It is easy to establish an upper bound $n$ for the bridge index of a knot by producing a diagram with the knot in $n$-bridge position.

Is there a known method to produce a reasonable lower bound on the bridge index?

For example, knot 11a1 has at most bridge index $4$, because a $4$-bridge position of this knot can be drawn: DT code (42, 66, 44, 56, 54, 46, 64, 40, -38, 86, 94, 76, 100, 80, 82, 98, 74, 96, 84, -110, -108, -68, -104, -22, -34, -32, -24, -102, -26, -30, -36, -20, -106, 90, 122, 60, 50, -116, -10, -6, -120, -70, -112, -14, -2, -4, -12, -114, -72, -118, -8, 78, 92, 88, 16, 18, 62, 48, 52, 58, -28) gives one such diagram. What is known about methods to eliminate the possibility that this knot has a $2$-bridge or $3$-bridge projection?

### Update

After posting this question I found a $3$-bridge projection (and because the knot is not rational, this is the lowest possible projection), given by DT code (12, 16, 58, 60, 14, -92, -90, -94, -32, -40, 120, 102, 108, 112, 98,116, 124, 106, 104, 122, 118, 100, 110, -26, -34, -38, -22, -42,-30, -96, -28, -44, -24, -36, 50, 18, 56, 62, 46, 64, 54, 20, 52, 66, 48, 128, 126, 114, -6, -74, -80, -86, -68, -88, -78, -76, -8, -4, -72, -82, -84, -70, -2, -10), but I am still interested in what is already known about establishing lower bounds more generally. Ryan Budney's suggestion of the Heegaard genus is a good one, but I haven't found a reference that shows this bound to be sharp.