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{{Rolfsen Knot Page|
n = 8 |
k = 4 |
KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-4,3,-7,5,-1,6,-8,2,-3,7,-5,4,-2,8,-6/goTop.html |
braid_table     = <table cellspacing=0 cellpadding=0 border=0>
<tr><td>[[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr>
<tr><td>[[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]]</td></tr>
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]]</td></tr>
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]]</td></tr>
</table> |
braid_crossings = 9 |
braid_width     = 4 |
braid_index     = 4 |
same_alexander  =  |
same_jones      =  |
khovanov_table  = <table border=1>
<tr align=center>
<td width=15.3846%><table cellpadding=0 cellspacing=0>
  <tr><td>\</td><td>&nbsp;</td><td>r</td></tr>
<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=7.69231%>-4</td    ><td width=7.69231%>-3</td    ><td width=7.69231%>-2</td    ><td width=7.69231%>-1</td    ><td width=7.69231%>0</td    ><td width=7.69231%>1</td    ><td width=7.69231%>2</td    ><td width=7.69231%>3</td    ><td width=7.69231%>4</td    ><td width=15.3846%>&chi;</td></tr>
<tr align=center><td>7</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</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>5</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</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>3</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>2</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>1</td></tr>
<tr align=center><td>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-1</td></tr>
<tr align=center><td>-1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>2</td><td bgcolor=yellow>2</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>0</td></tr>
<tr align=center><td>-3</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>2</td><td bgcolor=yellow>2</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>0</td></tr>
<tr align=center><td>-5</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>0</td></tr>
<tr align=center><td>-7</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>2</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
<tr align=center><td>-9</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-1</td></tr>
<tr align=center><td>-11</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
</table> |
coloured_jones_2 = <math>q^{10}-q^9-q^8+3 q^7-q^6-4 q^5+5 q^4-7 q^2+7 q+2-9 q^{-1} +7 q^{-2} +4 q^{-3} -10 q^{-4} +5 q^{-5} +5 q^{-6} -9 q^{-7} +4 q^{-8} +3 q^{-9} -6 q^{-10} +3 q^{-11} + q^{-12} -2 q^{-13} + q^{-14} </math> |
coloured_jones_3 = <math>q^{21}-q^{20}-q^{19}+3 q^{17}-3 q^{15}-3 q^{14}+5 q^{13}+3 q^{12}-3 q^{11}-7 q^{10}+4 q^9+7 q^8-q^7-9 q^6+q^5+9 q^4+q^3-9 q^2-2 q+9+3 q^{-1} -8 q^{-2} -5 q^{-3} +7 q^{-4} +6 q^{-5} -5 q^{-6} -8 q^{-7} +4 q^{-8} +9 q^{-9} -3 q^{-10} -8 q^{-11} + q^{-12} +8 q^{-13} -2 q^{-14} -5 q^{-15} +2 q^{-16} +4 q^{-17} -3 q^{-18} -2 q^{-19} +3 q^{-20} + q^{-21} -3 q^{-22} + q^{-24} + q^{-25} -2 q^{-26} + q^{-27} </math> |
coloured_jones_4 = <math>q^{36}-q^{35}-q^{34}+4 q^{31}-q^{30}-2 q^{29}-2 q^{28}-4 q^{27}+8 q^{26}+2 q^{25}-3 q^{23}-11 q^{22}+8 q^{21}+3 q^{20}+6 q^{19}+q^{18}-17 q^{17}+5 q^{16}-q^{15}+10 q^{14}+8 q^{13}-18 q^{12}+5 q^{11}-9 q^{10}+9 q^9+13 q^8-17 q^7+10 q^6-15 q^5+4 q^4+14 q^3-15 q^2+17 q-17- q^{-1} +13 q^{-2} -13 q^{-3} +24 q^{-4} -19 q^{-5} -7 q^{-6} +11 q^{-7} -8 q^{-8} +29 q^{-9} -22 q^{-10} -13 q^{-11} +8 q^{-12} -3 q^{-13} +34 q^{-14} -21 q^{-15} -18 q^{-16} +3 q^{-17} - q^{-18} +35 q^{-19} -16 q^{-20} -16 q^{-21} -6 q^{-23} +29 q^{-24} -9 q^{-25} -8 q^{-26} + q^{-27} -10 q^{-28} +18 q^{-29} -6 q^{-30} - q^{-31} +3 q^{-32} -9 q^{-33} +9 q^{-34} -4 q^{-35} + q^{-36} +3 q^{-37} -5 q^{-38} +3 q^{-39} -2 q^{-40} + q^{-41} + q^{-42} -2 q^{-43} + q^{-44} </math> |
