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[1]:= ?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[2]:=	K = Knot[9, 49]; {BraidLength[K], Crossings[BR[K]]}
Out[2]=	{11, 11}

9_49

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[3]:= ?BraidIndex BraidIndex[K] returns the braid index of the knot K, if known to KnotTheory`.

In[4]:= 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[5]:=	K = Knot[10, 136]; {BraidIndex[K], First@BR[K]}
Out[5]=	{4, 5}

In[7]:=	Show[BraidPlot[BR[K]]]
Out[7]=	-Graphics-