View source for 8 12

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{{Rolfsen Knot Page Header|n=8|k=12|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-4,3,-1,5,-8,2,-3,4,-2,6,-7,8,-5,7,-6/goTop.html}}

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{{:{{PAGENAME}} Further Notes and Views}}

{{Knot Presentations}}

<center><table border=1 cellpadding=10><tr align=center valign=top>
<td>
[[Braid Representatives|Minimum Braid Representative]]:
<table cellspacing=0 cellpadding=0 border=0>
<tr><td>[[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr>
<tr><td>[[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr>
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]]</td></tr>
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]]</td></tr>
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]]</td></tr>
</table>

[[Invariants from Braid Theory|Length]] is 8, width is 5.

[[Invariants from Braid Theory|Braid index]] is 5.
</td>
<td>
[[Lightly Documented Features|A Morse Link Presentation]]:

[[Image:{{PAGENAME}}_ML.gif]]
</td>
</tr></table></center>

{{3D Invariants}}
{{4D Invariants}}
{{Polynomial Invariants}}

=== "Similar" Knots (within the Atlas) ===

Same [[The Alexander-Conway Polynomial|Alexander/Conway Polynomial]]:
{...}

Same [[The Jones Polynomial|Jones Polynomial]] (up to mirroring, <math>q\leftrightarrow q^{-1}</math>): 
{...}

{{Vassiliev Invariants}}

{{Khovanov Homology|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>9</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>7</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 bgcolor=yellow>&nbsp;</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 bgcolor=yellow>3</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>2</td></tr>
<tr align=center><td>3</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>&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>3</td><td bgcolor=yellow>3</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>0</td></tr>
<tr align=center><td>-1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>3</td><td bgcolor=yellow>3</td><td>&nbsp;</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 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>-1</td></tr>
<tr align=center><td>-5</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>3</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>2</td></tr>
<tr align=center><td>-7</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>-9</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>}}

