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{{Rolfsen Knot Page| |
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n = 9 | |
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k = 12 | |
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KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,9,-2,1,-3,8,-6,7,-9,2,-4,5,-7,6,-8,3,-5,4/goTop.html | |
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<span id="top"></span> |
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braid_table = <table cellspacing=0 cellpadding=0 border=0> |
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{{Knot Navigation Links|ext=gif}} |
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{{Rolfsen Knot Page Header|n=9|k=12|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,9,-2,1,-3,8,-6,7,-9,2,-4,5,-7,6,-8,3,-5,4/goTop.html}} |
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<br style="clear:both" /> |
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{{:{{PAGENAME}} Further Notes and Views}} |
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{{Knot Presentations}} |
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<center><table border=1 cellpadding=10><tr align=center valign=top> |
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<td> |
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[[Braid Representatives|Minimum Braid Representative]]: |
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<table cellspacing=0 cellpadding=0 border=0> |
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<tr><td>[[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr> |
<tr><td>[[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr> |
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<tr><td>[[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr> |
<tr><td>[[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr> |
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<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]]</td></tr> |
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]]</td></tr> |
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<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]]</td></tr> |
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]]</td></tr> |
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</table> |
</table> | |
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braid_crossings = 10 | |
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braid_width = 5 | |
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[[Invariants from Braid Theory|Length]] is 10, width is 5. |
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braid_index = 5 | |
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same_alexander = [[K11n84]], | |
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[[Invariants from Braid Theory|Braid index]] is 5. |
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same_jones = [[K11n15]], | |
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</td> |
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khovanov_table = <table border=1> |
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<td> |
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[[Lightly Documented Features|A Morse Link Presentation]]: |
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[[Image:{{PAGENAME}}_ML.gif]] |
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</td> |
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</tr></table></center> |
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{{3D Invariants}} |
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{{4D Invariants}} |
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{{Polynomial Invariants}} |
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=== "Similar" Knots (within the Atlas) === |
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Same [[The Alexander-Conway Polynomial|Alexander/Conway Polynomial]]: |
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{[[K11n84]], ...} |
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Same [[The Jones Polynomial|Jones Polynomial]] (up to mirroring, <math>q\leftrightarrow q^{-1}</math>): |
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{[[K11n15]], ...} |
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{{Vassiliev Invariants}} |
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{{Khovanov Homology|table=<table border=1> |
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<tr align=center> |
<tr align=center> |
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<td width=14.2857%><table cellpadding=0 cellspacing=0> |
<td width=14.2857%><table cellpadding=0 cellspacing=0> |
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<tr><td>\</td><td> </td><td>r</td></tr> |
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<tr><td> </td><td> \ </td><td> </td></tr> |
<tr><td> </td><td> \ </td><td> </td></tr> |
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<tr><td>j</td><td> </td><td>\</td></tr> |
<tr><td>j</td><td> </td><td>\</td></tr> |
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</table></td> |
</table></td> |
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<td width=7.14286%>-7</td ><td width=7.14286%>-6</td ><td width=7.14286%>-5</td ><td width=7.14286%>-4</td ><td width=7.14286%>-3</td ><td width=7.14286%>-2</td ><td width=7.14286%>-1</td ><td width=7.14286%>0</td ><td width=7.14286%>1</td ><td width=7.14286%>2</td ><td width=14.2857%>χ</td></tr> |
