9 45: Difference between revisions
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{{Rolfsen Knot Page| |
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n = 9 | |
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k = 45 | |
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KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-4,3,-1,2,9,-5,-3,4,-2,7,-8,-9,5,6,-7,8,-6/goTop.html | |
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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=45|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-4,3,-1,2,9,-5,-3,4,-2,7,-8,-9,5,6,-7,8,-6/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:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr> |
<tr><td>[[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr> |
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<tr><td>[[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]]</td></tr> |
<tr><td>[[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]]</td></tr> |
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<tr><td>[[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: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: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:BraidPart4.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]]</td></tr> |
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</table> |
</table> | |
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braid_crossings = 9 | |
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braid_width = 4 | |
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[[Invariants from Braid Theory|Length]] is 9, width is 4. |
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braid_index = 4 | |
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same_alexander = | |
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[[Invariants from Braid Theory|Braid index]] is 4. |
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same_jones = | |
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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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{...} |
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Same [[The Jones Polynomial|Jones Polynomial]] (up to mirroring, <math>q\leftrightarrow q^{-1}</math>): |
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{...} |
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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=16.6667%><table cellpadding=0 cellspacing=0> |
<td width=16.6667%><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=8.33333%>-7</td ><td width=8.33333%>-6</td ><td width=8.33333%>-5</td ><td width=8.33333%>-4</td ><td width=8.33333%>-3</td ><td width=8.33333%>-2</td ><td width=8.33333%>-1</td ><td width=8.33333%>0</td ><td width=16.6667%>χ</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 bgcolor=yellow>2</td><td>2</td></tr> |
<tr align=center><td>-1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td>2</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 bgcolor=yellow>2</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 bgcolor=yellow>2</td><td bgcolor=yellow>1</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>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>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>-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>-1</td></tr> |
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</table> |
</table> | |
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coloured_jones_2 = <math> q^{-1} + q^{-2} -5 q^{-3} +4 q^{-4} +6 q^{-5} -13 q^{-6} +6 q^{-7} +12 q^{-8} -19 q^{-9} +5 q^{-10} +15 q^{-11} -18 q^{-12} + q^{-13} +15 q^{-14} -13 q^{-15} -3 q^{-16} +12 q^{-17} -6 q^{-18} -4 q^{-19} +6 q^{-20} - q^{-21} -2 q^{-22} + q^{-23} </math> | |
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coloured_jones_3 = <math>2 q^{-1} -2 q^{-2} - q^{-3} -3 q^{-4} +10 q^{-5} +2 q^{-6} -12 q^{-7} -10 q^{-8} +20 q^{-9} +17 q^{-10} -23 q^{-11} -28 q^{-12} +28 q^{-13} +35 q^{-14} -26 q^{-15} -43 q^{-16} +25 q^{-17} +45 q^{-18} -20 q^{-19} -48 q^{-20} +16 q^{-21} +45 q^{-22} -9 q^{-23} -43 q^{-24} +3 q^{-25} +38 q^{-26} +6 q^{-27} -34 q^{-28} -11 q^{-29} +26 q^{-30} +16 q^{-31} -18 q^{-32} -19 q^{-33} +11 q^{-34} +17 q^{-35} -3 q^{-36} -14 q^{-37} - q^{-38} +9 q^{-39} +3 q^{-40} -5 q^{-41} -2 q^{-42} + q^{-43} +2 q^{-44} - q^{-45} </math> | |
