Braid Representatives: Difference between revisions
No edit summary |
DrorsRobot (talk | contribs) No edit summary |
||
Line 132: | Line 132: | ||
<!--$$Show[BraidPlot[CollapseBraid[BR[pd]]]]$$--> |
<!--$$Show[BraidPlot[CollapseBraid[BR[pd]]]]$$--> |
||
<!--Robot Land, no human edits to "END"--> |
<!--Robot Land, no human edits to "END"--> |
||
{{ |
{{Graphics| |
||
n = |
n = 16 | |
||
in = <nowiki>Show[BraidPlot[CollapseBraid[BR[pd]]]</nowiki> | |
in = <nowiki>Show[BraidPlot[CollapseBraid[BR[pd]]]]</nowiki> | |
||
img= Braid_Representatives_Out_15.gif | |
|||
out= <nowiki> |
out= <nowiki>-Graphics-</nowiki>}} |
||
<!--END--> |
<!--END--> |
||
Revision as of 15:04, 2 September 2005
Every knot and every link is the closure of a braid. KnotTheory`
can also represent knots and links as braid closures:
(For In[1] see Setup)
|
|
|
Thus for example,
In[4]:=
|
br1 = BR[2, {-1, -1, -1}];
|
In[5]:=
|
PD[br1]
|
Out[5]=
|
PD[X[6, 3, 1, 4], X[4, 1, 5, 2], X[2, 5, 3, 6]]
|
In[6]:=
|
Jones[br1][q]
|
Out[6]=
|
-4 -3 1
-q + q + -
q
|
In[7]:=
|
Mirror[br1]
|
Out[7]=
|
BR[2, {1, 1, 1}]
|
K11a362 |
KnotTheory`
has the braid representatives of some knots and links pre-loaded, and for all other knots and links it will find a braid representative using Vogel's algorithm. Thus for example,
In[8]:=
|
BR[TorusKnot[5, 4]]
|
Out[8]=
|
BR[4, {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3}]
|
In[9]:=
|
BR[Knot[11, Alternating, 362]]
|
Out[9]=
|
BR[10, {1, 2, -3, -4, 5, 6, 5, 4, 3, -2, -1, -4, 3, -2, -4, 3, 5, 4,
-6, 7, -6, 5, 8, 7, 6, 5, -4, -3, 2, 5, -6, 9, -8, 7, -6, 5, 4, -3,
5, 6, 5, 4, 5, -7, 8, -7, 6, 5, -9, -8, -7}]
|
(As we see, Vogel's algorithm sometimes produces scary results. A $Failed-crossings braid representative for an 11-crossing knot, in this case).
The minimum braid representative of a given knot is a braid representative for that knot which has a minimal number of braid crossings and within those braid representatives with a minimal number of braid crossings, it has a minimal number of strands (full details are in [Gittings]). Thomas Gittings kindly provided us the minimum braid representatives for all knots with up to 10 crossings. Thus for example, the minimum braid representative for the knot 10_1 has length (number of crossings) 13 and width 6 (number of strands, also see Invariants from Braid Theory):
In[10]:=
|
br2 = BR[Knot[10, 1]]
|
Out[10]=
|
BR[6, {-1, -1, -2, 1, -2, -3, 2, -3, -4, 3, 5, -4, 5}]
|
In[12]:=
|
Show[BraidPlot[CollapseBraid[br2]]]
|
Out[12]=
|
-Graphics-
|
5_2 |
Already for the knot 5_2 the minimum braid is shorter than the braid produced by Vogel's algorithm. Indeed, the minimum braid is
In[13]:=
|
Show[BraidPlot[CollapseBraid[BR[Knot[5, 2]]]]
|
Out[13]=
|
$Failed
|
To force KnotTheory`
to run Vogel's algorithm on 5_2, we first convert it to its PD
form,
In[14]:=
|
pd = PD[Knot[5, 2]]
|
Out[14]=
|
PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 10, 6, 1], X[9, 6, 10, 7],
X[7, 2, 8, 3]]
|
and only then run BR
:
In[16]:=
|
Show[BraidPlot[CollapseBraid[BR[pd]]]]
|
Out[16]=
|
-Graphics-
|
(Check Drawing Braids for information about the command BraidPlot
and the related command CollapseBraid
.)
[Gittings] ^ T. A. Gittings, Minimum braids: a complete invariant of knots and links, arXiv:math.GT/0401051.