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<tr align=center><td>3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td>1</td></tr> |
<tr align=center><td>3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td>1</td></tr> |
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<tr align=center><td>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow> </td><td>-1</td></tr> |
<tr align=center><td>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow> </td><td>-1</td></tr> |
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<tr align=center><td>-15</td><td bgcolor=yellow> </td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> |
<tr align=center><td>-15</td><td bgcolor=yellow> </td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> |
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<tr align=center><td>-17</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr> |
<tr align=center><td>-17</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr> |
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</table> |
</table> | |
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coloured_jones_2 = <math>q^4-2 q^3+5 q-7+14 q^{-2} -16 q^{-3} -3 q^{-4} +26 q^{-5} -23 q^{-6} -8 q^{-7} +35 q^{-8} -25 q^{-9} -12 q^{-10} +34 q^{-11} -19 q^{-12} -13 q^{-13} +25 q^{-14} -9 q^{-15} -11 q^{-16} +14 q^{-17} -2 q^{-18} -7 q^{-19} +5 q^{-20} -2 q^{-22} + q^{-23} </math> | |
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coloured_jones_3 = <math>q^9-2 q^8+q^6+3 q^5-4 q^4-2 q^3+5 q^2+4 q-11-3 q^{-1} +15 q^{-2} +10 q^{-3} -27 q^{-4} -13 q^{-5} +34 q^{-6} +26 q^{-7} -46 q^{-8} -35 q^{-9} +50 q^{-10} +51 q^{-11} -57 q^{-12} -60 q^{-13} +56 q^{-14} +71 q^{-15} -56 q^{-16} -74 q^{-17} +49 q^{-18} +76 q^{-19} -41 q^{-20} -75 q^{-21} +31 q^{-22} +71 q^{-23} -20 q^{-24} -63 q^{-25} +5 q^{-26} +57 q^{-27} +3 q^{-28} -44 q^{-29} -13 q^{-30} +35 q^{-31} +16 q^{-32} -23 q^{-33} -16 q^{-34} +13 q^{-35} +14 q^{-36} -8 q^{-37} -9 q^{-38} +3 q^{-39} +6 q^{-40} -2 q^{-41} -2 q^{-42} +2 q^{-44} - q^{-45} </math> | |
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{{Display Coloured Jones|J2=<math>q^4-2 q^3+5 q-7+14 q^{-2} -16 q^{-3} -3 q^{-4} +26 q^{-5} -23 q^{-6} -8 q^{-7} +35 q^{-8} -25 q^{-9} -12 q^{-10} +34 q^{-11} -19 q^{-12} -13 q^{-13} +25 q^{-14} -9 q^{-15} -11 q^{-16} +14 q^{-17} -2 q^{-18} -7 q^{-19} +5 q^{-20} -2 q^{-22} + q^{-23} </math>|J3=<math>q^9-2 q^8+q^6+3 q^5-4 q^4-2 q^3+5 q^2+4 q-11-3 q^{-1} +15 q^{-2} +10 q^{-3} -27 q^{-4} -13 q^{-5} +34 q^{-6} +26 q^{-7} -46 q^{-8} -35 q^{-9} +50 q^{-10} +51 q^{-11} -57 q^{-12} -60 q^{-13} +56 q^{-14} +71 q^{-15} -56 q^{-16} -74 q^{-17} +49 q^{-18} +76 q^{-19} -41 q^{-20} -75 q^{-21} +31 q^{-22} +71 q^{-23} -20 q^{-24} -63 q^{-25} +5 q^{-26} +57 q^{-27} +3 q^{-28} -44 q^{-29} -13 q^{-30} +35 q^{-31} +16 q^{-32} -23 q^{-33} -16 q^{-34} +13 q^{-35} +14 q^{-36} -8 q^{-37} -9 q^{-38} +3 q^{-39} +6 q^{-40} -2 q^{-41} -2 q^{-42} +2 q^{-44} - q^{-45} </math>|J4=<math>q^{16}-2 q^{15}+q^{13}-q^{12}+6 q^{11}-6 q^{10}+q^8-7 q^7+15 q^6-12 q^5+7 q^4+7 q^3-22 q^2+18 q-28+24 q^{-1} +32 q^{-2} -32 q^{-3} +14 q^{-4} -78 q^{-5} +31 q^{-6} +83 q^{-7} -8 q^{-8} +20 q^{-9} -164 q^{-10} +2 q^{-11} +132 q^{-12} +52 q^{-13} +62 q^{-14} -253 q^{-15} -61 q^{-16} +150 q^{-17} +115 q^{-18} +128 q^{-19} -308 q^{-20} -121 q^{-21} +136 q^{-22} +148 q^{-23} +188 q^{-24} -320 q^{-25} -152 q^{-26} +107 q^{-27} +147 q^{-28} +221 q^{-29} -291 q^{-30} -155 q^{-31} +63 q^{-32} +123 q^{-33} +235 q^{-34} -227 q^{-35} -142 q^{-36} +6 q^{-37} +77 q^{-38} +230 q^{-39} -133 q^{-40} -106 q^{-41} -52 q^{-42} +12 q^{-43} +199 q^{-44} -41 q^{-45} -49 q^{-46} -77 q^{-47} -46 q^{-48} +136 q^{-49} +12 q^{-50} +8 q^{-51} -60 q^{-52} -68 q^{-53} +67 q^{-54} +17 q^{-55} +32 q^{-56} -25 q^{-57} -51 q^{-58} +23 q^{-59} +4 q^{-60} +24 q^{-61} -4 q^{-62} -24 q^{-63} +8 q^{-64} -2 q^{-65} +9 q^{-66} + q^{-67} -8 q^{-68} +3 q^{-69} - q^{-70} +2 q^{-71} -2 q^{-73} + q^{-74} </math>|J5=<math>q^{25}-2 q^{24}+q^{22}-q^{21}+2 q^{20}+4 q^{19}-4 q^{18}-4 q^{17}+q^{16}-5 q^{15}+3 q^{14}+13 q^{13}+q^{12}-4 q^{11}-5 q^{10}-16 q^9-9 q^8+18 q^7+23 q^6+17 q^5-41 q^3-49 q^2-7 q+41+82 q^{-1} +55 q^{-2} -42 q^{-3} -122 q^{-4} -104 q^{-5} -4 q^{-6} +152 q^{-7} +204 q^{-8} +58 q^{-9} -167 q^{-10} -279 q^{-11} -178 q^{-12} +138 q^{-13} +393 q^{-14} +300 q^{-15} -87 q^{-16} -444 q^{-17} -460 q^{-18} -18 q^{-19} +508 q^{-20} +594 q^{-21} +127 q^{-22} -502 q^{-23} -728 q^{-24} -255 q^{-25} +501 q^{-26} +810 q^{-27} +361 q^{-28} -451 q^{-29} -877 q^{-30} -458 q^{-31} +422 q^{-32} +896 q^{-33} +516 q^{-34} -359 q^{-35} -909 q^{-36} -565 q^{-37} +328 q^{-38} +886 q^{-39} +585 q^{-40} -274 q^{-41} -854 q^{-42} -603 q^{-43} +225 q^{-44} +807 q^{-45} +603 q^{-46} -161 q^{-47} -737 q^{-48} -599 q^{-49} +84 q^{-50} +646 q^{-51} +585 q^{-52} +2 q^{-53} -537 q^{-54} -549 q^{-55} -82 q^{-56} +393 q^{-57} +495 q^{-58} +171 q^{-59} -264 q^{-60} -413 q^{-61} -212 q^{-62} +110 q^{-63} +309 q^{-64} +250 q^{-65} -201 q^{-67} -223 q^{-68} -95 q^{-69} +83 q^{-70} +189 q^{-71} +141 q^{-72} +4 q^{-73} -120 q^{-74} -150 q^{-75} -68 q^{-76} +55 q^{-77} +130 q^{-78} +97 q^{-79} -6 q^{-80} -89 q^{-81} -96 q^{-82} -30 q^{-83} +53 q^{-84} +77 q^{-85} +38 q^{-86} -19 q^{-87} -53 q^{-88} -40 q^{-89} +9 q^{-90} +29 q^{-91} +24 q^{-92} +7 q^{-93} -16 q^{-94} -20 q^{-95} - q^{-96} +9 q^{-97} +4 q^{-98} +5 q^{-99} - q^{-100} -8 q^{-101} +4 q^{-103} - q^{-104} + q^{-106} -2 q^{-107} +2 q^{-109} - q^{-110} </math>|J6=<math>q^{36}-2 q^{35}+q^{33}-q^{32}+2 q^{31}+6 q^{29}-8 q^{28}-4 q^{27}+3 q^{26}-6 q^{25}+4 q^{24}+4 q^{23}+23 