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{{Display Coloured Jones|J2=<math> q^{-1} + q^{-2} -5 q^{-3} +4 q^{-4} +6 q^{-5} -13 q^{-6} +6 q^{-7} +12 q^{-8} -19 q^{-9} +5 q^{-10} +15 q^{-11} -18 q^{-12} + q^{-13} +15 q^{-14} -13 q^{-15} -3 q^{-16} +12 q^{-17} -6 q^{-18} -4 q^{-19} +6 q^{-20} - q^{-21} -2 q^{-22} + q^{-23} </math>|J3=<math>2 q^{-1} -2 q^{-2} - q^{-3} -3 q^{-4} +10 q^{-5} +2 q^{-6} -12 q^{-7} -10 q^{-8} +20 q^{-9} +17 q^{-10} -23 q^{-11} -28 q^{-12} +28 q^{-13} +35 q^{-14} -26 q^{-15} -43 q^{-16} +25 q^{-17} +45 q^{-18} -20 q^{-19} -48 q^{-20} +16 q^{-21} +45 q^{-22} -9 q^{-23} -43 q^{-24} +3 q^{-25} +38 q^{-26} +6 q^{-27} -34 q^{-28} -11 q^{-29} +26 q^{-30} +16 q^{-31} -18 q^{-32} -19 q^{-33} +11 q^{-34} +17 q^{-35} -3 q^{-36} -14 q^{-37} - q^{-38} +9 q^{-39} +3 q^{-40} -5 q^{-41} -2 q^{-42} + q^{-43} +2 q^{-44} - q^{-45} </math>|J4=<math>1+ q^{-1} -3 q^{-2} -4 q^{-3} +4 q^{-4} +5 q^{-5} +10 q^{-6} -8 q^{-7} -27 q^{-8} + q^{-9} +18 q^{-10} +47 q^{-11} -3 q^{-12} -74 q^{-13} -28 q^{-14} +21 q^{-15} +109 q^{-16} +33 q^{-17} -118 q^{-18} -80 q^{-19} - q^{-20} +162 q^{-21} +85 q^{-22} -133 q^{-23} -118 q^{-24} -38 q^{-25} +181 q^{-26} +123 q^{-27} -123 q^{-28} -126 q^{-29} -67 q^{-30} +170 q^{-31} +133 q^{-32} -99 q^{-33} -110 q^{-34} -88 q^{-35} +141 q^{-36} +129 q^{-37} -65 q^{-38} -83 q^{-39} -104 q^{-40} +97 q^{-41} +115 q^{-42} -21 q^{-43} -45 q^{-44} -113 q^{-45} +41 q^{-46} +87 q^{-47} +18 q^{-48} + q^{-49} -98 q^{-50} -8 q^{-51} +41 q^{-52} +31 q^{-53} +38 q^{-54} -57 q^{-55} -26 q^{-56} - q^{-57} +14 q^{-58} +44 q^{-59} -15 q^{-60} -14 q^{-61} -15 q^{-62} -5 q^{-63} +24 q^{-64} + q^{-65} -7 q^{-67} -7 q^{-68} +6 q^{-69} + q^{-70} +2 q^{-71} - q^{-72} -2 q^{-73} + q^{-74} </math>|J5=<math>2 q-2-3 q^{-2} -3 q^{-3} +6 q^{-4} +15 q^{-5} -2 q^{-6} -9 q^{-7} -20 q^{-8} -28 q^{-9} +19 q^{-10} +61 q^{-11} +41 q^{-12} -9 q^{-13} -83 q^{-14} -114 q^{-15} -15 q^{-16} +138 q^{-17} +177 q^{-18} +67 q^{-19} -152 q^{-20} -282 q^{-21} -147 q^{-22} +167 q^{-23} +369 q^{-24} +245 q^{-25} -143 q^{-26} -450 q^{-27} -352 q^{-28} +109 q^{-29} +497 q^{-30} +447 q^{-31} -49 q^{-32} -531 q^{-33} -517 q^{-34} -2 q^{-35} +524 q^{-36} +566 q^{-37} +58 q^{-38} -517 q^{-39} -588 q^{-40} -90 q^{-41} +484 q^{-42} +590 q^{-43} +125 q^{-44} -458 q^{-45} -579 q^{-46} -140 q^{-47} +415 q^{-48} +559 q^{-49} +164 q^{-50} -375 q^{-51} -531 q^{-52} -182 q^{-53} +316 q^{-54} +500 q^{-55} +213 q^{-56} -259 q^{-57} -457 q^{-58} -237 q^{-59} +179 q^{-60} +408 q^{-61} +264 q^{-62} -101 q^{-63} -345 q^{-64} -272 q^{-65} +15 q^{-66} +264 q^{-67} +274 q^{-68} +55 q^{-69} -177 q^{-70} -244 q^{-71} -111 q^{-72} +87 q^{-73} +194 q^{-74} +142 q^{-75} -10 q^{-76} -132 q^{-77} -137 q^{-78} -45 q^{-79} +60 q^{-80} +113 q^{-81} +74 q^{-82} -9 q^{-83} -68 q^{-84} -74 q^{-85} -28 q^{-86} +28 q^{-87} +54 q^{-88} +41 q^{-89} +4 q^{-90} -32 q^{-91} -36 q^{-92} -14 q^{-93} +7 q^{-94} +23 q^{-95} +20 q^{-96} + q^{-97} -13 q^{-98} -10 q^{-99} -5 q^{-100} +9 q^{-102} +5 q^{-103} -2 q^{-104} -2 q^{-105} - q^{-106} -2 q^{-107} + q^{-108} +2 q^{-109} - q^{-110} </math>|J6=<math>q^3+q^2-3 q-2+ q^{-2} +2 q^{-3} +9 q^{-4} +12 q^{-5} -9 q^{-6} -29 q^{-7} -21 q^{-8} -3 q^{-9} +12 q^{-10} +64 q^{-11} +80 q^{-12} +12 q^{-13} -96 q^{-14} -144 q^{-15} -105 q^{-16} -30 q^{-17} +189 q^{-18} +329 q^{-19} +209 q^{-20} -109 q^{-21} -385 q^{-22} -456 q^{-23} -320 q^{-24} +241 q^{-25} +743 q^{-26} +736 q^{-27} +164 q^{-28} -552 q^{-29} -989 q^{-30} -969 q^{-31} -14 q^{-32} +1061 q^{-33} +1437 q^{-34} +780 q^{-35} -406 q^{-36} -1393 q^{-37} -1736 q^{-38} -569 q^{-39} +1049 q^{-40} +1957 q^{-41} +1449 q^{-42} +7 q^{-43} -1459 q^{-44} -2262 q^{-45} -1126 q^{-46} +778 q^{-47} +2124 q^{-48} +1861 q^{-49} +417 q^{-50} -1283 q^{-51} -2432 q^{-52} -1447 q^{-53} +486 q^{-54} +2046 q^{-55} +1969 q^{-56} +654 q^{-57} -1066 q^{-58} -2371 q^{-59} -1536 q^{-60} +291 q^{-61} +1886 q^{-62} +1902 q^{-63} +755 q^{-64} -874 q^{-65} -2210 q^{-66} -1524 q^{-67} +129 q^{-68} +1679 q^{-69} +1773 q^{-70} +846 q^{-71} -640 q^{-72} -1974 q^{-73} -1505 q^{-74} -108 q^{-75} +1368 q^{-76} +1599 q^{-77} +993 q^{-78} -282 q^{-79} -1617 q^{-80} -1467 q^{-81} -445 q^{-82} +899 q^{-83} +1317 q^{-84} +1134 q^{-85} +183 q^{-86} -1086 q^{-87} -1302 q^{-88} -766 q^{-89} +312 q^{-90} +848 q^{-91} +1107 q^{-92} +600 q^{-93} -430 q^{-94} -901 q^{-95} -866 q^{-96} -208 q^{-97} +247 q^{-98} +788 q^{-99} +741 q^{-100} +132 q^{-101} -340 q^{-102} -623 q^{-103} -415 q^{-104} -240 q^{-105} +291 q^{-106} +515 q^{-107} +348 q^{-108} +98 q^{-109} -203 q^{-110} -256 q^{-111} -373 q^{-112} -74 q^{-113} +144 q^{-114} +215 q^{-115} +199 q^{-116} +72 q^{-117} +7 q^{-118} -209 q^{-119} -135 q^{-120} -63 q^{-121} +17 q^{-122} +73 q^{-123} +90 q^{-124} +110 q^{-125} -36 q^{-126} -38 q^{-127} -60 q^{-128} -42 q^{-129} -24 q^{-130} +14 q^{-131} +68 q^{-132} +11 q^{-133} +15 q^{-134} -9 q^{-135} -15 q^{-136} -27 q^{-137} -13 q^{-138} +18 q^{-139} +2 q^{-140} +11 q^{-141} +4 q^{-142} +3 q^{-143} -9 q^{-144} -7 q^{-145} +4 q^{-146} -2 q^{-147} +2 q^{-148} + q^{-149} +2 q^{-150} - q^{-151} -2 q^{-152} + q^{-153} </math>|J7=<math>2 q^5-2 q^4-2 q^2-3 q+1+6 q^{-1} +7 q^{-2} +6 q^{-3} -2 