Legendrian Non-simple Whitehead Doubles Of The Trefoil

Ozsváth and Stipsicz showed that some Eliashberg-Chekanov twist knots, which are Whitehead doubles of the unknot, are not Legendrian simple. We extend their result by considering some Whitehead doubles of the trefoil: Using properties of knot Floer homology and the distinguished surgery triangle, we show that this family of knots is Legendrian non-simple in the standard contact 3-sphere.

💡 Research Summary

The paper investigates the Legendrian simplicity of a family of knots obtained by taking positively clasped, (–n)-twisted Whitehead doubles of the right‑handed trefoil, denoted (Wh^{+}(S,-n)). Building on the earlier work of Ozsváth and Stipsicz, who showed that certain Eliashberg–Chekanov twist knots (Whitehead doubles of the unknot) are Legendrian non‑simple, the author extends the phenomenon to a more intricate class of knots.

The main theorem (Theorem 1.1) states that for every integer (n\ge 1), the knot (Wh^{+}(S,-n)) is Legendrian non‑simple: there exist at least (\big\lfloor\frac{n+3}{2}\big\rfloor) pairwise non‑Legendrian‑isotopic realizations that share the same classical invariants, namely Thurston–Bennequin number (tb=1) and rotation number (rot=0). The proof follows the strategy introduced in