q^{22}-14 q^{21}-10 q^{20}+q^{19}-23 q^{18}-5 q^{17}+7 q^{16}+63 q^{15}-8 q^{14}-6 q^{13}+q^{12}-66 q^{11}-48 q^{10}-9 q^9+128 q^8+34 q^7+41 q^6+30 q^5-140 q^4-167 q^3-103 q^2+179 q+123+203 q^{-1} +184 q^{-2} -185 q^{-3} -384 q^{-4} -388 q^{-5} +74 q^{-6} +164 q^{-7} +514 q^{-8} +622 q^{-9} +3 q^{-10} -566 q^{-11} -904 q^{-12} -390 q^{-13} -102 q^{-14} +801 q^{-15} +1362 q^{-16} +649 q^{-17} -403 q^{-18} -1420 q^{-19} -1200 q^{-20} -893 q^{-21} +729 q^{-22} +2109 q^{-23} +1675 q^{-24} +284 q^{-25} -1581 q^{-26} -2006 q^{-27} -2038 q^{-28} +171 q^{-29} +2492 q^{-30} +2666 q^{-31} +1258 q^{-32} -1295 q^{-33} -2446 q^{-34} -3078 q^{-35} -598 q^{-36} +2439 q^{-37} +3266 q^{-38} +2081 q^{-39} -819 q^{-40} -2475 q^{-41} -3703 q^{-42} -1212 q^{-43} +2177 q^{-44} +3457 q^{-45} +2535 q^{-46} -430 q^{-47} -2300 q^{-48} -3934 q^{-49} -1547 q^{-50} +1910 q^{-51} +3419 q^{-52} +2693 q^{-53} -174 q^{-54} -2078 q^{-55} -3930 q^{-56} -1709 q^{-57} +1630 q^{-58} +3254 q^{-59} +2709 q^{-60} +79 q^{-61} -1770 q^{-62} -3764 q^{-63} -1836 q^{-64} +1206 q^{-65} +2904 q^{-66} +2635 q^{-67} +435 q^{-68} -1252 q^{-69} -3366 q^{-70} -1943 q^{-71} +568 q^{-72} +2263 q^{-73} +2379 q^{-74} +848 q^{-75} -500 q^{-76} -2642 q^{-77} -1884 q^{-78} -154 q^{-79} +1349 q^{-80} +1810 q^{-81} +1092 q^{-82} +303 q^{-83} -1635 q^{-84} -1485 q^{-85} -652 q^{-86} +403 q^{-87} +959 q^{-88} +938 q^{-89} +810 q^{-90} -626 q^{-91} -773 q^{-92} -670 q^{-93} -204 q^{-94} +121 q^{-95} +428 q^{-96} +794 q^{-97} +6 q^{-98} -73 q^{-99} -295 q^{-100} -283 q^{-101} -336 q^{-102} -82 q^{-103} +412 q^{-104} +124 q^{-105} +265 q^{-106} +83 q^{-107} -43 q^{-108} -333 q^{-109} -281 q^{-110} +63 q^{-111} -25 q^{-112} +223 q^{-113} +196 q^{-114} +145 q^{-115} -138 q^{-116} -201 q^{-117} -49 q^{-118} -120 q^{-119} +69 q^{-120} +115 q^{-121} +152 q^{-122} -17 q^{-123} -75 q^{-124} -19 q^{-125} -94 q^{-126} -6 q^{-127} +28 q^{-128} +82 q^{-129} +6 q^{-130} -20 q^{-131} +10 q^{-132} -40 q^{-133} -12 q^{-134} - q^{-135} +31 q^{-136} + q^{-137} -8 q^{-138} +11 q^{-139} -11 q^{-140} -4 q^{-141} -3 q^{-142} +10 q^{-143} - q^{-144} -5 q^{-145} +5 q^{-146} -2 q^{-147} - q^{-149} +2 q^{-150} -2 q^{-152} + q^{-153} </math>|J7=<math>q^{49}-2 q^{48}+q^{46}-q^{45}+2 q^{44}+2 q^{42}+2 q^{41}-8 q^{40}-2 q^{39}+2 q^{38}-5 q^{37}+6 q^{36}+2 q^{35}+10 q^{34}+14 q^{33}-20 q^{32}-8 q^{31}-3 q^{30}-18 q^{29}+5 q^{28}+q^{27}+27 q^{26}+47 q^{25}-22 q^{24}-15 q^{23}-17 q^{22}-52 q^{21}-3 q^{20}-13 q^{19}+47 q^{18}+115 q^{17}+2 q^{16}-10 q^{15}-54 q^{14}-129 q^{13}-46 q^{12}-45 q^{11}+87 q^{10}+245 q^9+105 q^8+56 q^7-107 q^6-328 q^5-232 q^4-190 q^3+107 q^2+497 q+437+394 q^{-1} -21 q^{-2} -633 q^{-3} -751 q^{-4} -785 q^{-5} -213 q^{-6} +719 q^{-7} +1108 q^{-8} +1367 q^{-9} +734 q^{-10} -604 q^{-11} -1499 q^{-12} -2163 q^{-13} -1532 q^{-14} +211 q^{-15} +1679 q^{-16} +3035 q^{-17} +2740 q^{-18} +666 q^{-19} -1634 q^{-20} -3948 q^{-21} -4153 q^{-22} -1911 q^{-23} +1075 q^{-24} +4556 q^{-25} +5758 q^{-26} +3676 q^{-27} -81 q^{-28} -4912 q^{-29} -7247 q^{-30} -5570 q^{-31} -1417 q^{-32} +4701 q^{-33} +8508 q^{-34} +7619 q^{-35} +3201 q^{-36} -4138 q^{-37} -9366 q^{-38} -9411 q^{-39} -5108 q^{-40} +3157 q^{-41} +9801 q^{-42} +10930 q^{-43} +6913 q^{-44} -2041 q^{-45} -9850 q^{-46} -11991 q^{-47} -8469 q^{-48} +875 q^{-49} +9628 q^{-50} +12691 q^{-51} +9659 q^{-52} +131 q^{-53} -9240 q^{-54} -13006 q^{-55} -10516 q^{-56} -995 q^{-57} +8840 q^{-58} +13141 q^{-59} +11041 q^{-60} +1579 q^{-61} -8434 q^{-62} -13063 q^{-63} -11373 q^{-64} -2057 q^{-65} +8099 q^{-66} +12985 q^{-67} +11532 q^{-68} +2363 q^{-69} -7775 q^{-70} -12798 q^{-71} -11617 q^{-72} -2689 q^{-73} +7427 q^{-74} +12593 q^{-75} +11682 q^{-76} +3005 q^{-77} -7008 q^{-78} -12276 q^{-79} -11681 q^{-80} -3415 q^{-81} +6397 q^{-82} +11827 q^{-83} +11667 q^{-84} +3922 q^{-85} -5618 q^{-86} -11181 q^{-87} -11530 q^{-88} -4511 q^{-89} +4590 q^{-90} +10267 q^{-91} +11255 q^{-92} +5150 q^{-93} -3337 q^{-94} -9083 q^{-95} -10777 q^{-96} -5738 q^{-97} +1963 q^{-98} +7609 q^{-99} +9957 q^{-100} +6175 q^{-101} -463 q^{-102} -5890 q^{-103} -8898 q^{-104} -6365 q^{-105} -848 q^{-106} +4064 q^{-107} +7429 q^{-108} +6152 q^{-109} +2011 q^{-110} -2213 q^{-111} -5819 q^{-112} -5594 q^{-113} -2696 q^{-114} +633 q^{-115} +4038 q^{-116} +4596 q^{-117} +2962 q^{-118} +657 q^{-119} -2385 q^{-120} -3432 q^{-121} -2731 q^{-122} -1386 q^{-123} +993 q^{-124} +2130 q^{-125} +2124 q^{-126} +1672 q^{-127} -4 q^{-128} -999 q^{-129} -1338 q^{-130} -1489 q^{-131} -512 q^{-132} +135 q^{-133} +529 q^{-134} +1043 q^{-135} +632 q^{-136} +370 q^{-137} +107 q^{-138} -522 q^{-139} -459 q^{-140} -520 q^{-141} -489 q^{-142} +56 q^{-143} +163 q^{-144} +444 q^{-145} +608 q^{-146} +232 q^{-147} +118 q^{-148} -241 q^{-149} -546 q^{-150} -343 q^{-151} -278 q^{-152} +30 q^{-153} +369 q^{-154} +315 q^{-155} +358 q^{-156} +105 q^{-157} -224 q^{-158} -217 q^{-159} -294 q^{-160} -160 q^{-161} +65 q^{-162} +109 q^{-163} +245 q^{-164} +166 q^{-165} -26 q^{-166} -45 q^{-167} -134 q^{-168} -113 q^{-169} -24 q^{-170} -19 q^{-171} +92 q^{-172} +93 q^{-173} +11 q^{-174} +10 q^{-175} -42 q^{-176} -35 q^{-177} -10 q^{-178} -32 q^{-179} +18 q^{-180} +36 q^{-181} +6 q^{-182} +8 q^{-183} -14 q^{-184} -6 q^{-185} +6 q^{-186} -16 q^{-187} +10 q^{-189} +2 q^{-190} +3 q^{-191} -6 q^{-192} - q^{-193} +6 q^{-194} -4 q^{-195} -2 q^{-196} +2 q^{-197} + q^{-199} -2 q^{-200} +2 q^{-202} - q^{-203} </math>}} |