q^{-4} -8 q^{-5} -18 q^{-6} -34 q^{-7} -16 q^{-8} +29 q^{-9} +65 q^{-10} +73 q^{-11} +46 q^{-12} -11 q^{-13} -101 q^{-14} -202 q^{-15} -199 q^{-16} -19 q^{-17} +190 q^{-18} +369 q^{-19} +405 q^{-20} +224 q^{-21} -147 q^{-22} -632 q^{-23} -869 q^{-24} -589 q^{-25} +54 q^{-26} +846 q^{-27} +1385 q^{-28} +1241 q^{-29} +400 q^{-30} -963 q^{-31} -2082 q^{-32} -2135 q^{-33} -1084 q^{-34} +838 q^{-35} +2644 q^{-36} +3207 q^{-37} +2149 q^{-38} -366 q^{-39} -3085 q^{-40} -4318 q^{-41} -3411 q^{-42} -412 q^{-43} +3201 q^{-44} +5271 q^{-45} +4772 q^{-46} +1462 q^{-47} -2997 q^{-48} -5978 q^{-49} -6027 q^{-50} -2620 q^{-51} +2505 q^{-52} +6357 q^{-53} +7045 q^{-54} +3739 q^{-55} -1857 q^{-56} -6403 q^{-57} -7743 q^{-58} -4699 q^{-59} +1135 q^{-60} +6235 q^{-61} +8154 q^{-62} +5394 q^{-63} -510 q^{-64} -5911 q^{-65} -8262 q^{-66} -5857 q^{-67} -27 q^{-68} +5562 q^{-69} +8234 q^{-70} +6087 q^{-71} +373 q^{-72} -5224 q^{-73} -8059 q^{-74} -6166 q^{-75} -628 q^{-76} +4927 q^{-77} +7873 q^{-78} +6152 q^{-79} +773 q^{-80} -4673 q^{-81} -7635 q^{-82} -6090 q^{-83} -914 q^{-84} +4409 q^{-85} +7408 q^{-86} +6027 q^{-87} +1058 q^{-88} -4113 q^{-89} -7122 q^{-90} -5982 q^{-91} -1282 q^{-92} +3734 q^{-93} +6811 q^{-94} +5939 q^{-95} +1572 q^{-96} -3231 q^{-97} -6397 q^{-98} -5908 q^{-99} -1973 q^{-100} +2617 q^{-101} +5884 q^{-102} +5825 q^{-103} +2418 q^{-104} -1845 q^{-105} -5209 q^{-106} -5683 q^{-107} -2900 q^{-108} +974 q^{-109} +4399 q^{-110} +5383 q^{-111} +3312 q^{-112} -29 q^{-113} -3407 q^{-114} -4897 q^{-115} -3620 q^{-116} -892 q^{-117} +2314 q^{-118} +4207 q^{-119} +3675 q^{-120} +1685 q^{-121} -1153 q^{-122} -3295 q^{-123} -3471 q^{-124} -2273 q^{-125} +84 q^{-126} +2259 q^{-127} +2971 q^{-128} +2515 q^{-129} +799 q^{-130} -1181 q^{-131} -2230 q^{-132} -2421 q^{-133} -1390 q^{-134} +237 q^{-135} +1378 q^{-136} +2015 q^{-137} +1592 q^{-138} +460 q^{-139} -531 q^{-140} -1398 q^{-141} -1480 q^{-142} -847 q^{-143} -114 q^{-144} +745 q^{-145} +1108 q^{-146} +892 q^{-147} +516 q^{-148} -180 q^{-149} -650 q^{-150} -709 q^{-151} -637 q^{-152} -181 q^{-153} +233 q^{-154} +421 q^{-155} +540 q^{-156} +318 q^{-157} +48 q^{-158} -124 q^{-159} -348 q^{-160} -310 q^{-161} -168 q^{-162} -49 q^{-163} +158 q^{-164} +181 q^{-165} +159 q^{-166} +145 q^{-167} -10 q^{-168} -86 q^{-169} -111 q^{-170} -125 q^{-171} -33 q^{-172} -2 q^{-173} +26 q^{-174} +92 q^{-175} +56 q^{-176} +30 q^{-177} -3 q^{-178} -47 q^{-179} -24 q^{-180} -26 q^{-181} -26 q^{-182} +11 q^{-183} +18 q^{-184} +23 q^{-185} +17 q^{-186} -9 q^{-187} -4 q^{-189} -13 q^{-190} -4 q^{-191} -2 q^{-192} +6 q^{-193} +7 q^{-194} -2 q^{-195} +2 q^{-197} -2 q^{-198} - q^{-199} -2 q^{-200} + q^{-201} +2 q^{-202} - q^{-203} </math>}} |
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coloured_jones_4 = <math>1+ q^{-1} -3 q^{-2} -4 q^{-3} +4 q^{-4} +5 q^{-5} +10 q^{-6} -8 q^{-7} -27 q^{-8} + q^{-9} +18 q^{-10} +47 q^{-11} -3 q^{-12} -74 q^{-13} -28 q^{-14} +21 q^{-15} +109 q^{-16} +33 q^{-17} -118 q^{-18} -80 q^{-19} - q^{-20} +162 q^{-21} +85 q^{-22} -133 q^{-23} -118 q^{-24} -38 q^{-25} +181 q^{-26} +123 q^{-27} -123 q^{-28} -126 q^{-29} -67 q^{-30} +170 q^{-31} +133 q^{-32} -99 q^{-33} -110 q^{-34} -88 q^{-35} +141 q^{-36} +129 q^{-37} -65 q^{-38} -83 q^{-39} -104 q^{-40} +97 q^{-41} +115 q^{-42} -21 q^{-43} -45 q^{-44} -113 q^{-45} +41 q^{-46} +87 q^{-47} +18 q^{-48} + q^{-49} -98 q^{-50} -8 q^{-51} +41 q^{-52} +31 q^{-53} +38 q^{-54} -57 q^{-55} -26 q^{-56} - q^{-57} +14 q^{-58} +44 q^{-59} -15 q^{-60} -14 q^{-61} -15 q^{-62} -5 q^{-63} +24 q^{-64} + q^{-65} -7 q^{-67} -7 q^{-68} +6 q^{-69} + q^{-70} +2 q^{-71} - q^{-72} -2 q^{-73} + q^{-74} </math> | |
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coloured_jones_5 = <math>2 q-2-3 q^{-2} -3 q^{-3} +6 q^{-4} +15 q^{-5} -2 q^{-6} -9 q^{-7} -20 q^{-8} -28 q^{-9} +19 q^{-10} +61 q^{-11} +41 q^{-12} -9 q^{-13} -83 q^{-14} -114 q^{-15} -15 q^{-16} +138 q^{-17} +177 q^{-18} +67 q^{-19} -152 q^{-20} -282 q^{-21} -147 q^{-22} +167 q^{-23} +369 q^{-24} +245 q^{-25} -143 q^{-26} -450 q^{-27} -352 q^{-28} +109 q^{-29} +497 q^{-30} +447 q^{-31} -49 q^{-32} -531 q^{-33} -517 q^{-34} -2 q^{-35} +524 q^{-36} +566 q^{-37} +58 q^{-38} -517 q^{-39} -588 q^{-40} -90 q^{-41} +484 q^{-42} +590 q^{-43} +125 q^{-44} -458 q^{-45} -579 q^{-46} -140 q^{-47} +415 q^{-48} +559 q^{-49} +164 q^{-50} -375 q^{-51} -531 q^{-52} -182 q^{-53} +316 q^{-54} +500 q^{-55} +213 q^{-56} -259 q^{-57} -457 q^{-58} -237 q^{-59} +179 q^{-60} +408 q^{-61} +264 q^{-62} -101 q^{-63} -345 q^{-64} -272 q^{-65} +15 q^{-66} +264 q^{-67} +274 q^{-68} +55 q^{-69} -177 q^{-70} -244 q^{-71} -111 q^{-72} +87 q^{-73} +194 q^{-74} +142 q^{-75} -10 q^{-76} -132 q^{-77} -137 q^{-78} -45 q^{-79} +60 q^{-80} +113 q^{-81} +74 q^{-82} -9 q^{-83} -68 q^{-84} -74 q^{-85} -28 q^{-86} +28 q^{-87} +54 q^{-88} +41 q^{-89} +4 q^{-90} -32 q^{-91} -36 q^{-92} -14 q^{-93} +7 q^{-94} +23 q^{-95} +20 q^{-96} + q^{-97} -13 q^{-98} -10 q^{-99} -5 q^{-100} +9 q^{-102} +5 q^{-103} -2 q^{-104} -2 q^{-105} - q^{-106} -2 q^{-107} + q^{-108} +2 q^{-109} - q^{-110} </math> | |