coloured_jones_5 = <math>q^{55}-q^{54}-q^{53}+q^{50}+3 q^{49}-3 q^{47}-2 q^{46}-2 q^{45}-q^{44}+6 q^{43}+5 q^{42}-3 q^{40}-6 q^{39}-7 q^{38}+3 q^{37}+8 q^{36}+6 q^{35}+4 q^{34}-5 q^{33}-12 q^{32}-5 q^{31}+2 q^{30}+7 q^{29}+12 q^{28}+4 q^{27}-9 q^{26}-9 q^{25}-6 q^{24}-2 q^{23}+11 q^{22}+11 q^{21}+q^{20}-5 q^{19}-7 q^{18}-10 q^{17}+8 q^{15}+7 q^{14}+6 q^{13}-9 q^{11}-10 q^{10}-5 q^9+5 q^8+14 q^7+12 q^6-15 q^4-18 q^3-6 q^2+16 q+23+12 q^{-1} -15 q^{-2} -29 q^{-3} -17 q^{-4} +16 q^{-5} +31 q^{-6} +22 q^{-7} -15 q^{-8} -37 q^{-9} -24 q^{-10} +16 q^{-11} +40 q^{-12} +29 q^{-13} -17 q^{-14} -47 q^{-15} -32 q^{-16} +18 q^{-17} +52 q^{-18} +37 q^{-19} -18 q^{-20} -57 q^{-21} -43 q^{-22} +18 q^{-23} +60 q^{-24} +44 q^{-25} -10 q^{-26} -58 q^{-27} -50 q^{-28} +7 q^{-29} +54 q^{-30} +46 q^{-31} -43 q^{-33} -45 q^{-34} -5 q^{-35} +36 q^{-36} +36 q^{-37} +7 q^{-38} -25 q^{-39} -29 q^{-40} -7 q^{-41} +17 q^{-42} +20 q^{-43} +7 q^{-44} -12 q^{-45} -14 q^{-46} -2 q^{-47} +6 q^{-48} +7 q^{-49} +3 q^{-50} -3 q^{-51} -6 q^{-52} + q^{-53} +2 q^{-54} - q^{-55} +2 q^{-56} + q^{-57} -3 q^{-58} + q^{-59} + q^{-60} -2 q^{-61} + q^{-62} + q^{-63} -2 q^{-64} + q^{-65} </math> |
coloured_jones_6 = <math>q^{78}-q^{77}-q^{76}+q^{73}+4 q^{71}-q^{70}-3 q^{69}-2 q^{68}-2 q^{67}-2 q^{65}+10 q^{64}+3 q^{63}-2 q^{61}-5 q^{60}-5 q^{59}-12 q^{58}+11 q^{57}+6 q^{56}+7 q^{55}+5 q^{54}+2 q^{53}-5 q^{52}-24 q^{51}+3 q^{50}-3 q^{49}+7 q^{48}+9 q^{47}+17 q^{46}+8 q^{45}-24 q^{44}+q^{43}-17 q^{42}-5 q^{41}-3 q^{40}+22 q^{39}+21 q^{38}-12 q^{37}+15 q^{36}-17 q^{35}-12 q^{34}-24 q^{33}+11 q^{32}+16 q^{31}-9 q^{30}+34 q^{29}-33 q^{26}-q^{25}-3 q^{24}-27 q^{23}+37 q^{22}+18 q^{21}+26 q^{20}-22 q^{19}+3 q^{18}-20 q^{17}-57 q^{16}+19 q^{15}+20 q^{14}+48 q^{13}+25 q^{11}-21 q^{10}-84 q^9-11 q^8+6 q^7+58 q^6+21 q^5+55 q^4-7 q^3-98 q^2-41 q-16+56 q^{-1} +33 q^{-2} +83 q^{-3} +13 q^{-4} -100 q^{-5} -63 q^{-6} -37 q^{-7} +47 q^{-8} +38 q^{-9} +104 q^{-10} +31 q^{-11} -98 q^{-12} -80 q^{-13} -51 q^{-14} +40 q^{-15} +45 q^{-16} +120 q^{-17} +40 q^{-18} -102 q^{-19} -99 q^{-20} -61 q^{-21} +41 q^{-22} +61 q^{-23} +138 q^{-24} +44 q^{-25} -111 q^{-26} -123 q^{-27} -75 q^{-28} +41 q^{-29} +78 q^{-30} +158 q^{-31} +57 q^{-32} -110 q^{-33} -140 q^{-34} -94 q^{-35} +27 q^{-36} +77 q^{-37} +164 q^{-38} +77 q^{-39} -87 q^{-40} -129 q^{-41} -101 q^{-42} +2 q^{-43} +51 q^{-44} +143 q^{-45} +83 q^{-46} -54 q^{-47} -93 q^{-48} -83 q^{-49} -10 q^{-50} +19 q^{-51} +103 q^{-52} +66 q^{-53} -34 q^{-54} -53 q^{-55} -53 q^{-56} -5 q^{-57} +2 q^{-58} +62 q^{-59} +39 q^{-60} -27 q^{-61} -24 q^{-62} -25 q^{-63} +3 q^{-64} -4 q^{-65} +32 q^{-66} +17 q^{-67} -20 q^{-68} -7 q^{-69} -8 q^{-70} +5 q^{-71} -6 q^{-72} +14 q^{-73} +6 q^{-74} -11 q^{-75} + q^{-76} -2 q^{-77} +3 q^{-78} -5 q^{-79} +5 q^{-80} +2 q^{-81} -5 q^{-82} +3 q^{-83} - q^{-84} + q^{-85} -2 q^{-86} + q^{-87} + q^{-88} -2 q^{-89} + q^{-90} </math> |
coloured_jones_7 = <math>q^{105}-q^{104}-q^{103}+q^{100}+q^{98}+3 q^{97}-q^{96}-3 q^{95}-2 q^{94}-3 q^{93}+q^{92}+9 q^{89}+4 q^{88}-2 q^{86}-8 q^{85}-3 q^{84}-6 q^{83}-8 q^{82}+9 q^{81}+9 q^{80}+9 q^{79}+10 q^{78}-5 q^{77}-9 q^{75}-22 q^{74}-4 q^{73}-2 q^{72}+7 q^{71}+20 q^{70}+7 q^{69}+16 q^{68}+6 q^{67}-21 q^{66}-10 q^{65}-21 q^{64}-16 q^{63}+10 q^{62}+q^{61}+25 q^{60}+27 q^{59}+10 q^{57}-16 q^{56}-30 q^{55}-9 q^{54}-26 q^{53}+5 q^{52}+24 q^{51}+6 q^{50}+37 q^{49}+14 q^{48}-9 q^{47}-41 q^{45}-23 q^{44}-5 q^{43}-22 q^{42}+28 q^{41}+31 q^{40}+24 q^{39}+43 q^{38}-17 q^{37}-21 q^{36}-21 q^{35}-67 q^{34}-15 q^{33}+3 q^{32}+26 q^{31}+80 q^{30}+32 q^{29}+21 q^{28}-85 q^{26}-59 q^{25}-52 q^{24}-20 q^{23}+81 q^{22}+67 q^{21}+76 q^{20}+55 q^{19}-63 q^{18}-71 q^{17}-102 q^{16}-85 q^{15}+40 q^{14}+67 q^{13}+114 q^{12}+116 q^{11}-13 q^{10}-51 q^9-122 q^8-143 q^7-17 q^6+34 q^5+123 q^4+163 q^3+44 q^2-9 q-115-180 q^{-1} -74 q^{-2} -14 