L4a1: Difference between revisions

Latest revision as of 16:20, 27 May 2009

See the full Thistlethwaite Link Table (up to 11 crossings).

L4a1 is $4_{1}^{2}$ in the Rolfsen table of links. It frequently occurs in late Roman mosaics and some medieval decorations. In this context, it is called the "Solomon's knot" (sigillum Salomonis) or "guilloche knot". It is also the "Kramo-bone" symbol (meaning "one being bad makes all appear to be bad") of the Adinkra symbol system. Link L10a101 contains multiple L4a1 configurations.

[edit L4a1 Further Notes and Views]

Simple squared depiction	A Kolam with two cycles[1]	Hearst Castle tile [2]	Mosaic seen at Kibbutz Lahav [3]
Carving above door of church in Italy	Decorative depiction	(crossings along one side)	Linked hearts used as symbol of Vendée region of France
Heraldic ornament.	Composed of intersecting circles.	Decorative fitting closely within square.	Ancient Roman mosaic.
Made of two (impossible) Penrose rectangles.	Array of "Solomon's knots" forming overall circular patterns.	Configuration of three L4a1	Configuration of four L4a1
Medieval manuscript	Medieval manuscript	Medieval manuscript	Rotated knotwork cross with eight L4a1 sub-configurations
Knotopologynn-diagram for "Solomon's knots"	Commercial logo

Link Presentations

[edit Notes on L4a1's Link Presentations] Why such an ugly Braid Representative?

Planar diagram presentation	X₆₁₇₂ X₈₃₅₄ X₂₅₃₆ X₄₇₁₈
Gauss code	{1, -3, 2, -4}, {3, -1, 4, -2}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-u-v}{{\sqrt {u}}{\sqrt {v}}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{3/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{-1}-a^{3}z-a^{3}z^{-1}-az$ (db)
Kauffman polynomial	$a^{5}z^{3}-3a^{5}z+a^{5}z^{-1}+a^{4}z^{2}-a^{4}+a^{3}z^{3}-2a^{3}z+a^{3}z^{-1}+a^{2}z^{2}+az$ (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		\

-4

-3

-2

-1

0

χ

0

1

-2

1

0

-4

0

-6

1

-8

1

-10

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-4$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-3$
$r=-2$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$

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.

L2a1

L5a1

@@ Line 1: / Line 1: @@
+<!--                       WARNING! WARNING! WARNING!
-<!-- This page was  generated from the splice template "Link_Splice_Template". Please do not edit! -->
+<!-- This page was generated from the splice template [[Link_Splice_Base]]. Please do not edit!
-<!--  --> <!--
+<!-- You probably want to edit the template referred to immediately below. (See [[Category:Knot Page Template]].)
- -->
+<!-- This page itself was created by running [[Media:KnotPageSpliceRobot.nb]] on [[Link_Splice_Base]]. -->
+<!-- <math>\text{Null}</math> -->
+<!-- <math>\text{Null}</math> -->
+<!--                       WARNING! WARNING! WARNING!
+<!-- This page was generated from the splice template [[Link Splice Template]]. Please do not edit!
+<!-- Almost certainly, you want to edit [[Template:Link Page]], which actually produces this page.
+<!-- The text below simply calls [[Template:Link Page]] setting the values of all the parameters appropriately.
+<!-- This page itself was created by running [[Media:KnotPageSpliceRobot.nb]] on [[Link Splice Template]]. -->
+<!-- <math>\text{Null}</math> -->
 {{Link Page|
 n = 4 |
@@ Line 7: / Line 16: @@
 k = 1 |
 KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-3,2,-4:3,-1,4,-2/goTop.html |
+braid_table     = <table cellspacing=0 cellpadding=0 border=0 style="white-space: pre">
+<tr><td>[[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr>
+<tr><td>[[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart4.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]]</td></tr>
+<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart3.gif]][[Image:BraidPart4.gif]]</td></tr>
+<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart0.gif]]</td></tr>
+</table> |
 khovanov_table  = <table border=1>
 <tr align=center>
@@ Line 28: / 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 29, 2005, 15:33:11)...</td></tr>
+         <tr valign=top><td colspan=2>Loading KnotTheory` (version of September 3, 2005, 2:11:43)...</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[4, 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>4</nowiki></pre></td></tr>
@@ Line 38: / Line 53: @@
 <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, -3, 2, -4}, {3, -1, 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>BR[Link[4, 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[Link[4, 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[4, {1, -2, 3, -2, -1, -2, -3, -2}]</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[4, Alternating, 1]]]</nowiki></pre></td></tr><tr><td></td><td  align=left>[[Image:L4a1_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>alex = Alexander[Link[4, Alternating, 1]][t]</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[4, 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>ComplexInfinity</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>-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>Conway[Link[4, Alternating, 1]][z]</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[4, 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>ComplexInfinity</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>  -(9/2)    -(5/2)    -(3/2)      1
-         <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>Select[AllKnots[], (alex === Alexander[#][t])&]</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>{}</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>{KnotDet[Link[4, Alternating, 1]], KnotSignature[Link[4, Alternating, 1]]}</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>{Infinity, -1}</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>J=Jones[Link[4, Alternating, 1]][q]</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>  -(9/2)    -(5/2)    -(3/2)      1
 -q       - q       + q       - -------
                                Sqrt[q]</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>A2Invariant[Link[4, 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[4, Alternating, 1]][q]</nowiki></pre></td></tr>
-<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>     -16    2     2     2     -8
+<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>     -16    2     2     2     -8
 + q    + --- + --- + --- + q
 12    10
            q     q     q</nowiki></pre></td></tr>
-         <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[14]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>HOMFLYPT[Link[4, 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[4, Alternating, 1]][a, z]</nowiki></pre></td></tr>
-<tr valign=top><td><pre   style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>   3     5
+<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     5
   a     a           3
 -(--) + -- - a z - a  z
   z     z</nowiki></pre></td></tr>
-         <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[15]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kauffman[Link[4, 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[4, Alternating, 1]][a, z]</nowiki></pre></td></tr>
-<tr valign=top><td><pre   style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>      3    5
+<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    5
 a    a             3        5      2  2    4  2    3  3    5  3
 a  - -- - -- - a z + 2 a  z + 3 a  z - a  z  - a  z  - a  z  - a  z
      z    z</nowiki></pre></td></tr>
-         <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[16]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{Vassiliev[2][Link[4, Alternating, 1]], Vassiliev[3][Link[4, Alternating, 1]]}</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[4, Alternating, 1]][q, t]</nowiki></pre></td></tr>
-<tr valign=top><td><pre   style="color: blue; border: 0px; padding: 0em"><nowiki>Out[16]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>      65
+<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
-{0, -(--)}
-</nowiki></pre></td></tr>
-         <tr valign=top><td><pre style="color:    blue; border: 0px; padding: 0em"><nowiki>In[17]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kh[Link[4, Alternating, 1]][q, t]</nowiki></pre></td></tr>
-<tr valign=top><td><pre   style="color: blue; border: 0px; padding: 0em"><nowiki>Out[17]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>     -2     1        1       1      1
 + q   + ------ + ----- + ----- + ----
 4    8  4    6  2    2

L4a1: Difference between revisions

Latest revision as of 16:20, 27 May 2009

Contents

Link Presentations

Polynomial invariants

Khovanov Homology

Computer Talk

Modifying This Page

Navigation menu

Page actions

Page actions

Personal tools

Navigation

Search

Tools