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{{Computer Talk Header}} |
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coloured_jones_6 = <math>q^3+q^2-3 q-2+ q^{-2} +2 q^{-3} +9 q^{-4} +12 q^{-5} -9 q^{-6} -29 q^{-7} -21 q^{-8} -3 q^{-9} +12 q^{-10} +64 q^{-11} +80 q^{-12} +12 q^{-13} -96 q^{-14} -144 q^{-15} -105 q^{-16} -30 q^{-17} +189 q^{-18} +329 q^{-19} +209 q^{-20} -109 q^{-21} -385 q^{-22} -456 q^{-23} -320 q^{-24} +241 q^{-25} +743 q^{-26} +736 q^{-27} +164 q^{-28} -552 q^{-29} -989 q^{-30} -969 q^{-31} -14 q^{-32} +1061 q^{-33} +1437 q^{-34} +780 q^{-35} -406 q^{-36} -1393 q^{-37} -1736 q^{-38} -569 q^{-39} +1049 q^{-40} +1957 q^{-41} +1449 q^{-42} +7 q^{-43} -1459 q^{-44} -2262 q^{-45} -1126 q^{-46} +778 q^{-47} +2124 q^{-48} +1861 q^{-49} +417 q^{-50} -1283 q^{-51} -2432 q^{-52} -1447 q^{-53} +486 q^{-54} +2046 q^{-55} +1969 q^{-56} +654 q^{-57} -1066 q^{-58} -2371 q^{-59} -1536 q^{-60} +291 q^{-61} +1886 q^{-62} +1902 q^{-63} +755 q^{-64} -874 q^{-65} -2210 q^{-66} -1524 q^{-67} +129 q^{-68} +1679 q^{-69} +1773 q^{-70} +846 q^{-71} -640 q^{-72} -1974 q^{-73} -1505 q^{-74} -108 q^{-75} +1368 q^{-76} +1599 q^{-77} +993 q^{-78} -282 q^{-79} -1617 q^{-80} -1467 q^{-81} -445 q^{-82} +899 q^{-83} +1317 q^{-84} +1134 q^{-85} +183 q^{-86} -1086 q^{-87} -1302 q^{-88} -766 q^{-89} +312 q^{-90} +848 q^{-91} +1107 q^{-92} +600 q^{-93} -430 q^{-94} -901 q^{-95} -866 q^{-96} -208 q^{-97} +247 q^{-98} +788 q^{-99} +741 q^{-100} +132 q^{-101} -340 q^{-102} -623 q^{-103} -415 q^{-104} -240 q^{-105} +291 q^{-106} +515 q^{-107} +348 q^{-108} +98 q^{-109} -203 q^{-110} -256 q^{-111} -373 q^{-112} -74 q^{-113} +144 q^{-114} +215 q^{-115} +199 q^{-116} +72 q^{-117} +7 q^{-118} -209 q^{-119} -135 q^{-120} -63 q^{-121} +17 q^{-122} +73 q^{-123} +90 q^{-124} +110 q^{-125} -36 q^{-126} -38 q^{-127} -60 q^{-128} -42 q^{-129} -24 q^{-130} +14 q^{-131} +68 q^{-132} +11 q^{-133} +15 q^{-134} -9 q^{-135} -15 q^{-136} -27 q^{-137} -13 q^{-138} +18 q^{-139} +2 q^{-140} +11 q^{-141} +4 q^{-142} +3 q^{-143} -9 q^{-144} -7 q^{-145} +4 q^{-146} -2 q^{-147} +2 q^{-148} + q^{-149} +2 q^{-150} - q^{-151} -2 q^{-152} + q^{-153} </math> | |
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coloured_jones_7 = <math>2 q^5-2 q^4-2 q^2-3 q+1+6 q^{-1} +7 q^{-2} +6 q^{-3} -2 q^{-4} -8 q^{-5} -18 q^{-6} -34 q^{-7} -16 q^{-8} +29 q^{-9} +65 q^{-10} +73 q^{-11} +46 q^{-12} -11 q^{-13} -101 q^{-14} -202 q^{-15} -199 q^{-16} -19 q^{-17} +190 q^{-18} +369 q^{-19} +405 q^{-20} +224 q^{-21} -147 q^{-22} -632 q^{-23} -869 q^{-24} -589 q^{-25} +54 q^{-26} +846 q^{-27} +1385 q^{-28} +1241 q^{-29} +400 q^{-30} -963 q^{-31} -2082 q^{-32} -2135 q^{-33} -1084 q^{-34} +838 q^{-35} +2644 q^{-36} +3207 q^{-37} +2149 q^{-38} -366 q^{-39} -3085 q^{-40} -4318 q^{-41} -3411 q^{-42} -412 q^{-43} +3201 q^{-44} +5271 q^{-45} +4772 q^{-46} +1462 q^{-47} -2997 q^{-48} -5978 q^{-49} -6027 q^{-50} -2620 q^{-51} +2505 q^{-52} +6357 q^{-53} +7045 q^{-54} +3739 q^{-55} -1857 q^{-56} -6403 q^{-57} -7743 q^{-58} -4699 q^{-59} +1135 q^{-60} +6235 q^{-61} +8154 q^{-62} +5394 q^{-63} -510 q^{-64} -5911 q^{-65} -8262 q^{-66} -5857 q^{-67} -27 q^{-68} +5562 q^{-69} +8234 q^{-70} +6087 q^{-71} +373 q^{-72} -5224 q^{-73} -8059 q^{-74} -6166 q^{-75} -628 q^{-76} +4927 q^{-77} +7873 q^{-78} +6152 q^{-79} +773 q^{-80} -4673 q^{-81} -7635 q^{-82} -6090 q^{-83} -914 q^{-84} +4409 q^{-85} +7408 q^{-86} +6027 q^{-87} +1058 q^{-88} -4113 q^{-89} -7122 q^{-90} -5982 q^{-91} -1282 q^{-92} +3734 q^{-93} +6811 q^{-94} +5939 q^{-95} +1572 q^{-96} -3231 q^{-97} -6397 q^{-98} -5908 q^{-99} -1973 q^{-100} +2617 q^{-101} +5884 q^{-102} +5825 q^{-103} +2418 q^{-104} -1845 q^{-105} -5209 q^{-106} -5683 q^{-107} -2900 q^{-108} +974 q^{-109} +4399 q^{-110} +5383 q^{-111} +3312 q^{-112} -29 q^{-113} -3407 q^{-114} -4897 q^{-115} -3620 q^{-116} -892 q^{-117} +2314 q^{-118} +4207 q^{-119} +3675 q^{-120} +1685 q^{-121} -1153 q^{-122} -3295 q^{-123} -3471 q^{-124} -2273 q^{-125} +84 q^{-126} +2259 q^{-127} +2971 q^{-128} +2515 q^{-129} +799 q^{-130} -1181 q^{-131} -2230 q^{-132} -2421 q^{-133} -1390 q^{-134} +237 q^{-135} +1378 q^{-136} +2015 q^{-137} +1592 q^{-138} +460 q^{-139} -531 q^{-140} -1398 q^{-141} -1480 q^{-142} -847 q^{-143} -114 q^{-144} +745 q^{-145} +1108 q^{-146} +892 q^{-147} +516 q^{-148} -180 q^{-149} -650 q^{-150} -709 q^{-151} -637 q^{-152} -181 q^{-153} +233 q^{-154} +421 q^{-155} +540 q^{-156} +318 q^{-157} +48 q^{-158} -124 q^{-159} -348 q^{-160} -310 q^{-161} -168 q^{-162} -49 q^{-163} +158 q^{-164} +181 q^{-165} +159 q^{-166} +145 q^{-167} -10 q^{-168} -86 q^{-169} -111 q^{-170} -125 q^{-171} -33 q^{-172} -2 q^{-173} +26 q^{-174} +92 q^{-175} +56 q^{-176} +30 q^{-177} -3 q^{-178} -47 q^{-179} -24 q^{-180} -26 q^{-181} -26 q^{-182} +11 q^{-183} +18 q^{-184} +23 q^{-185} +17 q^{-186} -9 q^{-187} -4 q^{-189} -13 q^{-190} -4 q^{-191} -2 q^{-192} +6 q^{-193} +7 q^{-194} -2 q^{-195} +2 q^{-197} -2 q^{-198} - q^{-199} -2 q^{-200} + q^{-201} +2 q^{-202} - q^{-203} </math> | |
