© | Dror Bar-Natan: Knot Atlas: KnotTheory`:               This page is passe. Go here instead!

7.1 Invariants from Braid Theory

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:

In[1]

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 Section 4.4):

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

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]]]
Out[7]=	-Graphics-