4 1: Difference between revisions

From Knot Atlas
Jump to navigationJump to search
m (Reverted edit of 60.190.79.18, changed back to last version by ScottTestRobot)
No edit summary
Line 22: Line 22:
<tr align=center>
<tr align=center>
<td width=22.2222%><table cellpadding=0 cellspacing=0>
<td width=22.2222%><table cellpadding=0 cellspacing=0>
<tr><td>\</td><td>&nbsp;</td><td>r</td></tr>
<tr><td>\</td><td>
<tr><td>&nbsp;</td><td>&nbsp;\&nbsp;</td><td>&nbsp;</td></tr>
<tr><td>j</td><td>&nbsp;</td><td>\</td></tr>
</table></td>
<td width=11.1111%>-2</td ><td width=11.1111%>-1</td ><td width=11.1111%>0</td ><td width=11.1111%>1</td ><td width=11.1111%>2</td ><td width=22.2222%>&chi;</td></tr>
<tr align=center><td>5</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td>1</td></tr>
<tr align=center><td>3</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>&nbsp;</td><td>0</td></tr>
<tr align=center><td>1</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>0</td></tr>
<tr align=center><td>-1</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>0</td></tr>
<tr align=center><td>-3</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>0</td></tr>
<tr align=center><td>-5</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
</table> |
coloured_jones_2 = <math>q^6-q^5-q^4+2 q^3-q^2-q+3- q^{-1} - q^{-2} +2 q^{-3} - q^{-4} - q^{-5} + q^{-6} </math> |
coloured_jones_3 = <math>q^{12}-q^{11}-q^{10}+2 q^8-2 q^6+3 q^4-3 q^2+3-3 q^{-2} +3 q^{-4} -2 q^{-6} +2 q^{-8} - q^{-10} - q^{-11} + q^{-12} </math> |
coloured_jones_4 = <math>q^{20}-q^{19}-q^{18}+3 q^{15}-q^{14}-q^{13}-q^{12}-q^{11}+5 q^{10}-q^9-2 q^8-2 q^7-q^6+6 q^5-q^4-2 q^3-2 q^2-q+7- q^{-1} -2 q^{-2} -2 q^{-3} - q^{-4} +6 q^{-5} - q^{-6} -2 q^{-7} -2 q^{-8} - q^{-9} +5 q^{-10} - q^{-11} - q^{-12} - q^{-13} - q^{-14} +3 q^{-15} - q^{-18} - q^{-19} + q^{-20} </math> |
coloured_jones_5 = <math>q^{30}-q^{29}-q^{28}+q^{25}+2 q^{24}-2 q^{22}-q^{21}-q^{20}+q^{19}+3 q^{18}+q^{17}-2 q^{16}-3 q^{15}-2 q^{14}+2 q^{13}+4 q^{12}+2 q^{11}-2 q^{10}-4 q^9-2 q^8+2 q^7+5 q^6+2 q^5-2 q^4-5 q^3-2 q^2+2 q+5+2 q^{-1} -2 q^{-2} -5 q^{-3} -2 q^{-4} +2 q^{-5} +5 q^{-6} +2 q^{-7} -2 q^{-8} -4 q^{-9} -2 q^{-10} +2 q^{-11} +4 q^{-12} +2 q^{-13} -2 q^{-14} -3 q^{-15} -2 q^{-16} + q^{-17} +3 q^{-18} + q^{-19} - q^{-20} - q^{-21} -2 q^{-22} +2 q^{-24} + q^{-25} - q^{-28} - q^{-29} + q^{-30} </math> |
coloured_jones_6 = <math>q^{42}-q^{41}-q^{40}+q^{37}+3 q^{35}-q^{34}-2 q^{33}-q^{32}-q^{31}+6 q^{28}-q^{27}-2 q^{26}-2 q^{25}-2 q^{24}-q^{23}+9 q^{21}-2 q^{19}-3 q^{18}-3 q^{17}-2 q^{16}+11 q^{14}-2 q^{12}-4 q^{11}-4 q^{10}-2 q^9+12 q^7-2 q^5-4 q^4-4 q^3-2 q^2+13-2 q^{-2} -4 q^{-3} -4 q^{-4} -2 q^{-5} +12 q^{-7} -2 q^{-9} -4 q^{-10} -4 q^{-11} -2 q^{-12} +11 q^{-14} -2 q^{-16} -3 q^{-17} -3 q^{-18} -2 q^{-19} +9 q^{-21} - q^{-23} -2 q^{-24} -2 q^{-25} -2 q^{-26} - q^{-27} +6 q^{-28} - q^{-31} - q^{-32} -2 q^{-33} - q^{-34} +3 q^{-35} + q^{-37} - q^{-40} - q^{-41} + q^{-42} </math> |
coloured_jones_7 = <math>q^{56}-q^{55}-q^{54}+q^{51}+q^{49}+2 q^{48}-q^{47}-2 q^{46}-q^{45}-2 q^{44}+q^{43}+q^{41}+5 q^{40}-2 q^{38}-2 q^{37}-4 q^{36}+2 q^{33}+7 q^{32}+q^{31}-q^{30}-2 q^{29}-7 q^{28}-2 q^{27}+2 q^{25}+9 q^{24}+2 q^{23}-3 q^{21}-9 q^{20}-3 q^{19}+3 q^{17}+10 q^{16}+3 q^{15}-3 q^{13}-10 q^{12}-3 q^{11}+3 q^9+11 q^8+3 q^7-3 q^5-11 q^4-3 q^3+3 q+11+3 q^{-1} -3 q^{-3} -11 q^{-4} -3 q^{-5} +3 q^{-7} +11 q^{-8} +3 q^{-9} -3 q^{-11} -10 q^{-12} -3 q^{-13} +3 q^{-15} +10 q^{-16} +3 q^{-17} -3 q^{-19} -9 q^{-20} -3 q^{-21} +2 q^{-23} +9 q^{-24} +2 q^{-25} -2 q^{-27} -7 q^{-28} -2 q^{-29} - q^{-30} + q^{-31} +7 q^{-32} +2 q^{-33} -4 q^{-36} -2 q^{-37} -2 q^{-38} +5 q^{-40} + q^{-41} + q^{-43} -2 q^{-44} - q^{-45} -2 q^{-46} - q^{-47} +2 q^{-48} + q^{-49} + q^{-51} - q^{-54} - q^{-55} + q^{-56} </math> |
computer_talk =
<table>
<tr valign=top>
<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:=&nbsp;&nbsp;&nbsp;&nbsp;</pre></td>
<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
</tr>
<tr valign=top><td colspan=2><nowiki>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</nowiki></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[2]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[Knot[4, 1]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[2]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[X[4, 2, 5, 1], X[8, 6, 1, 5], X[6, 3, 7, 4], X[2, 7, 3, 8]]</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[3]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[Knot[4, 1]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[3]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[1, -4, 3, -1, 2, -3, 4, -2]</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[4]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[Knot[4, 1]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[4]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[4, 6, 8, 2]</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[5]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>br = BR[Knot[4, 