  We consider self-avoiding polymers attached to the tip of an impenetrable
probe. The scaling exponents $\gamma_1$ and $\gamma_2$, characterizing the
number of configurations for the attachment of the polymer by one end, or at
its midpoint, vary continuously with the tip's angle. These apex exponents are
calculated analytically by $\epsilon$-expansion, and numerically by simulations
in three dimensions. We find that when the polymer can move through the
attachment point, it typically slides to one end; the apex exponents quantify
the entropic barrier to threading the eye of the probe.
