Invariants from Braid Theory
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The braid length of a knot or a link is the smallest number of crossings in a braid whose closure is . KnotTheory`
has some braid lengths preloaded:
(For In[1] see Setup)

Note that the braid length of is simply the length of the minimum braid representing (see Braid Representatives):
In[3]:=

K = Knot[9, 49]; {BraidLength[K], Crossings[BR[K]]}

Out[3]=

{11, 11}

9_49 
10_136 
The braid index of a knot or a link is the smallest number of strands in a braid whose closure is . KnotTheory`
has some braid indices preloaded:


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