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computer_talk = |
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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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<td align=left><pre style="color: red; border: 0px; padding: 0em"><< KnotTheory`</pre></td> |
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<tr valign=top><td colspan=2><nowiki>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</nowiki></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, 45]]</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[4, 2, 5, 1], X[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10], |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[2]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[Knot[9, 45]]</nowiki></code></td></tr> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[2]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[X[4, 2, 5, 1], X[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10], |
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X[7, 14, 8, 15], X[18, 15, 1, 16], X[16, 11, 17, 12], |
X[7, 14, 8, 15], X[18, 15, 1, 16], X[16, 11, 17, 12], |
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X[12, 17, 13, 18], X[13, 6, 14, 7]]</nowiki></ |
X[12, 17, 13, 18], X[13, 6, 14, 7]]</nowiki></code></td></tr> |
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</table> |
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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, 45]]</nowiki></pre></td></tr> |
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< |
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[3]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[Knot[9, 45]]</nowiki></code></td></tr> |
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<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, 45]]</nowiki></pre></td></tr> |
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< |
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[3]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[1, -4, 3, -1, 2, 9, -5, -3, 4, -2, 7, -8, -9, 5, 6, -7, 8, -6]</nowiki></code></td></tr> |
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</table> |
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<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, 45]]</nowiki></pre></td></tr> |
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<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[4, {-1, -1, -2, 1, -2, -1, -3, 2, -3}]</nowiki></pre></td></tr> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[4]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[Knot[9, 45]]</nowiki></code></td></tr> |
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<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>{4, 9}</nowiki></pre></td></tr> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[4]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[4, 8, 10, -14, 2, 16, -6, 18, 12]</nowiki></code></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>4</nowiki></pre></td></tr> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[5]:=</code></td> |
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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, 45]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:9_45_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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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>br = BR[Knot[9, 45]]</nowiki></code></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, 45]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}</nowiki></pre></td></tr> |
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< |
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[5]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>BR[4, {-1, -1, -2, 1, -2, -1, -3, 2, -3}]</nowiki></code></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, 45]][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 6 2 |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[6]:=</code></td> |
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<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> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[6]:=</code></td> |
