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. 



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> Is there a known method to produce a reasonable lower bound on the bridge index? 
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For example, knot [11a1](http://katlas.org/wiki/Image:K11a1.gif) 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.



Is there a known method for finding the minimum bridge index of a knot?
I imagine bridge index is readily computable, at least for "small enough" knots. Bridge index is a lot like Heegaard genus of a 3-manifold. Heegaard splitting surfaces can be found via (almost) normal surface theory, provided you have a triangulation of the 3-manifold, and the triangulation isn't so big that the search for the surface doesn't take too much memory or time. 


Bridge index is analogous to Heegaard genus for closed manifolds. In that, if a knot has bridge index $k$ there is a $2k$-punctured sphere properly embedded in the complement, separating the complement into the union of two 3-manifolds that are handlebodies. Heegaard surfaces can generally be found via almost normal surface theory, so presumably the bridge index can be found in an analogous way. Off the top of my head I don't know the proper references for this but I believe the results for Heegaard splittings go back to Rubinstein (and perhaps other results using normal surface theory that go back to Waldhausen). If you don't get an answer here in the next few days I can ask someone who likely knows. 


