# Invariants from Braid Theory

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The braid length of a knot or a link ${\displaystyle K}$ is the smallest number of crossings in a braid whose closure is ${\displaystyle K}$. `KnotTheory`` has some braid lengths preloaded:

(For In[1] see Setup)

 In[2]:= ?BraidLength BraidLength[K] returns the braid length of the knot K, if known to KnotTheory`.

Note that the braid length of ${\displaystyle K}$ is simply the length of the minimum braid representing ${\displaystyle K}$ (see Braid Representatives):

 `In[3]:=` `K = Knot[9, 49]; {BraidLength[K], Crossings[BR[K]]}` `Out[3]=` `{11, 11}`

The braid index of a knot or a link ${\displaystyle K}$ is the smallest number of strands in a braid whose closure is ${\displaystyle K}$. `KnotTheory`` has some braid indices preloaded:

 In[4]:= ?BraidIndex BraidIndex[K] returns the braid index of the knot K, if known to KnotTheory`.
 In[5]:= BraidIndex::about The braid index data known to KnotTheory` is taken from Charles Livingston's "Table of Knot Invariants", http://www.indiana.edu/~knotinfo/.

Of the 250 knots with up to 10 crossings, only 10_136 has braid index smaller than the width of its minimum braid:

 `In[6]:=` `K = Knot[10, 136]; {BraidIndex[K], First@BR[K]}` `Out[6]=` `{4, 5}`
 `In[7]:=` `Show[BraidPlot[BR[K]]]` `Out[7]=` `-Graphics-`