{{Display Coloured Jones|J2=<math>q^{12}-2 q^{11}+6 q^9-8 q^8-3 q^7+18 q^6-15 q^5-10 q^4+30 q^3-18 q^2-16 q+35-16 q^{-1} -18 q^{-2} +30 q^{-3} -10 q^{-4} -15 q^{-5} +18 q^{-6} -3 q^{-7} -8 q^{-8} +6 q^{-9} -2 q^{-11} + q^{-12} </math>|J3=<math>q^{24}-2 q^{23}+2 q^{21}+3 q^{20}-7 q^{19}-5 q^{18}+11 q^{17}+13 q^{16}-18 q^{15}-22 q^{14}+20 q^{13}+40 q^{12}-26 q^{11}-54 q^{10}+23 q^9+73 q^8-20 q^7-88 q^6+15 q^5+100 q^4-9 q^3-108 q^2+4 q+109+4 q^{-1} -108 q^{-2} -9 q^{-3} +100 q^{-4} +15 q^{-5} -88 q^{-6} -20 q^{-7} +73 q^{-8} +23 q^{-9} -54 q^{-10} -26 q^{-11} +40 q^{-12} +20 q^{-13} -22 q^{-14} -18 q^{-15} +13 q^{-16} +11 q^{-17} -5 q^{-18} -7 q^{-19} +3 q^{-20} +2 q^{-21} -2 q^{-23} + q^{-24} </math>|J4=<math>q^{40}-2 q^{39}+2 q^{37}-q^{36}+4 q^{35}-9 q^{34}-q^{33}+10 q^{32}+q^{31}+13 q^{30}-32 q^{29}-14 q^{28}+25 q^{27}+19 q^{26}+46 q^{25}-72 q^{24}-58 q^{23}+23 q^{22}+54 q^{21}+130 q^{20}-105 q^{19}-134 q^{18}-21 q^{17}+81 q^{16}+255 q^{15}-101 q^{14}-209 q^{13}-104 q^{12}+77 q^{11}+381 q^{10}-64 q^9-257 q^8-187 q^7+52 q^6+464 q^5-20 q^4-267 q^3-244 q^2+18 q+493+18 q^{-1} -244 q^{-2} -267 q^{-3} -20 q^{-4} +464 q^{-5} +52 q^{-6} -187 q^{-7} -257 q^{-8} -64 q^{-9} +381 q^{-10} +77 q^{-11} -104 q^{-12} -209 q^{-13} -101 q^{-14} +255 q^{-15} +81 q^{-16} -21 q^{-17} -134 q^{-18} -105 q^{-19} +130 q^{-20} +54 q^{-21} +23 q^{-22} -58 q^{-23} -72 q^{-24} +46 q^{-25} +19 q^{-26} +25 q^{-27} -14 q^{-28} -32 q^{-29} +13 q^{-30} + q^{-31} +10 q^{-32} - q^{-33} -9 q^{-34} +4 q^{-35} - q^{-36} +2 q^{-37} -2 q^{-39} + q^{-40} </math>|J5=<math>q^{60}-2 q^{59}+2 q^{57}-q^{56}+2 q^{54}-5 q^{53}-2 q^{52}+9 q^{51}+3 q^{50}-2 q^{49}-3 q^{48}-18 q^{47}-8 q^{46}+22 q^{45}+32 q^{44}+14 q^{43}-20 q^{42}-63 q^{41}-52 q^{40}+30 q^{39}+99 q^{38}+101 q^{37}-q^{36}-149 q^{35}-185 q^{34}-39 q^{33}+177 q^{32}+285 q^{31}+144 q^{30}-202 q^{29}-411 q^{28}-251 q^{27}+169 q^{26}+520 q^{25}+428 q^{24}-118 q^{23}-632 q^{22}-588 q^{21}+23 q^{20}+695 q^{19}+777 q^{18}+91 q^{17}-748 q^{16}-931 q^{15}-217 q^{14}+766 q^{13}+1064 q^{12}+339 q^{11}-761 q^{10}-1171 q^9-444 q^8+743 q^7+1238 q^6+535 q^5-705 q^4-1286 q^3-605 q^2+666 q+1291+666 q^{-1} -605 q^{-2} -1286 q^{-3} -705 q^{-4} +535 q^{-5} +1238 q^{-6} +743 q^{-7} -444 q^{-8} -1171 q^{-9} -761 q^{-10} +339 q^{-11} +1064 q^{-12} +766 q^{-13} -217 q^{-14} -931 q^{-15} -748 q^{-16} +91 q^{-17} +777 q^{-18} +695 q^{-19} +23 q^{-20} -588 q^{-21} -632 q^{-22} -118 q^{-23} +428 q^{-24} +520 q^{-25} +169 q^{-26} -251 q^{-27} -411 