L5a1: Difference between revisions

Revision as of 18:28, 2 September 2005

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L5a1 at Knotilus!
	[edit L5a1 Quick Notes] L5a1 is $5_{1}^{2}$ in Rolfsen's Table of Links. It is also known as the "Whitehead Link".

[edit L5a1 Further Notes and Views]

Basic depiction		Drawing of "Thor's hammer" or Mjölnir found in Sweden		Wolfgang Staubach's medallion based on this [1]
A kolam with two cycles, one of which is twisted[2]		A simplest closed-loop version of heraldic "fret" / "fretty" ornamentation.		Bisexuality symbol.

Link Presentations

[edit Notes on L5a1's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_10,7,5,8 X₄₅₁₆ X_2,10,3,9 X₈₄₉₃
Gauss code	{1, -4, 5, -3}, {3, -1, 2, -5, 4, -2}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)}{{\sqrt {u}}{\sqrt {v}}}}$ (db)
Jones polynomial	${\frac {1}{q^{7/2}}}-{\frac {2}{q^{5/2}}}-q^{3/2}+{\frac {1}{q^{3/2}}}+{\sqrt {q}}-{\frac {2}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{3}+z^{3}a+2za+az^{-1}-za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{2}a^{4}-2z^{3}a^{3}+2za^{3}-z^{4}a^{2}-3z^{3}a+4za-az^{-1}-z^{4}+z^{2}+1-z^{3}a^{-1}+2za^{-1}-a^{-1}z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

\		r
	\
j		\

-3

-2

-1

0

1

2

χ

4

1

2

0

2

1

-2

1

2

1

-4

1

-6

1

-8

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }^{2}$	${\mathbb {Z} }^{2}$
$r=1$	${\mathbb {Z} }$
$r=2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

Back to the top.

