Infinitely Many Knots With NonIntegral Trace

Abstract

We prove that there are infinitely many non-homeomorphic hyperbolic knot complements $S^3\setminus K_i={\mathbb{H}}^3/{\Gamma }_i$ for which Γ_i contains elements whose trace is an algebraic non-integer.

Keywords:

• Hyperbolic knot
• nonintegral trace
• closed embedded essential surface

Mathematics Subject Classification:

• 57M25

Acknowledgments

We are very grateful to Shelly Harvey for pointing out the reference [21] to us. We are also very grateful to Ken Baker, Neil Hoffman and Josh Howie for comments on an earlier version of this paper that led to the revised Section 7. We thank to Howie for allowing us to include his proof that the knot complements constructed in the proof of Theorem 1.1 contain a closed embedded essential surface that carries an essential simple closed curve isotopic to a meridian, as well as for the tangle decompositions shown in Figure 4. Finally we thank the referees for several useful comments and suggestions.

Declaration of Interest

No potential conflict of interest was reported by the author(s).

Additional information

Funding

First author supported by NSF grant DMS 1812397. The second author was partially supported by NSF DMS-1745670.