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<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> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[7]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>BraidIndex[Knot[9, 45]]</nowiki></code></td></tr> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[7]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>4</nowiki></code></td></tr> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[8]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Show[DrawMorseLink[Knot[9, 45]]]</nowiki></code></td></tr> |
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<tr align=left><td></td><td>[[Image:9_45_ML.gif]]</td></tr><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[8]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>-Graphics-</nowiki></code></td></tr> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[9]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> (#[Knot[9, 45]]&) /@ { |
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SymmetryType, UnknottingNumber, ThreeGenus, |
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BridgeIndex, SuperBridgeIndex, NakanishiIndex |
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}</nowiki></code></td></tr> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[9]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Reversible, 1, 2, 3, {4, 5}, 1}</nowiki></code></td></tr> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[10]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>alex = Alexander[Knot[9, 45]][t]</nowiki></code></td></tr> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[10]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -2 6 2 |
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-9 - t + - + 6 t - t |
-9 - t + - + 6 t - t |
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t</nowiki></ |
t</nowiki></code></td></tr> |
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</table> |
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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, 45]][z]</nowiki></pre></td></tr> |
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< |
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[11]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Conway[Knot[9, 45]][z]</nowiki></code></td></tr> |
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1 + 2 z - z</nowiki></pre></td></tr> |
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< |
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[11]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 4 |
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1 + 2 z - z</nowiki></code></td></tr> |
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<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, 45]], KnotSignature[Knot[9, 45]]}</nowiki></pre></td></tr> |
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<table><tr align=left> |
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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>{23, -2}</nowiki></pre></td></tr> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[12]:=</code></td> |
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<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> |
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<tr align=left> |
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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 4 4 4 3 2 |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[12]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Knot[9, 45]}</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[13]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{KnotDet[Knot[9, 45]], KnotSignature[Knot[9, 45]]}</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[13]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{23, -2}</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[14]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Jones[Knot[9, 45]][q]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[14]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -8 2 3 4 4 4 3 2 |
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-q + -- - -- + -- - -- + -- - -- + - |
-q + -- - -- + -- - -- + -- - -- + - |
||
7 6 5 4 3 2 q |
7 6 5 4 3 2 q |
||
q q q q q q</nowiki></ |
q q q q q q</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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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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< |
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[15]:=</code></td> |
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<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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<tr align=left> |