L4a1

L6a1

@@ Line 16: / Line 16: @@
 k = 1 |
 KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-4,5,-3:3,-1,2,-5,4,-2/goTop.html |
+braid_table     = <table cellspacing=0 cellpadding=0 border=0>
+<tr><td>[[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]]</td></tr>
+<tr><td>[[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]]</td></tr>
+<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]]</td></tr>
+</table> |
 khovanov_table  = <table border=1>
 <tr align=center>
@@ Line 38: / Line 43: @@
          <td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
          </tr>
-         <tr valign=top><td colspan=2>Loading KnotTheory` (version of August 28, 2005, 22:58:49)...</td></tr>
+         <tr valign=top><td colspan=2>Loading KnotTheory` (version of September 2, 2005, 15:8:39)...</td></tr>
          <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[2]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Crossings[Link[5, Alternating, 1]]</nowiki></pre></td></tr>
 <tr valign=top><td><pre   style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>5</nowiki></pre></td></tr>
@@ Line 49: / Line 54: @@
          <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[5]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>GaussCode[Link[5, Alternating, 1]]</nowiki></pre></td></tr>
 <tr valign=top><td><pre   style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[{1, -4, 5, -3}, {3, -1, 2, -5, 4, -2}]</nowiki></pre></td></tr>
-         <tr  valign=top><td><pre style="color: blue; border: 0px; padding:  0em"><nowiki>In[6]:=</nowiki></pre></td><td><pre style="color: red;  border: 0px; padding:  0em"><nowiki>Show[DrawMorseLink[Link[5, Alternating, 1]]]</nowiki></pre></td></tr><tr><td></td><td  align=left>[[Image:L5a1_ML.gif]]</td></tr><tr valign=top><td><tt><font  color=blue>Out[6]=</font></tt><td><tt><font  color=black>-Graphics-</font></tt></td></tr>
+         <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[6]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>BR[Link[5, Alternating, 1]]</nowiki></pre></td></tr>
-         <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[7]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>KnotSignature[Link[5, Alternating, 1]]</nowiki></pre></td></tr>
+<tr valign=top><td><pre   style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[3, {-1, 2, -1, 2, -1}]</nowiki></pre></td></tr>
-<tr valign=top><td><pre   style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>-1</nowiki></pre></td></tr>
+         <tr  valign=top><td><pre style="color: blue; border: 0px; padding:  0em"><nowiki>In[7]:=</nowiki></pre></td><td><pre style="color: red;  border: 0px; padding:  0em"><nowiki>Show[DrawMorseLink[Link[5, Alternating, 1]]]</nowiki></pre></td></tr><tr><td></td><td  align=left>[[Image:L5a1_ML.gif]]</td></tr><tr valign=top><td><tt><font  color=blue>Out[7]=</font></tt><td><tt><font  color=black>-Graphics-</font></tt></td></tr>
-         <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[8]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>J=Jones[Link[5, Alternating, 1]][q]</nowiki></pre></td></tr>
+         <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[8]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>KnotSignature[Link[5, Alternating, 1]]</nowiki></pre></td></tr>
-<tr valign=top><td><pre   style="color: blue; border: 0px; padding: 0em"><nowiki>Out[8]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -(7/2)    2      -(3/2)      2                 3/2
+<tr valign=top><td><pre   style="color: blue; border: 0px; padding: 0em"><nowiki>Out[8]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>-1</nowiki></pre></td></tr>
+         <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[9]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>J=Jones[Link[5, Alternating, 1]][q]</nowiki></pre></td></tr>
+<tr valign=top><td><pre   style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -(7/2)    2      -(3/2)      2                 3/2
 q       - ---- + q       - ------- + Sqrt[q] - q
 /2             Sqrt[q]
           q</nowiki></pre></td></tr>
-         <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[9]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[Link[5, Alternating, 1]][q]</nowiki></pre></td></tr>
+         <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[10]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[Link[5, Alternating, 1]][q]</nowiki></pre></td></tr>
-<tr valign=top><td><pre   style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>     -12    -8    -6   2     -2    2    4    6
+<tr valign=top><td><pre   style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>     -12    -8    -6   2     -2    2    4    6
 - q    + q   + q   + -- + q   + q  + q  + q
                        q</nowiki></pre></td></tr>
-         <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[10]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>HOMFLYPT[Link[5, Alternating, 1]][a, z]</nowiki></pre></td></tr>
+         <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[11]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>HOMFLYPT[Link[5, Alternating, 1]][a, z]</nowiki></pre></td></tr>
-<tr valign=top><td><pre   style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>   1     a   z            3        3
+<tr valign=top><td><pre   style="color: blue; border: 0px; padding: 0em"><nowiki>Out[11]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>   1     a   z            3        3
 -(---) + - - - + 2 a z - a  z + a z
   a z    z   a</nowiki></pre></td></tr>
-         <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[11]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kauffman[Link[5, Alternating, 1]][a, z]</nowiki></pre></td></tr>
+         <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[12]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kauffman[Link[5, Alternating, 1]][a, z]</nowiki></pre></td></tr>
-<tr valign=top><td><pre   style="color: blue; border: 0px; padding: 0em"><nowiki>Out[11]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>                                                   3
+<tr valign=top><td><pre   style="color: blue; border: 0px; padding: 0em"><nowiki>Out[12]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>                                                   3
 a   2 z              3      2    4  2   z         3
 - --- - - + --- + 4 a z + 2 a  z + z  - a  z  - -- - 3 a z  -
@@ Line 74: / Line 81: @@
 3    4    2  4
 a  z  - z  - a  z</nowiki></pre></td></tr>
-         <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[12]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kh[Link[5, Alternating, 1]][q, t]</nowiki></pre></td></tr>
+         <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[13]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kh[Link[5, Alternating, 1]][q, t]</nowiki></pre></td></tr>
-<tr valign=top><td><pre   style="color: blue; border: 0px; padding: 0em"><nowiki>Out[12]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>    2      1       1       1      1          4  2
+<tr valign=top><td><pre   style="color: blue; border: 0px; padding: 0em"><nowiki>Out[13]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>    2      1       1       1      1          4  2
 + -- + ----- + ----- + ----- + ---- + t + q  t
 8  3    6  2    4  2    2

L5a1: Difference between revisions

Revision as of 18:28, 2 September 2005

Contents

Link Presentations

Polynomial invariants

Khovanov Homology

Computer Talk

Modifying This Page

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