q^{-28} -202 q^{-29} +144 q^{-30} +285 q^{-31} +177 q^{-32} -39 q^{-33} -185 q^{-34} -149 q^{-35} - q^{-36} +101 q^{-37} +99 q^{-38} +30 q^{-39} -52 q^{-40} -63 q^{-41} -20 q^{-42} +14 q^{-43} +32 q^{-44} +22 q^{-45} -8 q^{-46} -18 q^{-47} -3 q^{-48} -2 q^{-49} +3 q^{-50} +9 q^{-51} -2 q^{-52} -5 q^{-53} +2 q^{-54} - q^{-56} +2 q^{-57} -2 q^{-59} + q^{-60} </math>|J6=<math>q^{84}-2 q^{83}+2 q^{81}-q^{80}-2 q^{78}+6 q^{77}-6 q^{76}-3 q^{75}+11 q^{74}-q^{73}-2 q^{72}-12 q^{71}+11 q^{70}-16 q^{69}-7 q^{68}+38 q^{67}+16 q^{66}+3 q^{65}-41 q^{64}+q^{63}-68 q^{62}-33 q^{61}+98 q^{60}+93 q^{59}+77 q^{58}-58 q^{57}-37 q^{56}-237 q^{55}-183 q^{54}+124 q^{53}+255 q^{52}+331 q^{51}+93 q^{50}+5 q^{49}-556 q^{48}-604 q^{47}-102 q^{46}+356 q^{45}+782 q^{44}+595 q^{43}+402 q^{42}-827 q^{41}-1294 q^{40}-786 q^{39}+85 q^{38}+1183 q^{37}+1419 q^{36}+1336 q^{35}-712 q^{34}-1957 q^{33}-1851 q^{32}-715 q^{31}+1194 q^{30}+2244 q^{29}+2655 q^{28}-93 q^{27}-2262 q^{26}-2933 q^{25}-1845 q^{24}+746 q^{23}+2752 q^{22}+3943 q^{21}+806 q^{20}-2153 q^{19}-3708 q^{18}-2911 q^{17}+70 q^{16}+2891 q^{15}+4876 q^{14}+1634 q^{13}-1812 q^{12}-4100 q^{11}-3656 q^{10}-556 q^9+2791 q^8+5381 q^7+2212 q^6-1428 q^5-4191 q^4-4055 q^3-1036 q^2+2568 q+5533+2568 q^{-1} -1036 q^{-2} -4055 q^{-3} -4191 q^{-4} -1428 q^{-5} +2212 q^{-6} +5381 q^{-7} +2791 q^{-8} -556 q^{-9} -3656 q^{-10} -4100 q^{-11} -1812 q^{-12} +1634 q^{-13} +4876 q^{-14} +2891 q^{-15} +70 q^{-16} -2911 q^{-17} -3708 q^{-18} -2153 q^{-19} +806 q^{-20} +3943 q^{-21} +2752 q^{-22} +746 q^{-23} -1845 q^{-24} -2933 q^{-25} -2262 q^{-26} -93 q^{-27} +2655 q^{-28} +2244 q^{-29} +1194 q^{-30} -715 q^{-31} -1851 q^{-32} -1957 q^{-33} -712 q^{-34} +1336 q^{-35} +1419 q^{-36} +1183 q^{-37} +85 q^{-38} -786 q^{-39} -1294 q^{-40} -827 q^{-41} +402 q^{-42} +595 q^{-43} +782 q^{-44} +356 q^{-45} -102 q^{-46} -604 q^{-47} -556 q^{-48} +5 q^{-49} +93 q^{-50} +331 q^{-51} +255 q^{-52} +124 q^{-53} -183 q^{-54} -237 q^{-55} -37 q^{-56} -58 q^{-57} +77 q^{-58} +93 q^{-59} +98 q^{-60} -33 q^{-61} -68 q^{-62} + q^{-63} -41 q^{-64} +3 q^{-65} +16 q^{-66} +38 q^{-67} -7 q^{-68} -16 q^{-69} +11 q^{-70} -12 q^{-71} -2 q^{-72} - q^{-73} +11 q^{-74} -3 q^{-75} -6 q^{-76} +6 q^{-77} -2 q^{-78} - q^{-80} +2 q^{-81} -2 q^{-83} + q^{-84} </math>|J7=<math>q^{112}-2 q^{111}+2 q^{109}-q^{108}-2 q^{106}+2 q^{105}+5 q^{104}-7 q^{103}-q^{102}+7 q^{101}-q^{100}-12 q^{98}-2 q^{97}+17 q^{96}-13 q^{95}+3 q^{94}+22 q^{93}+7 q^{92}+7 q^{91}-43 q^{90}-35 