The Franciscan Tau cross. The Tau cross itself ( T ) predates the Crucifixion cross. In the book of Ezekiel, God instructs an angel to mark the heads of the faithful with a Tau (Ezekiel 9:4) . St. Anthony the Abbot, the father of Western monasticism, is said to have carried a Tau cross, but it is mostly associated with the Franciscans. It is believed St. Francis embraced the Tau after Pope Innocent spoke about the it in referencing Ezekiel 9:4 at the Fourth Lateran Council. There was a religious community (Antonians) active in Assisi at St. Francis's time that followed in the footsteps of St. Anthony, so St. Francis could have first seen the Tau in encountering them.
Here I have a modified Franciscan symbol, which includes the Tau cross under the crossed arms of Our Lord and St. Francis.