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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, 45]][q]</nowiki></pre></td></tr> |
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< |
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[15]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Knot[9, 45]}</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[16]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>A2Invariant[Knot[9, 45]][q]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[16]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -26 -24 -22 -18 -16 -14 -10 -8 -6 2 |
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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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2 |
2 |
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q</nowiki></ |
q</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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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, 45]][a, z]</nowiki></pre></td></tr> |
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< |
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[17]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>HOMFLYPT[Knot[9, 45]][a, z]</nowiki></code></td></tr> |
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2 a - 2 a + 2 a - a + 2 a z - 2 a z + 2 a z - a z</nowiki></pre></td></tr> |
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<tr align=left> |
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< |
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[17]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 4 6 8 2 2 4 2 6 2 4 4 |
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2 a - 2 a + 2 a - a + 2 a z - 2 a z + 2 a z - a z</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[18]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kauffman[Knot[9, 45]][a, z]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[18]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 4 6 8 7 9 2 2 4 2 |
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-2 a - 2 a - 2 a - a + 2 a z + 2 a z + 3 a z + 6 a z + |
-2 a - 2 a - 2 a - a + 2 a z + 2 a z + 3 a z + 6 a z + |
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Line 154: | Line 196: | ||
5 7 7 7 |
5 7 7 7 |
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a z + a z</nowiki></ |
a z + a z</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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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, 45]], Vassiliev[3][Knot[9, 45]]}</nowiki></pre></td></tr> |
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< |
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[19]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Vassiliev[2][Knot[9, 45]], Vassiliev[3][Knot[9, 45]]}</nowiki></code></td></tr> |
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<tr align=left> |
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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, 45]][q, t]</nowiki></pre></td></tr> |
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< |
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[19]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{2, -4}</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[20]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kh[Knot[9, 45]][q, t]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[20]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -3 2 1 1 1 2 1 2 2 |
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q + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + |
q + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + |
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q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 |
q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 |
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Line 168: | Line 218: | ||
----- + ----- + ----- + ----- + ---- + ---- |
----- + ----- + ----- + ----- + ---- + ---- |
||
9 3 7 3 7 2 5 2 5 3 |
9 3 7 3 7 2 5 2 5 3 |
||
q t q t q t q t q t q t</nowiki></ |
q t q t q t q t q t q t</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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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, 45], 2][q]</nowiki></pre></td></tr> |
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< |
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[21]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>ColouredJones[Knot[9, 45], 2][q]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[21]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -23 2 -21 6 4 6 12 3 13 15 -13 |
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q - --- - q + --- - --- - --- + --- - --- - --- + --- + q - |
