Invariants from Braid Theory

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The braid length of a knot or a link K is the smallest number of crossings in a braid whose closure is 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 K is simply the length of the minimum braid representing K (see Braid Representatives):

In[3]:= K = Knot[9, 49]; {BraidLength[K], Crossings[BR[K]]}
Out[3]= {11, 11}
9 49.gif
9_49
10 136.gif
10_136

The braid index of a knot or a link K is the smallest number of strands in a braid whose closure is 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]]]
Invariants from Braid Theory Out 7.gif
Out[7]= -Graphics-