ−3
2 is not orderpreserving, a new finding in the field. Furthermore, guided by the algorithm's output, the researchers were able to generalize this result, proving that the infinite family of simple 3braids σ1σ2m+1
2 are not orderpreserving for any integer m.

Main Conclusions: The paper presents a novel algorithm for determining the nonorderpreserving property of braids. This algorithm provides a practical tool for studying braid groups and their connection to biorderability in free groups and link complements. The authors demonstrate the algorithm's effectiveness by proving new results about specific braid families.

Significance: This research contributes significantly to the study of braid groups and their orderpreserving properties. The algorithm provides a new tool for researchers to explore open questions in the field, such as the classification of biorderable links.

Limitations and Future Research: While the algorithm effectively identifies nonorderpreserving braids, it may not terminate for orderpreserving braids. Further research could explore ways to improve the algorithm's efficiency and potentially develop a method for identifying orderpreserving braids. Additionally, investigating the relationship between the braid word length and the minimal k required for the algorithm to find a contradiction could provide valuable insights.
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Searching for nonorderpreserving braids algorithmically
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The braid σ1σ−3
2 was found to be nonorderpreserving using the algorithm with k=4.
Quotes
"A group G is called biorderable if there is a strict total ordering on G that is invariant under both left and right multiplication."
"An nstrand braid β is called orderpreserving if there exists a positive cone P of Fn preserved by β. That is β(P) = P, setwise."
"An nstrand braid β is orderpreserving if and only if β preserves a kprecone of the free group Fn for every positive integer k."
Deeper Inquiries
How might this algorithm be adapted to explore other algebraic properties beyond orderpreservingness in braid groups?
This algorithm, at its core, leverages the interplay between the structure of the braid group $B_n$ and its action on the free group $F_n$. By cleverly reducing an infinite problem (finding a biorder preserved by a braid) to a finite, albeit computationally intensive, one (finding preserved kprecones), it provides a concrete method for obstructing orderpreservingness. This framework can potentially be extended to investigate other algebraic properties of braids by:
Identifying Suitable Group Actions: The key lies in finding other group actions of $B_n$ on groups where the desired algebraic property translates into a recognizable feature of the action. For instance, instead of biorders and positive cones, one could consider:
Leftorderability: Investigate actions that preserve leftorders, potentially leading to an algorithm for detecting braids that do not preserve any leftorder on the group they act upon.
Subgroup Distortions: Analyze actions on groups with geometric structure (e.g., hyperbolic groups) and study how the braid action distorts lengths of elements in certain subgroups. This could provide insights into the dynamical properties of the braid.
Defining Analogous Finite Structures: The concept of kprecones provides a finite, truncated representation of a biorder. Similarly, one would need to define analogous finite structures that capture the essence of the desired algebraic property in the context of the chosen group action.
Adapting the Saturation and Search: The saturation operation Sβ(P) ensures closure under relevant operations (products, conjugations, braid action). This operation would need to be modified to reflect the closure properties associated with the new algebraic property and the chosen finite structures. The search algorithm would then need to be adapted to efficiently explore the space of these finite structures.
However, challenges remain:
Finding suitable group actions and finite structures that effectively capture the desired algebraic property might be nontrivial.
Computational complexity is a significant hurdle. Even for orderpreservingness, the algorithm's runtime grows exponentially with k. Adapting it to other properties might further exacerbate this issue.
Could there be a fundamentally different approach to classifying orderpreserving braids that doesn't rely on searching for kprecones, potentially leading to a more efficient algorithm?
Yes, a fundamentally different approach is certainly possible. The current algorithm's reliance on searching through kprecones, while effective for obstruction, inherently involves a computationally expensive combinatorial explosion as k increases. Alternative avenues could explore:
Geometric Techniques:
Braids as Mapping Class Groups: Leverage the connection between braids and mapping class groups of punctured disks. Properties of the braid's action on the curve complex or other geometric objects associated with the punctured disk might provide insights into orderpreservingness.
Train Track Representatives: Utilize train track representatives of braids to analyze their dynamics. Certain dynamical properties of the train track map might be indicative of orderpreservingness.
Representation Theory:
Faithful Representations: Find faithful representations of braid groups into groups where orderpreservingness has a more tractable characterization. Analyzing the image of the braid under such representations might provide an efficient way to determine orderpreservingness.
Connections to Other Fields:
Dynamical Systems: Explore connections between orderpreserving braids and dynamical systems. For instance, the braid's action on the real line (via its Burau representation) might reveal dynamical obstructions to orderpreservingness.
Combinatorics on Words: Investigate combinatorial properties of braid words that are directly related to orderpreservingness. This could lead to efficient algorithms based on word manipulations.
Finding such a fundamentally different approach is a challenging open problem. However, the potential rewards are significant. A more efficient algorithm would not only provide a deeper understanding of orderpreserving braids but also open doors to exploring the orderability properties of more complex braids and their associated topological objects.
What are the implications of this research for understanding the topological properties of knots and links, particularly in the context of biorderability?
This research significantly contributes to our understanding of the relationship between braids, knots and links, and biorderability. The key bridge is the concept of a braided link, formed by the closure of a braid and its axis.
Here's a breakdown of the implications:
Obstructing Biorderability of Braided Links: The algorithm provides a concrete method to prove that certain braided links are not biorderable. This is significant because:
Biorderability is a powerful property: It imposes strong restrictions on the topology and geometry of a link complement. For example, a biorderable knot cannot be slice (bounding a disk in the 4ball) and its complement admits a taut foliation.
Many links are braided: Braided links form a large and important class of links, so understanding their biorderability has broader implications.
New Examples and Infinite Families: The algorithm led to the discovery of new examples of nonorderpreserving braids, such as σ1σ−3
2, and, more importantly, the infinite family σ1σ2m+1
This is crucial for:
Testing Conjectures: Having a larger and more diverse set of examples allows mathematicians to test conjectures about orderpreserving braids and biorderable links.
Refining Classifications: The identification of infinite families with specific orderability properties contributes to the ongoing effort to classify knots and links based on the orderability of their groups.
Future Directions: This research opens up several exciting avenues for future exploration:
Sharper Obstructions: Can the algorithm's efficiency be improved to tackle more complex braids and links? Can we find bounds on the minimal k required to find a contradiction?
Relationship with Other Properties: How does the orderpreserving property of a braid relate to other topological invariants of the corresponding link, such as its genus, fiberedness, or hyperbolic volume?
Generalizations: Can the techniques be extended to study orderability in other contexts, such as links in other 3manifolds or higherdimensional braids?
In summary, this research provides valuable tools and insights for studying biorderability in the context of knot theory. It highlights the fruitful interplay between algebra, topology, and computation, paving the way for a deeper understanding of the orderability properties of knots, links, and their associated groups.