q^{89}+10 q^{88}-26 q^{87}+25 q^{86}+81 q^{85}+63 q^{84}+69 q^{83}-77 q^{82}-148 q^{81}-104 q^{80}-156 q^{79}+17 q^{78}+206 q^{77}+282 q^{76}+356 q^{75}+55 q^{74}-268 q^{73}-435 q^{72}-661 q^{71}-340 q^{70}+204 q^{69}+654 q^{68}+1130 q^{67}+788 q^{66}+49 q^{65}-750 q^{64}-1693 q^{63}-1545 q^{62}-598 q^{61}+680 q^{60}+2252 q^{59}+2499 q^{58}+1547 q^{57}-198 q^{56}-2717 q^{55}-3672 q^{54}-2860 q^{53}-655 q^{52}+2844 q^{51}+4768 q^{50}+4537 q^{49}+2104 q^{48}-2562 q^{47}-5828 q^{46}-6373 q^{45}-3891 q^{44}+1784 q^{43}+6441 q^{42}+8212 q^{41}+6132 q^{40}-511 q^{39}-6764 q^{38}-9928 q^{37}-8380 q^{36}-1108 q^{35}+6552 q^{34}+11294 q^{33}+10685 q^{32}+2988 q^{31}-6052 q^{30}-12328 q^{29}-12727 q^{28}-4870 q^{27}+5238 q^{26}+12969 q^{25}+14476 q^{24}+6663 q^{23}-4286 q^{22}-13308 q^{21}-15867 q^{20}-8241 q^{19}+3344 q^{18}+13388 q^{17}+16876 q^{16}+9543 q^{15}-2421 q^{14}-13285 q^{13}-17618 q^{12}-10578 q^{11}+1640 q^{10}+13082 q^9+18052 q^8+11361 q^7-915 q^6-12789 q^5-18322 q^4-11965 q^3+311 q^2+12426 q+18379+12426 q^{-1} +311 q^{-2} -11965 q^{-3} -18322 q^{-4} -12789 q^{-5} -915 q^{-6} +11361 q^{-7} +18052 q^{-8} +13082 q^{-9} +1640 q^{-10} -10578 q^{-11} -17618 q^{-12} -13285 q^{-13} -2421 q^{-14} +9543 q^{-15} +16876 q^{-16} +13388 q^{-17} +3344 q^{-18} -8241 q^{-19} -15867 q^{-20} -13308 q^{-21} -4286 q^{-22} +6663 q^{-23} +14476 q^{-24} +12969 q^{-25} +5238 q^{-26} -4870 q^{-27} -12727 q^{-28} -12328 q^{-29} -6052 q^{-30} +2988 q^{-31} +10685 q^{-32} +11294 q^{-33} +6552 q^{-34} -1108 q^{-35} -8380 q^{-36} -9928 q^{-37} -6764 q^{-38} -511 q^{-39} +6132 q^{-40} +8212 q^{-41} +6441 q^{-42} +1784 q^{-43} -3891 q^{-44} -6373 q^{-45} -5828 q^{-46} -2562 q^{-47} +2104 q^{-48} +4537 q^{-49} +4768 q^{-50} +2844 q^{-51} -655 q^{-52} -2860 q^{-53} -3672 q^{-54} -2717 q^{-55} -198 q^{-56} +1547 q^{-57} +2499 q^{-58} +2252 q^{-59} +680 q^{-60} -598 q^{-61} -1545 q^{-62} -1693 q^{-63} -750 q^{-64} +49 q^{-65} +788 q^{-66} +1130 q^{-67} +654 q^{-68} +204 q^{-69} -340 q^{-70} -661 q^{-71} -435 q^{-72} -268 q^{-73} +55 q^{-74} +356 q^{-75} +282 q^{-76} +206 q^{-77} +17 q^{-78} -156 q^{-79} -104 q^{-80} -148 q^{-81} -77 q^{-82} +69 q^{-83} +63 q^{-84} +81 q^{-85} +25 q^{-86} -26 q^{-87} +10 q^{-88} -35 q^{-89} -43 q^{-90} +7 q^{-91} +7 q^{-92} +22 q^{-93} +3 q^{-94} -13 q^{-95} +17 q^{-96} -2 q^{-97} -12 q^{-98} - q^{-100} +7 q^{-101} - q^{-102} -7 q^{-103} +5 q^{-104} +2 q^{-105} -2 q^{-106} - q^{-108} +2 q^{-109} -2 q^{-111} + q^{-112} </math>}}