q - --- - q + --- - --- - --- + --- - --- - --- + --- + q - |
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22 20 19 18 17 16 15 14 |
22 20 19 18 17 16 15 14 |
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Line 179: | Line 233: | ||
--- + --- + --- - -- + -- + -- - -- + -- + -- - -- + q + - |
--- + --- + --- - -- + -- + -- - -- + -- + -- - -- + q + - |
||
12 11 10 9 8 7 6 5 4 3 q |
12 11 10 9 8 7 6 5 4 3 q |
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q q q q q q q q q q</nowiki></ |
q q q q q q q q q q</nowiki></code></td></tr> |
||
</table> }} |
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</table> |
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See/edit the [[Rolfsen_Splice_Template]]. |
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[[Category:Knot Page]] |
Latest revision as of 17:04, 1 September 2005
|
|
(KnotPlot image) |
See the full Rolfsen Knot Table. Visit 9 45's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
Planar diagram presentation | X4251 X10,6,11,5 X8394 X2,9,3,10 X7,14,8,15 X18,15,1,16 X16,11,17,12 X12,17,13,18 X13,6,14,7 |
Gauss code | 1, -4, 3, -1, 2, 9, -5, -3, 4, -2, 7, -8, -9, 5, 6, -7, 8, -6 |
Dowker-Thistlethwaite code | 4 8 10 -14 2 16 -6 18 12 |
Conway Notation | [211,21,2-] |
Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||
Length is 9, width is 4, Braid index is 4 |
[{2, 11}, {1, 7}, {10, 4}, {11, 9}, {6, 8}, {7, 10}, {3, 5}, {4, 6}, {5, 2}, {8, 3}, {9, 1}] |
[edit Notes on presentations of 9 45]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
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K = Knot["9 45"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
|
X4251 X10,6,11,5 X8394 X2,9,3,10 X7,14,8,15 X18,15,1,16 X16,11,17,12 X12,17,13,18 X13,6,14,7 |
In[5]:=
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GaussCode[K]
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Out[5]=
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1, -4, 3, -1, 2, 9, -5, -3, 4, -2, 7, -8, -9, 5, 6, -7, 8, -6 |
In[6]:=
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DTCode[K]
|
Out[6]=
|
4 8 10 -14 2 16 -6 18 12 |
(The path below may be different on your system)
In[7]:=
|
AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
|
ConwayNotation[K]
|
Out[8]=
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[211,21,2-] |
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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{ 4, 9, 4 } |
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[{2, 11}, {1, 7}, {10, 4}, {11, 9}, {6, 8}, {7, 10}, {3, 5}, {4, 6}, {5, 2}, {8, 3}, {9, 1}] |
In[14]:=
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Draw[ap]
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Out[14]=
|
-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 |
A4 Invariants.
Weight | Invariant |
---|---|
0,1,0,0 | |
1,0,0,0 |
B2 Invariants.
Weight | Invariant |
---|---|
0,1 | |
1,0 |
D4 Invariants.
Weight | Invariant |
---|---|
1,0,0,0 |
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]:=
|
AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
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K = Knot["9 45"];
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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]=
|
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.
|
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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{ 23, -2 } |
In[8]:=
|
Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
|
In[9]:=
|
HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
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In[10]:=
|
Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
|
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, ): {}
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`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
|
K = Knot["9 45"];
|
In[4]:=
|
{A = Alexander[K][t], J = Jones[K][q]}
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[4]=
|
{ , } |
In[5]:=
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DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
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KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
|
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
|
Out[5]=
|
{} |
In[6]:=
|
DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
|
Out[6]=
|
{} |
Vassiliev invariants
V2 and V3: | (2, -4) |
V2,1 through V6,9: |
|
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 45. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
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Integral Khovanov Homology
(db, data source) |
|
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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