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coloured_jones_4 = <math>q^{16}-2 q^{15}+q^{13}-q^{12}+6 q^{11}-6 q^{10}+q^8-7 q^7+15 q^6-12 q^5+7 q^4+7 q^3-22 q^2+18 q-28+24 q^{-1} +32 q^{-2} -32 q^{-3} +14 q^{-4} -78 q^{-5} +31 q^{-6} +83 q^{-7} -8 q^{-8} +20 q^{-9} -164 q^{-10} +2 q^{-11} +132 q^{-12} +52 q^{-13} +62 q^{-14} -253 q^{-15} -61 q^{-16} +150 q^{-17} +115 q^{-18} +128 q^{-19} -308 q^{-20} -121 q^{-21} +136 q^{-22} +148 q^{-23} +188 q^{-24} -320 q^{-25} -152 q^{-26} +107 q^{-27} +147 q^{-28} +221 q^{-29} -291 q^{-30} -155 q^{-31} +63 q^{-32} +123 q^{-33} +235 q^{-34} -227 q^{-35} -142 q^{-36} +6 q^{-37} +77 q^{-38} +230 q^{-39} -133 q^{-40} -106 q^{-41} -52 q^{-42} +12 q^{-43} +199 q^{-44} -41 q^{-45} -49 q^{-46} -77 q^{-47} -46 q^{-48} +136 q^{-49} +12 q^{-50} +8 q^{-51} -60 q^{-52} -68 q^{-53} +67 q^{-54} +17 q^{-55} +32 q^{-56} -25 q^{-57} -51 q^{-58} +23 q^{-59} +4 q^{-60} +24 q^{-61} -4 q^{-62} -24 q^{-63} +8 q^{-64} -2 q^{-65} +9 q^{-66} + q^{-67} -8 q^{-68} +3 q^{-69} - q^{-70} +2 q^{-71} -2 q^{-73} + q^{-74} </math> | |
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coloured_jones_5 = <math>q^{25}-2 q^{24}+q^{22}-q^{21}+2 q^{20}+4 q^{19}-4 q^{18}-4 q^{17}+q^{16}-5 q^{15}+3 q^{14}+13 q^{13}+q^{12}-4 q^{11}-5 q^{10}-16 q^9-9 q^8+18 q^7+23 q^6+17 q^5-41 q^3-49 q^2-7 q+41+82 q^{-1} +55 q^{-2} -42 q^{-3} -122 q^{-4} -104 q^{-5} -4 q^{-6} +152 q^{-7} +204 q^{-8} +58 q^{-9} -167 q^{-10} -279 q^{-11} -178 q^{-12} +138 q^{-13} +393 q^{-14} +300 q^{-15} -87 q^{-16} -444 q^{-17} -460 q^{-18} -18 q^{-19} +508 q^{-20} +594 q^{-21} +127 q^{-22} -502 q^{-23} -728 q^{-24} -255 q^{-25} +501 q^{-26} +810 q^{-27} +361 q^{-28} -451 q^{-29} -877 q^{-30} -458 q^{-31} +422 q^{-32} +896 q^{-33} +516 q^{-34} -359 q^{-35} -909 q^{-36} -565 q^{-37} +328 q^{-38} +886 q^{-39} +585 q^{-40} -274 q^{-41} -854 q^{-42} -603 q^{-43} +225 q^{-44} +807 q^{-45} +603 q^{-46} -161 q^{-47} -737 q^{-48} -599 q^{-49} +84 q^{-50} +646 q^{-51} +585 q^{-52} +2 q^{-53} -537 q^{-54} -549 q^{-55} -82 q^{-56} +393 q^{-57} +495 q^{-58} +171 q^{-59} -264 q^{-60} -413 q^{-61} -212 q^{-62} +110 q^{-63} +309 q^{-64} +250 q^{-65} -201 q^{-67} -223 q^{-68} -95 q^{-69} +83 q^{-70} +189 q^{-71} +141 q^{-72} +4 q^{-73} -120 q^{-74} -150 q^{-75} -68 q^{-76} +55 q^{-77} +130 q^{-78} +97 q^{-79} -6 q^{-80} -89 q^{-81} -96 q^{-82} -30 q^{-83} +53 q^{-84} +77 q^{-85} +38 q^{-86} -19 q^{-87} -53 q^{-88} -40 q^{-89} +9 q^{-90} +29 q^{-91} +24 q^{-92} +7 q^{-93} -16 q^{-94} -20 q^{-95} - q^{-96} +9 q^{-97} +4 q^{-98} +5 q^{-99} - q^{-100} -8 q^{-101} +4 q^{-103} - q^{-104} + q^{-106} -2 q^{-107} +2 q^{-109} - q^{-110} </math> | |
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{{Computer Talk Header}} |
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coloured_jones_6 = <math>q^{36}-2 q^{35}+q^{33}-q^{32}+2 q^{31}+6 q^{29}-8 q^{28}-4 q^{27}+3 q^{26}-6 q^{25}+4 q^{24}+4 q^{23}+23 q^{22}-14 q^{21}-10 q^{20}+q^{19}-23 q^{18}-5 q^{17}+7 q^{16}+63 q^{15}-8 q^{14}-6 q^{13}+q^{12}-66 q^{11}-48 q^{10}-9 q^9+128 q^8+34 q^7+41 q^6+30 q^5-140 q^4-167 q^3-103 q^2+179 q+123+203 q^{-1} +184 q^{-2} -185 q^{-3} -384 q^{-4} -388 q^{-5} +74 q^{-6} +164 q^{-7} +514 q^{-8} +622 q^{-9} +3 q^{-10} -566 q^{-11} -904 q^{-12} -390 q^{-13} -102 q^{-14} +801 q^{-15} +1362 q^{-16} +649 q^{-17} -403 q^{-18} -1420 q^{-19} -1200 q^{-20} -893 q^{-21} +729 q^{-22} +2109 q^{-23} +1675 q^{-24} +284 q^{-25} -1581 q^{-26} -2006 q^{-27} -2038 q^{-28} +171 q^{-29} +2492 q^{-30} +2666 q^{-31} +1258 q^{-32} -1295 q^{-33} -2446 q^{-34} -3078 q^{-35} -598 q^{-36} +2439 q^{-37} +3266 q^{-38} +2081 q^{-39} -819 q^{-40} -2475 q^{-41} -3703 q^{-42} -1212 q^{-43} +2177 q^{-44} +3457 q^{-45} +2535 q^{-46} -430 q^{-47} -2300 q^{-48} -3934 q^{-49} -1547 q^{-50} +1910 q^{-51} +3419 q^{-52} +2693 q^{-53} -174 q^{-54} -2078 q^{-55} -3930 q^{-56} -1709 q^{-57} +1630 q^{-58} +3254 q^{-59} +2709 q^{-60} +79 q^{-61} -1770 q^{-62} -3764 q^{-63} -1836 q^{-64} +1206 q^{-65} +2904 q^{-66} +2635 q^{-67} +435 q^{-68} -1252 q^{-69} -3366 q^{-70} -1943 q^{-71} +568 q^{-72} +2263 q^{-73} +2379 q^{-74} +848 q^{-75} -500 q^{-76} -2642 q^{-77} -1884 q^{-78} -154 q^{-79} +1349 q^{-80} +1810 q^{-81} +1092 q^{-82} +303 q^{-83} -1635 q^{-84} -1485 q^{-85} -652 q^{-86} +403 q^{-87} +959 q^{-88} +938 q^{-89} +810 q^{-90} -626 q^{-91} -773 q^{-92} -670 q^{-93} -204 q^{-94} +121 q^{-95} +428 q^{-96} +794 q^{-97} +6 q^{-98} -73 q^{-99} -295 q^{-100} -283 q^{-101} -336 q^{-102} -82 q^{-103} +412 q^{-104} +124 q^{-105} +265 q^{-106} +83 q^{-107} -43 q^{-108} -333 q^{-109} -281 q^{-110} +63 q^{-111} -25 q^{-112} +223 q^{-113} +196 q^{-114} +145 q^{-115} -138 q^{-116} -201 q^{-117} -49 q^{-118} -120 q^{-119} +69 q^{-120} +115 q^{-121} +152 q^{-122} -17 q^{-123} -75 q^{-124} -19 q^{-125} -94 q^{-126} -6 q^{-127} +28 q^{-128} +82 q^{-129} +6 q^{-130} -20 q^{-131} +10 q^{-132} -40 q^{-133} -12 q^{-134} - q^{-135} +31 q^{-136} + q^{-137} -8 q^{-138} +11 q^{-139} -11 q^{-140} -4 q^{-141} -3 q^{-142} +10 q^{-143} - q^{-144} -5 q^{-145} +5 q^{-146} -2 q^{-147} - q^{-149} +2 q^{-150} -2 q^{-152} + q^{-153} </math> | |
