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Description

Knots and links are very commonly described via \emph{knot diagrams}.  The \emph{crossing number} $c(K)$ of a knot $K$ is the smallest crossing number among all diagrams $D$ representing $K$. A more modern -- by thousands of years -- description of knots comes from three-dimensional triangulations.  Suppose that $T$ is a triangulation of the three-sphere.  We say that $T$ \emph{supports} a knot $K$ if we can isotope $K$ to lie in the one-skeleton of T.  The \emph{tetrahedron number} $t(K)$ of a knot $K$ is the smallest tetrahedron number among all triangulations $T$ supporting $K$. It is an exercise to prove that there is a constant $A \geq 1$ so that $t(K) \leq A \cdot c(K)$.  Our main theorem is that there is a constant $B > 1$ so that \[ c(K) \leq \exp(B \cdot t(K)) \] There are examples showing this cannot be improved (beyond reducing $B$).  It follows that, for some knots, representing $K$ by a triangulation is exponentially more efficient than representing $K$ by a diagram.  The previous best bound, of the form $c(K) \leq \exp(B \cdot t(K)^2)$, is due to King [2003].  Our techniques also show that the decision problem \textsc{Knot complement recognition} lies in NP. This is joint work with Robert Haraway, Neil Hoffman, and Eric Sedgwick.