q^{-3} +108 q^{-4} +190 q^{-5} +95 q^{-6} +41 q^{-7} -92 q^{-8} -200 q^{-9} -120 q^{-10} -61 q^{-11} +82 q^{-12} +201 q^{-13} +136 q^{-14} +84 q^{-15} -68 q^{-16} -210 q^{-17} -153 q^{-18} -94 q^{-19} +62 q^{-20} +210 q^{-21} +166 q^{-22} +108 q^{-23} -60 q^{-24} -223 q^{-25} -178 q^{-26} -110 q^{-27} +63 q^{-28} +232 q^{-29} +194 q^{-30} +118 q^{-31} -72 q^{-32} -250 q^{-33} -214 q^{-34} -122 q^{-35} +80 q^{-36} +267 q^{-37} +233 q^{-38} +134 q^{-39} -77 q^{-40} -281 q^{-41} -258 q^{-42} -151 q^{-43} +74 q^{-44} +284 q^{-45} +268 q^{-46} +171 q^{-47} -50 q^{-48} -270 q^{-49} -278 q^{-50} -191 q^{-51} +25 q^{-52} +248 q^{-53} +265 q^{-54} +196 q^{-55} +4 q^{-56} -205 q^{-57} -239 q^{-58} -200 q^{-59} -29 q^{-60} +171 q^{-61} +205 q^{-62} +179 q^{-63} +37 q^{-64} -128 q^{-65} -161 q^{-66} -154 q^{-67} -43 q^{-68} +101 q^{-69} +123 q^{-70} +122 q^{-71} +32 q^{-72} -79 q^{-73} -87 q^{-74} -88 q^{-75} -19 q^{-76} +61 q^{-77} +56 q^{-78} +62 q^{-79} +12 q^{-80} -53 q^{-81} -39 q^{-82} -36 q^{-83} +3 q^{-84} +40 q^{-85} +16 q^{-86} +23 q^{-87} -2 q^{-88} -32 q^{-89} -10 q^{-90} -14 q^{-91} +9 q^{-92} +23 q^{-93} + q^{-94} +5 q^{-95} -4 q^{-96} -14 q^{-97} + q^{-98} -6 q^{-99} +4 q^{-100} +12 q^{-101} -4 q^{-102} + q^{-103} - q^{-104} -4 q^{-105} +3 q^{-106} -4 q^{-107} + q^{-108} +5 q^{-109} -4 q^{-110} + q^{-111} + q^{-112} - q^{-113} + q^{-114} -2 q^{-115} + q^{-116} + q^{-117} -2 q^{-118} + q^{-119} </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[8, 4]]</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[6, 2, 7, 1], X[14, 10, 15, 9], X[10, 3, 11, 4], X[2, 13, 3, 14], 
 
  X[12, 5, 13, 6], X[16, 8, 1, 7], X[4, 11, 5, 12], X[8, 16, 9, 15]]</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[8, 4]]</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, -7, 5, -1, 6, -8, 2, -3, 7, -5, 4, -2, 8, -6]</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[8, 4]]</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[6, 10, 12, 16, 14, 4, 2, 8]</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[8, 4]]</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[4, {-1, -1, -1, 2, -1, 2, 3, -2, 3}]</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>{4, 9}</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[8, 4]]</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>4</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[8, 4]]]</nowiki></code></td></tr>
<tr align=left><td></td><td>[[Image:8_4_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[8, 4]]&) /@ {
                 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>{Reversible, 2, 2, 2, {3, 5}, 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[8, 4]][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>     2    5            2
-5 - -- + - + 5 t - 2 t
      2   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[8, 4]][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      4
1 - 3 z  - 2 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[8, 4]}</nowiki></code></td></tr>
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<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[8, 4]], KnotSignature[Knot[8, 4]]}</nowiki></code></td></tr>
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<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>{19, -2}</nowiki></code></td></tr>
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<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[8, 4]][q]</nowiki></code></td></tr>
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<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>      -5   2    3    3    3          2    3
-3 + q   - -- + -- - -- + - + 2 q - q  + q
            4    3    2   q
           q    q    q</nowiki></code></td></tr>
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<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>