1]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[5]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>BR[3, {-1, 2, -1, 2}]</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[6]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{First[br], Crossings[br]}</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[6]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{3, 4}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[7]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>BraidIndex[Knot[4, 1]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[7]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>3</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[8]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Show[DrawMorseLink[Knot[4, 1]]]</nowiki></code></td></tr>
<tr align=left><td></td><td>[[Image:4_1_ML.gif]]</td></tr><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[8]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>-Graphics-</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[9]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> (#[Knot[4, 1]]&) /@ {
SymmetryType, UnknottingNumber, ThreeGenus,
BridgeIndex, SuperBridgeIndex, NakanishiIndex
}</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[9]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{FullyAmphicheiral, 1, 1, 2, 3, 1}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[10]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>alex = Alexander[Knot[4, 1]][t]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[10]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 1
3 - - - t
t</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[11]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Conway[Knot[4, 1]][z]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[11]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2
1 - z</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[12]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[12]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Knot[4, 1]}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[13]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{KnotDet[Knot[4, 1]], KnotSignature[Knot[4, 1]]}</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[13]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{5, 0}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[14]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Jones[Knot[4, 1]][q]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[14]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -2 1 2
1 + q - - - q + q
q</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[15]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[15]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Knot[4, 1], Knot[11, NonAlternating, 19]}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[16]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>A2Invariant[Knot[4, 1]][q]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[16]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -8 -6 6 8
-1 + q + q + q + q</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[17]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>HOMFLYPT[Knot[4, 1]][a, z]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[17]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -2 2 2
-1 + a + a - z</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[18]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kauffman[Knot[4, 1]][a, z]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[18]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 3
-2 2 z 2 z 2 2 z 3
-1 - a - a - - - a z + 2 z + -- + a z + -- + a z
a 2 a
a</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[19]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Vassiliev[2][Knot[4, 1]], Vassiliev[3][Knot[4, 1]]}</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[19]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{-1, 0}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[20]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kh[Knot[4, 1]][q, t]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[20]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>1 1 1 5 2
- + q + ----- + --- + q t + q t
q 5 2 q t
q t</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[21]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>ColouredJones[Knot[4, 1], 2][q]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[21]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -6 -5 -4 2 -2 1 2 3 4 5 6
3 + q - q - q + -- - q - - - q - q + 2 q - q - q + q
3 q
q</nowiki></code></td></tr>
</table> }}

Revision as of 13:44, 14 July 2007

{{Rolfsen Knot Page| n = 4 | k = 1 | KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-4,3,-1,2,-3,4,-2/goTop.html |

braid_table =

BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif

|

braid_crossings = 4 | braid_width = 3 | braid_index = 3 | same_alexander = | same_jones = K11n19, |

khovanov_table =

\
Retrieved from "https://katlas.org/index.php?title=4_1&oldid=62277"