Drawing Braids

KnotTheory`/Navigation  The Mathematica Package KnotTheory` Acknowledgement Setup Naming and Enumeration Presentations Planar Diagrams Some further details Gauss Codes DT (Dowker-Thistlethwaite) Codes Braid Representatives MorseLink Presentations Arc Presentations Conway Notation Graphical Input Graphical Output Drawing Planar Diagrams How does it work? Drawing MorseLink Presentations Drawing Braids Structure and Operations Invariants Invariants from Braid Theory Three Dimensional Invariants The Alexander-Conway Polynomial The Multivariable Alexander Polynomial The Determinant and the Signature The Jones Polynomial How is the Jones polynomial computed? The Coloured Jones Polynomials The A2 Invariant Quantum knot invariants The HOMFLY-PT Polynomial The Kauffman Polynomial Finite Type (Vassiliev) Invariants Khovanov Homology Heegaard Floer Knot Homology R-Matrix Invariants Extras Included with KnotTheory` Drawing with TubePlot Standalone_TubePlot Using the LinKnot package WikiLink QuantumGroups` Lightly Documented Features A Sample KnotTheory` Session Further Usage Example Prime Links with a Non-Prime Component "Rubberband" Brunnian Links Identifying Knots within a List Cabling Burau's Theorem Threading a link by a polynomial Further Knot Theory Software KnotPlot Knotscape Xknots Extending/Modifying KnotTheory` Bugs How to Edit this Manual...

(For In[1] see Setup)

In[2]:= ?BraidPlot BraidPlot[br, opts] produces a plot of the braid br. Possible options are Mode, HTMLOpts, WikiOpts and Images.

Thus for example,

In[3]:=

br = BR[5, {{1,3}, {-2,-4}, {1, 3}}];

In[4]:=	Show[BraidPlot[br]]
Out[4]=	-Graphics-

BraidPlot takes several options:

In[5]:=	Options[BraidPlot]
Out[5]=	{Mode -> Graphics, Images -> {0.gif, 1.gif, 2.gif, 3.gif, 4.gif}, HTMLOpts -> , WikiOpts -> }

The Mode option to BraidPlot defaults to "Graphics", which produces output as above. An alternative is setting Mode -> "HTML", which produces an HTML <table> that can be readily inserted into HTML documents:

In[6]:=	BraidPlot[br, Mode -> "HTML"]
Out[6]=	<table cellspacing=0 cellpadding=0 border=0> <tr><td><img src=1.gif><img src=0.gif><img src=1.gif></td></tr> <tr><td><img src=2.gif><img src=3.gif><img src=2.gif></td></tr> <tr><td><img src=1.gif><img src=4.gif><img src=1.gif></td></tr> <tr><td><img src=2.gif><img src=3.gif><img src=2.gif></td></tr> <tr><td><img src=0.gif><img src=4.gif><img src=0.gif></td></tr> </table>

The table produced contains an array of image inclusions that together draws the braid using 5 fundamental building blocks: a horizontal "unbraided" line (0.gif above), the upper and lower halves of an overcrossing (1.gif and 2.gif above) and the upper and lower halves of an underfcrossing (3.gif and 4.gif above).

Assuming 0.gif through 4.gif are , , , and , the above table is rendered as follows:

The meaning of the Images option to BraidPlot should be clear from reading its default definition:

In[7]:=	Images /. Options[BraidPlot]
Out[7]=	{0.gif, 1.gif, 2.gif, 3.gif, 4.gif}

The HTMLOpts option to BraidPlot allows to insert options within the HTML <img> tags. Thus

In[8]:=	BraidPlot[BR[2, {1, 1}], Mode -> "HTML", HTMLOpts -> "border=1"]
Out[8]=	<table cellspacing=0 cellpadding=0 border=0> <tr><td><img border=1 src=1.gif><img border=1 src=1.gif></td></tr> <tr><td><img border=1 src=2.gif><img border=1 src=2.gif></td></tr> </table>

The above table is rendered as follows:

In[9]:= ?CollapseBraid CollapseBraid[br] groups together commuting generators in the braid br. Useful in conjunction with BraidPlot to produce compact braid plots.

Thus compare the plots of br1 and br2 below:

In[10]:=	br1 = BR[TorusKnot[5, 4]]
Out[10]=	BR[4, {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3}]

In[11]:=	Show[BraidPlot[br1]]
Out[11]=	-Graphics-

In[12]:=	br2 = CollapseBraid[BR[TorusKnot[5, 4]]]
Out[12]=	BR[4, {{1}, {2}, {3, 1}, {2}, {3, 1}, {2}, {3, 1}, {2}, {3, 1}, {2}, {3}}]

In[13]:=	Show[BraidPlot[br2]]
Out[13]=	-Graphics-