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<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[8, 4]}</nowiki></code></td></tr>
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<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[8, 4]][q]</nowiki></code></td></tr>
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<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>      -16    -10    -6    -4    -2    2    4    6    8    10
-1 + q    + q    + q   - q   - q   - q  + q  + q  + q  + q</nowiki></code></td></tr>
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<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[8, 4]][a, z]</nowiki></code></td></tr>
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<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     4      2   z       2  2    4  2    4    2  4
-2 + -- + a  - 3 z  + -- - 2 a  z  + a  z  - z  - a  z
      2                2
     a                a</nowiki></code></td></tr>
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<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[8, 4]][a, z]</nowiki></code></td></tr>
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<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
     2     4   z            3         2   7 z     2  2      4  2
-2 - -- + a  - - + a z + 2 a  z + 10 z  + ---- - a  z  - 3 a  z  + 
      2        a                            2
     a                                     a
 
             3                                           4
   6  2   4 z         3      3  3      5  3       4   5 z       2  4
  a  z  + ---- - 3 a z  - 5 a  z  + 2 a  z  - 11 z  - ---- - 3 a  z  + 
           a                                            2
                                                       a
 
               5                            6              7
     4  4   4 z       5      3  5      6   z       2  6   z       7
  3 a  z  - ---- - a z  + 3 a  z  + 3 z  + -- + 2 a  z  + -- + a z
             a                              2             a
                                           a</nowiki></code></td></tr>
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<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[8, 4]], Vassiliev[3][Knot[8, 4]]}</nowiki></code></td></tr>
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<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>{-3, 1}</nowiki></code></td></tr>
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<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[8, 4]][q, t]</nowiki></code></td></tr>
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<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>2    2     1        1       1       2       1      1      2     2 t
-- + - + ------ + ----- + ----- + ----- + ----- + ---- + ---- + --- + 
 3   q    11  4    9  3    7  3    7  2    5  2    5      3      q
q        q   t    q  t    q  t    q  t    q  t    q  t   q  t
 
           3  2    3  3    7  4
  q t + 2 q  t  + q  t  + q  t</nowiki></code></td></tr>
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<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[8, 4], 2][q]</nowiki></code></td></tr>
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<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>     -14    2     -12    3     6    3    4    9    5    5    10   4
2 + q    - --- + q    + --- - --- + -- + -- - -- + -- + -- - -- + -- + 
            13           11    10    9    8    7    6    5    4    3
           q            q     q     q    q    q    q    q    q    q
 
  7    9            2      4      5    6      7    8    9    10
  -- - - + 7 q - 7 q  + 5 q  - 4 q  - q  + 3 q  - q  - q  + q
   2   q
  q</nowiki></code></td></tr>
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