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While the shape of the Ly$\alpha$ profile is viewed as one of the besttracers of ionizing-photon escape fraction ($f_{esc}$) within low redshift(z~0.3) surveys of the Lyman continuum, this connection remains untested athigh redshift. Here, we combine deep, rest-UV Keck/LRIS spectra of 80 objectsfrom the Keck Lyman Continuum Spectroscopic Survey with rest-opticalKeck/MOSFIRE spectroscopy in order to examine potential correlations betweenLy$\alpha$ profile shape and the escape of ionizing radiation within z~3star-forming galaxies. We measure the velocity separation between double-peakedLy$\alpha$ emission structure (v$_{\rm sep}$), between red-side Ly$\alpha$emission peaks and systemic (v$_{\rm Ly\alpha,red}$), and between red-sideemission peaks and low-ionization interstellar absorption lines (v$_{\rmLy\alpha-LIS}$). We find that the IGM-corrected ratio of ionizing tonon-ionizing flux density is significantly higher in KLCS objects with lowerv$_{\rm Ly\alpha,red}$. We find no significant trend between measures ofionizing-photon escape and v$_{\rm Ly\alpha-LIS}$. We compare our results tomeasurements of z~0.3 "Green Peas" from the literature and find that KLCSobjects have larger v$_{\rm sep}$ at fixed v$_{\rm Ly\alpha,red}$, larger$f_{esc}$ at fixed v$_{\rm Ly\alpha,red}$, and higher v$_{\rm Ly\alpha,red}$overall than z~0.3 analogs. We conclude that the Ly$\alpha$ profile shapes ofour high-redshift sources are fundamentally different, and that measurements ofprofile shape such as v$_{\rm Ly\alpha,red}$ map on to $f_{esc}$ in differentways. We caution against building reionization-era $f_{esc}$ diagnostics basedpurely on Ly$\alpha$ profiles of low-redshift dwarf galaxies. Tracing v$_{\rmsep}$, v$_{\rm Ly\alpha,red}$, and $f_{esc}$ in a larger sample of z~3 galaxieswill reveal how these variables may be connected for galaxies at the epoch ofreionization.

Ly{\alpha} profile shape as an escape-fraction diagnostic at high redshift