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coloured_jones_7 = <math>q^{49}-2 q^{48}+q^{46}-q^{45}+2 q^{44}+2 q^{42}+2 q^{41}-8 q^{40}-2 q^{39}+2 q^{38}-5 q^{37}+6 q^{36}+2 q^{35}+10 q^{34}+14 q^{33}-20 q^{32}-8 q^{31}-3 q^{30}-18 q^{29}+5 q^{28}+q^{27}+27 q^{26}+47 q^{25}-22 q^{24}-15 q^{23}-17 q^{22}-52 q^{21}-3 q^{20}-13 q^{19}+47 q^{18}+115 q^{17}+2 q^{16}-10 q^{15}-54 q^{14}-129 q^{13}-46 q^{12}-45 q^{11}+87 q^{10}+245 q^9+105 q^8+56 q^7-107 q^6-328 q^5-232 q^4-190 q^3+107 q^2+497 q+437+394 q^{-1} -21 q^{-2} -633 q^{-3} -751 q^{-4} -785 q^{-5} -213 q^{-6} +719 q^{-7} +1108 q^{-8} +1367 q^{-9} +734 q^{-10} -604 q^{-11} -1499 q^{-12} -2163 q^{-13} -1532 q^{-14} +211 q^{-15} +1679 q^{-16} +3035 q^{-17} +2740 q^{-18} +666 q^{-19} -1634 q^{-20} -3948 q^{-21} -4153 q^{-22} -1911 q^{-23} +1075 q^{-24} +4556 q^{-25} +5758 q^{-26} +3676 q^{-27} -81 q^{-28} -4912 q^{-29} -7247 q^{-30} -5570 q^{-31} -1417 q^{-32} +4701 q^{-33} +8508 q^{-34} +7619 q^{-35} +3201 q^{-36} -4138 q^{-37} -9366 q^{-38} -9411 q^{-39} -5108 q^{-40} +3157 q^{-41} +9801 q^{-42} +10930 q^{-43} +6913 q^{-44} -2041 q^{-45} -9850 q^{-46} -11991 q^{-47} -8469 q^{-48} +875 q^{-49} +9628 q^{-50} +12691 q^{-51} +9659 q^{-52} +131 q^{-53} -9240 q^{-54} -13006 q^{-55} -10516 q^{-56} -995 q^{-57} +8840 q^{-58} +13141 q^{-59} +11041 q^{-60} +1579 q^{-61} -8434 q^{-62} -13063 q^{-63} -11373 q^{-64} -2057 q^{-65} +8099 q^{-66} +12985 q^{-67} +11532 q^{-68} +2363 q^{-69} -7775 q^{-70} -12798 q^{-71} -11617 q^{-72} -2689 q^{-73} +7427 q^{-74} +12593 q^{-75} +11682 q^{-76} +3005 q^{-77} -7008 q^{-78} -12276 q^{-79} -11681 q^{-80} -3415 q^{-81} +6397 q^{-82} +11827 q^{-83} +11667 q^{-84} +3922 q^{-85} -5618 q^{-86} -11181 q^{-87} -11530 q^{-88} -4511 q^{-89} +4590 q^{-90} +10267 q^{-91} +11255 q^{-92} +5150 q^{-93} -3337 q^{-94} -9083 q^{-95} -10777 q^{-96} -5738 q^{-97} +1963 q^{-98} +7609 q^{-99} +9957 q^{-100} +6175 q^{-101} -463 q^{-102} -5890 q^{-103} -8898 q^{-104} -6365 q^{-105} -848 q^{-106} +4064 q^{-107} +7429 q^{-108} +6152 q^{-109} +2011 q^{-110} -2213 q^{-111} -5819 q^{-112} -5594 q^{-113} -2696 q^{-114} +633 q^{-115} +4038 q^{-116} +4596 q^{-117} +2962 q^{-118} +657 q^{-119} -2385 q^{-120} -3432 q^{-121} -2731 q^{-122} -1386 q^{-123} +993 q^{-124} +2130 q^{-125} +2124 q^{-126} +1672 q^{-127} -4 q^{-128} -999 q^{-129} -1338 q^{-130} -1489 q^{-131} -512 q^{-132} +135 q^{-133} +529 q^{-134} +1043 q^{-135} +632 q^{-136} +370 q^{-137} +107 q^{-138} -522 q^{-139} -459 q^{-140} -520 q^{-141} -489 q^{-142} +56 q^{-143} +163 q^{-144} +444 q^{-145} +608 q^{-146} +232 q^{-147} +118 q^{-148} -241 q^{-149} -546 q^{-150} -343 q^{-151} -278 q^{-152} +30 q^{-153} +369 q^{-154} +315 q^{-155} +358 q^{-156} +105 q^{-157} -224 q^{-158} -217 q^{-159} -294 q^{-160} -160 q^{-161} +65 q^{-162} +109 q^{-163} +245 q^{-164} +166 q^{-165} -26 q^{-166} -45 q^{-167} -134 q^{-168} -113 q^{-169} -24 q^{-170} -19 q^{-171} +92 q^{-172} +93 q^{-173} +11 q^{-174} +10 q^{-175} -42 q^{-176} -35 q^{-177} -10 q^{-178} -32 q^{-179} +18 q^{-180} +36 q^{-181} +6 q^{-182} +8 q^{-183} -14 q^{-184} -6 q^{-185} +6 q^{-186} -16 q^{-187} +10 q^{-189} +2 q^{-190} +3 q^{-191} -6 q^{-192} - q^{-193} +6 q^{-194} -4 q^{-195} -2 q^{-196} +2 q^{-197} + q^{-199} -2 q^{-200} +2 q^{-202} - q^{-203} </math> | |
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<table> |
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computer_talk = |
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<table> |
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<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:= </pre></td> |
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<td align=left><pre style="color: red; border: 0px; padding: 0em"><< KnotTheory`</pre></td> |
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<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:= </pre></td> |
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</tr> |
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<td align=left><pre style="color: red; border: 0px; padding: 0em"><< KnotTheory`</pre></td> |
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</tr> |
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<tr valign=top><td colspan=2>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</td></tr> |
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<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[9, 12]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre |
<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[9, 12]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[5, 16, 6, 17], X[11, 1, 12, 18], |
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X[17, 13, 18, 12], X[7, 14, 8, 15], X[13, 8, 14, 9], X[15, 6, 16, 7], |
X[17, 13, 18, 12], X[7, 14, 8, 15], X[13, 8, 14, 9], X[15, 6, 16, 7], |
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X[9, 2, 10, 3]]</nowiki></pre></td></tr> |
X[9, 2, 10, 3]]</nowiki></pre></td></tr> |
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<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[9, 12]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[3]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[-1, 9, -2, 1, -3, 8, -6, 7, -9, 2, -4, 5, -7, 6, -8, 3, -5, 4]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre |
<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[9, 12]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>DTCode[4, 10, 16, 14, 2, 18, 8, 6, 12]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[ |
<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[9, 12]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[ |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[5, {-1, -1, 2, -1, -3, 2, -3, -4, 3, -4}]</nowiki></pre></td></tr> |
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<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> |
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<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{5, 10}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre |
