On Legendrian representatives of non-fibered knots

We show that in $(S^3,{\xi }_{std})$ if $K$ is a non-trivial knot that realizes the three-dimensional Thurston-Bennequin bound (i.e. $K$ has a Legendrian representative $\mathrm{\Lambda }$ with $tb(\mathrm{\Lambda })-rot(\mathrm{\Lambda })=2g(K)-1$), then $K$ has a Legendrian representative $L$ with $tb=0$. Moreover, this result can be easily generalized to contact manifolds that uniquely represent the associated contact invariants. This is the first result on Legendrian representatives of non-fibered knots in $3-$manifolds other than $S^3$. We also show that if $K$ is a nearly fibered knot in $S^3$ then $\tau (K)=g(K)$ implies that $K$ realizes the three-dimensional Thurston-Bennequin bound.