{{Computer Talk Header}}

<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><pre style="border: 0px; padding: 0em">Loading KnotTheory` (version of August 29, 2005, 15:27:48)...</pre></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>PD[Knot[8, 12]]</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>PD[X[4, 2, 5, 1], X[10, 8, 11, 7], X[8, 3, 9, 4], X[2, 9, 3, 10], 
 
  X[14, 6, 15, 5], X[16, 11, 1, 12], X[12, 15, 13, 16], X[6, 14, 7, 13]]</nowiki></pre></td></tr>

<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[3]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>GaussCode[Knot[8, 12]]</nowiki></pre></td></tr>
<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[3]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[1, -4, 3, -1, 5, -8, 2, -3, 4, -2, 6, -7, 8, -5, 7, -6]</nowiki></pre></td></tr>

<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[4]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>DTCode[Knot[8, 12]]</nowiki></pre></td></tr>
<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>DTCode[4, 8, 14, 10, 2, 16, 6, 12]</nowiki></pre></td></tr>

<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>br = BR[Knot[8, 12]]</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>BR[5, {-1, 2, -1, -3, 2, 4, -3, 4}]</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>{First[br], Crossings[br]}</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>{5, 8}</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>BraidIndex[Knot[8, 12]]</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>5</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>Show[DrawMorseLink[Knot[8, 12]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:8_12_ML.gif]]</td></tr><tr valign=top><td><tt><font color=blue>Out[8]=</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[9]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>(#[Knot[8, 12]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}</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>{FullyAmphicheiral, 2, 2, 2, {4, 6}, 1}</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>alex = Alexander[Knot[8, 12]][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>      -2   7          2
13 + t   - - - 7 t + t
           t</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>Conway[Knot[8, 12]][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>       2    4
1 - 3 z  + 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>Select[AllKnots[], (alex === Alexander[#][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>{Knot[8, 12]}</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>{KnotDet[Knot[8, 12]], KnotSignature[Knot[8, 12]]}</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>{29, 0}</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>Jones[Knot[8, 12]][q]</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>     -4   2    4    5            2      3    4
5 + q   - -- + -- - - - 5 q + 4 q  - 2 q  + q
           3    2   q
          q    q</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>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]</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>{Knot[8, 12]}</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>A2Invariant[Knot[8, 12]][q]</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>      -14    -12    -10    -8    -4    -2    2    4    8    10    12
-1 + q    + q    - q    + q   - q   + q   + q  - q  + q  - q   + q   + 
 
   14
  q</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>HOMFLYPT[Knot[8, 12]][a, z]</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
     -4    -2    2    4    2   2 z       2  2    4
1 + a   - a   - a  + a  + z  - ---- - 2 a  z  + z
                                 2
                                a</nowiki></pre></td></tr>

<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[18]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kauffman[Knot[8, 12]][a, z]</nowiki></pre></td></tr>
<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[18]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>                                         2      2
     -4    -2    2    4   z     3     2 z    2 z       2  2      4  2
1 + a   + a   + a  + a  + -- + a  z - ---- - ---- - 2 a  z  - 2 a  z  - 
                           3            4      2
                          a            a      a
 
     3      3                              4    4
  3 z    3 z         3      3  3      4   z    z     2  4    4  4
  ---- - ---- - 3 a z  - 3 a  z  - 4 z  + -- - -- - a  z  + a  z  + 
    3     a                                4    2
   a                                      a    a
 
     5      5                                6              7
  2 z    2 z         5      3  5      6   2 z       2  6   z       7
  ---- + ---- + 2 a z  + 2 a  z  + 4 z  + ---- + 2 a  z  + -- + a z
    3     a                                 2              a
   a                                       a</nowiki></pre></td></tr>

<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[19]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{Vassiliev[2][Knot[8, 12]], Vassiliev[3][Knot[8, 12]]}</nowiki></pre></td></tr>
<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[19]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{-3, 0}</nowiki></pre></td></tr>

<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[20]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kh[Knot[8, 12]][q, t]</nowiki></pre></td></tr>
<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[20]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>3           1       1       1       3       1      2      3
- + 3 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 3 q t + 
q          9  4    7  3    5  3    5  2    3  2    3     q t
          q  t    q  t    q  t    q  t    q  t    q  t
 
     3      3  2      5  2    5  3    7  3    9  4
  2 q  t + q  t  + 3 q  t  + q  t  + q  t  + q  t</nowiki></pre></td></tr>

<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[21]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>ColouredJones[Knot[8, 12], 2][q]</nowiki></pre></td></tr>
<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[21]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>      -12    2    6    8    3    18   15   10   30   18   16
35 + q    - --- + -- - -- - -- + -- - -- - -- + -- - -- - -- - 16 q - 
             11    9    8    7    6    5    4    3    2   q
            q     q    q    q    q    q    q    q    q
 
      2       3       4       5       6      7      8      9      11
  18 q  + 30 q  - 10 q  - 15 q  + 18 q  - 3 q  - 8 q  + 6 q  - 2 q   + 
 
   12
  q</nowiki></pre></td></tr>

</table>

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