<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[9, 12]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>5</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: |
<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[9, 12]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:9_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> |
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<tr valign=top><td><pre |
<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[9, 12]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Reversible, 1, 2, 2, {4, 6}, 1}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[ |
<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[9, 12]][t]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[ |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 9 2 |
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<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[9, 12]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:9_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> |
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<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[9, 12]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Reversible, 1, 2, 2, {4, 6}, 1}</nowiki></pre></td></tr> |
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<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[9, 12]][t]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 9 2 |
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-13 - -- + - + 9 t - 2 t |
-13 - -- + - + 9 t - 2 t |
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2 t |
2 t |
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t</nowiki></pre></td></tr> |
t</nowiki></pre></td></tr> |
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<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[9, 12]][z]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[11]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 4 |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[11]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 4 |
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1 + z - 2 z</nowiki></pre></td></tr> |
1 + z - 2 z</nowiki></pre></td></tr> |
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<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> |
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<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[12]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[9, 12], Knot[11, NonAlternating, 84]}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre |
<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[9, 12]], KnotSignature[Knot[9, 12]]}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[13]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{35, -2}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[ |
<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[9, 12]][q]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[ |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -8 2 3 5 6 6 5 4 |
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<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[9, 12]][q]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -8 2 3 5 6 6 5 4 |
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-2 - q + -- - -- + -- - -- + -- - -- + - + q |
-2 - q + -- - -- + -- - -- + -- - -- + - + q |
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7 6 5 4 3 2 q |
7 6 5 4 3 2 q |
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q q q q q q</nowiki></pre></td></tr> |
q q q q q q</nowiki></pre></td></tr> |
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<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> |
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<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[9, 12], Knot[11, NonAlternating, 15]}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre |
<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[9, 12]][q]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[16]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -26 -24 -22 -18 2 -14 -10 -6 -4 2 4 |
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<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[9, 12]][q]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[16]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -26 -24 -22 -18 2 -14 -10 -6 -4 2 4 |
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-q - q + q + q + --- - q - q + q - q + -- + q |
-q - q + q + q + --- - q - q + q - q + -- + q |
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16 2 |
16 2 |
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q q</nowiki></pre></td></tr> |
q q</nowiki></pre></td></tr> |
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<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[9, 12]][a, z]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[17]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 4 6 8 2 2 2 4 2 6 2 2 4 4 4 |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[17]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 4 6 8 2 2 2 4 2 6 2 2 4 4 4 |
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1 - a + 2 a - a + z - a z - a z + 2 a z - a z - a z</nowiki></pre></td></tr> |
1 - a + 2 a - a + z - a z - a z + 2 a z - a z - a z</nowiki></pre></td></tr> |
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<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[9, 12]][a, z]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[18]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 4 6 8 3 5 7 9 2 2 2 |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[18]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 4 6 8 3 5 7 9 2 2 2 |
