Knots with infinitely many non-characterizing slopes

Abstract: Using the techniques on annulus twists, we observe that $6_3$ has infinitely many non-characterizing slopes, which affirmatively answers a question by Baker and Motegi. Furthermore, we prove that the knots $6_2$, $6_3$, $7_6$, $7_7$, $8_1$, $8_3$, $8_4$, $8_6$, $8_7$, $8_9$, $8_{10}$, $8_{11}$, $8_{12}$, $8_{13}$, $8_{14}$, $8_{17}$,$8_{20}$ and $8_{21}$ have infinitely many non-characterizing slopes. We also introduce the notion of trivial annulus twists and give some possible applications. Finally, we completely determine which knots have special annulus presentations up to 8-crossings.

• The paper proves that several knots, including 6_3, have infinitely many non-characterizing slopes, answering an open question in knot theory.
• The authors introduce and utilize the concept of trivial annulus twists as a key technique for identifying knots with this property.
• This research enhances understanding of knot equivalences and improves the classification of 3-manifolds arising from Dehn surgery.

Knots with Infinitely Many Non-Characterizing Slopes

The paper under consideration presents significant insights into the field of knot theory and 3-manifolds, specifically exploring the concept of non-characterizing slopes for knots. The authors, Tetsuya Abe and Keiji Tagami, investigate the conditions under which certain knots possess infinitely many non-characterizing slopes, addressing a question posed by Baker and Motegi. Their analysis leverages the concept of annulus twists, an operation that facilitates modifications on knots leading to the same 3-manifold post Dehn surgery.

Main Contributions

The paper establishes several key results, amongst which the most notable are:

1. Infinitely Many Non-Characterizing Slopes: The paper confirms that the knot $6_3$ has infinitely many non-characterizing slopes, definitively answering a question previously put forward in the literature. Furthermore, it extends this conclusion to a variety of other knots including $6_2$, $7_6$, $7_7$, and many others up to eight crossings, demonstrating a broader applicability of the result.
2. Annulus Twists: The authors introduce the notion of trivial annulus twists and discuss their potential applications in the broader context of knot theory. This technique is pivotal in identifying knots that have non-unique characterizing slopes.
3. Tabulation of Special Annulus Presentations: The paper provides a comprehensive analysis of special annulus presentations for knots up to eight crossings, which helps in identifying whether these knots possess the non-characterizing slope property.

Theoretical and Practical Implications

From a theoretical standpoint, the insights provided into characterizing slopes enrich our understanding of knot equivalences and the uniqueness of 3-manifold descriptions obtained via knot theory. This has implications for the classification of knots and the study of 3-manifolds that arise through Dehn surgery, a classical process in geometric topology.

Practically, the methods and conditions defined for annulus presentations could assist researchers in constructing examples of knots with desirable properties, including those that demonstrate specific isotopy distinctions in 3-manifolds. By utilizing trivial annulus twist operations, one could generate a family of examples that challenge current classifications, thereby propelling the study of isotopically non-equivalent knots in 3-manifold spaces.

Future Speculations

The investigation opens several avenues for future research. Exploring the applicability of the developed techniques on knots beyond eight crossings poses a direct challenge. Additionally, the interaction of these knot properties with other topological invariants could yield novel insights. It’s plausible that deeper analyses could uncover new families of knots exhibiting similar or contradictory behaviors in more complex manifold spaces.

Furthermore, the techniques discussed could be intertwined with computational methodologies to algorithmically identify non-characterizing slopes across a broader class of knots. This could significantly impact the computational topology landscape, aiding in automatic classifications and property identifications.

In summary, Abe and Tagami's work makes substantial contributions to knot theory, providing both new theoretical insights and practical techniques for studying the properties of knots and 3-manifolds. Their work not only answers significant open questions but also lays a rich foundation for future explorations in mathematical topology.