Strongly Invertible Legendrian Knots
This presentation explores strongly invertible Legendrian knots, a class of mathematical objects combining topological symmetry with contact geometry. We examine how an involution constraint creates new rigidity, define the equivariant Thurston–Bennequin number, and investigate when symmetry preserves or obstructs classical maximal invariants through explicit families and open conjectures.Script
A Legendrian knot lives in three-dimensional contact space, but some of these knots possess a hidden symmetry. A strongly invertible Legendrian knot is one that remains unchanged under a specific reflection, fixing exactly two points on the knot and flipping the contact structure's orientation.
When we project such a knot onto the x-z plane, its symmetry becomes visible as a transvergent front diagram, perfectly mirrored across the horizontal axis with all cusps on that axis pointing vertically. This projection reveals the combinatorial structure we need to compute invariants.
The equivariant Thurston–Bennequin number counts crossings minus half the cusps, just like the classical invariant, but now measured on a symmetric diagram. The maximal equivariant number can never exceed the classical maximum, but the fascinating question is whether symmetry forces it to be strictly smaller.
For all torus knots of type T 2, 2n plus 1 and all twist knots with even parameter, the answer is no. Explicit symmetric constructions achieve the classical maximum Thurston–Bennequin number, proving that infinite families of knots lose nothing to the symmetry constraint.
Yet computational evidence suggests knot 9 sub 42 tells a different story. Every symmetric representative found has Thurston–Bennequin number at most minus 5, while the classical maximum is minus 3. If confirmed, this would prove that symmetry can impose genuine contact-geometric cost, not just topological constraint.
Strongly invertible Legendrian knots reveal that symmetry and contact geometry interact in subtle, still-mysterious ways. To explore more cutting-edge mathematics and create videos on topics you care about, visit EmergentMind.com.