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1 - a - 2 a - a - 2 a z - 4 a z - a z + a z - 2 z - 2 a z + |
1 - a - 2 a - a - 2 a z - 4 a z - a z + a z - 2 z - 2 a z + |
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Line 163: | Line 112: | ||
7 7 4 8 6 8 |
7 7 4 8 6 8 |
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2 a z + a z + a z</nowiki></pre></td></tr> |
2 a z + a z + a z</nowiki></pre></td></tr> |
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<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[9, 12]], Vassiliev[3][Knot[9, 12]]}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[19]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{1, -3}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre |
<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[9, 12]][q, t]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[20]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>2 3 1 1 1 2 1 3 2 |
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<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[9, 12]][q, t]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[20]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>2 3 1 1 1 2 1 3 2 |
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-- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + |
-- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + |
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3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 |
3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 |
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Line 177: | Line 124: | ||
9 3 7 3 7 2 5 2 5 3 q |
9 3 7 3 7 2 5 2 5 3 q |
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q t q t q t q t q t q t</nowiki></pre></td></tr> |
q t q t q t q t q t q t</nowiki></pre></td></tr> |
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<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[9, 12], 2][q]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[21]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -23 2 5 7 2 14 11 9 25 13 19 |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[21]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -23 2 5 7 2 14 11 9 25 13 19 |
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-7 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + |
-7 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + |
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22 20 19 18 17 16 15 14 13 12 |
22 20 19 18 17 16 15 14 13 12 |
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Line 188: | Line 134: | ||
11 10 9 8 7 6 5 4 3 2 |
11 10 9 8 7 6 5 4 3 2 |
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q q q q q q q q q q</nowiki></pre></td></tr> |
q q q q q q q q q q</nowiki></pre></td></tr> |
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</table> }} |
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</table> |
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{| width=100% |
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|align=left|See/edit the [[Rolfsen_Splice_Template]]. |
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Back to the [[#top|top]]. |
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|align=right|{{Knot Navigation Links|ext=gif}} |
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|} |
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[[Category:Knot Page]] |
Revision as of 10:42, 30 August 2005
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![]() (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 9 12's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
Planar diagram presentation | X1425 X3,10,4,11 X5,16,6,17 X11,1,12,18 X17,13,18,12 X7,14,8,15 X13,8,14,9 X15,6,16,7 X9,2,10,3 |
Gauss code | -1, 9, -2, 1, -3, 8, -6, 7, -9, 2, -4, 5, -7, 6, -8, 3, -5, 4 |
Dowker-Thistlethwaite code | 4 10 16 14 2 18 8 6 12 |
Conway Notation | [4212] |
Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||||
Length is 10, width is 5, Braid index is 5 |
![]() |
![]() [{12, 2}, {1, 10}, {6, 11}, {10, 12}, {9, 3}, {2, 8}, {7, 9}, {8, 6}, {4, 7}, {3, 5}, {11, 4}, {5, 1}] |
[edit Notes on presentations of 9 12]
KnotTheory`
. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["9 12"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X1425 X3,10,4,11 X5,16,6,17 X11,1,12,18 X17,13,18,12 X7,14,8,15 X13,8,14,9 X15,6,16,7 X9,2,10,3 |
In[5]:=
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GaussCode[K]
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Out[5]=
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-1, 9, -2, 1, -3, 8, -6, 7, -9, 2, -4, 5, -7, 6, -8, 3, -5, 4 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 10 16 14 2 18 8 6 12 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
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ConwayNotation[K]
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Out[8]=
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[4212] |
In[9]:=
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br = BR[K]
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KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
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Out[9]=
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In[10]:=
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{First[br], Crossings[br], BraidIndex[K]}
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KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
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KnotTheory::loading: Loading precomputed data in IndianaData`.
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Out[10]=
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{ 5, 10, 5 } |
In[11]:=
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Show[BraidPlot[br]]
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Out[11]=
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-Graphics- |
In[12]:=
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Show[DrawMorseLink[K]]
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KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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![]() |
Out[12]=
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-Graphics- |
In[13]:=
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ap = ArcPresentation[K]
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Out[13]=
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ArcPresentation[{12, 2}, {1, 10}, {6, 11}, {10, 12}, {9, 3}, {2, 8}, {7, 9}, {8, 6}, {4, 7}, {3, 5}, {11, 4}, {5, 1}] |
In[14]:=
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Draw[ap]
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![]() |
Out[14]=
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-Graphics- |
Three dimensional invariants
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Four dimensional invariants
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Polynomial invariants
A1 Invariants.
Weight | Invariant |
---|---|
1 | |
2 | |
3 | |
4 | |
5 |
A2 Invariants.
Weight | Invariant |
---|---|
1,0 | |
1,1 | |
2,0 |
A3 Invariants.
Weight | Invariant |
---|---|
0,1,0 | |
1,0,0 |
B2 Invariants.
Weight | Invariant |
---|---|
0,1 | |
1,0 |
G2 Invariants.
Weight | Invariant |
---|---|
1,0 |
.
KnotTheory`
, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
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In[3]:=
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K = Knot["9 12"];
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In[4]:=
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Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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In[5]:=
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Conway[K][z]
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Out[5]=
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In[6]:=
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Alexander[K, 2][t]
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KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
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Out[6]=
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In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 35, -2 } |
In[8]:=
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Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
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In[9]:=
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HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
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In[10]:=
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Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
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"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11n84,}
Same Jones Polynomial (up to mirroring, ): {K11n15,}
KnotTheory`
. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
|
AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["9 12"];
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In[4]:=
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{A = Alexander[K][t], J = Jones[K][q]}
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[4]=
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{ , } |
In[5]:=
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DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
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KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
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KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
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Out[5]=
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{K11n84,} |
In[6]:=
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DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
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KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
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Out[6]=
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{K11n15,} |
Vassiliev invariants
V2 and V3: | (1, -3) |
V2,1 through V6,9: |
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V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.
Khovanov Homology
The coefficients of the monomials are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where -2 is the signature of 9 12. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
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Integral Khovanov Homology
(db, data source) |
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The Coloured Jones Polynomials
2 | |
3 | |
4 | |
5 | |
6 | |
7 |
Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`
. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.
Modifying This Page
Read me first: Modifying Knot Pages
See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top. |
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