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{{Rolfsen Knot Page| |
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n = 9 | |
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k = 46 | |
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KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-4,3,-1,-5,6,-2,9,-8,-3,4,2,-6,5,-7,8,-9,7/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=46|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-4,3,-1,-5,6,-2,9,-8,-3,4,2,-6,5,-7,8,-9,7/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:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr> |
<tr><td>[[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr> |
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<tr><td>[[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]]</td></tr> |
<tr><td>[[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]]</td></tr> |
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<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart4.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart3.gif]]</td></tr> |
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart4.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.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:BraidPart4.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]]</td></tr> |
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.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 = [[6_1]], [[K11n67]], [[K11n97]], [[K11n139]], | |
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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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{[[6_1]], [[K11n67]], [[K11n97]], [[K11n139]], ...} |
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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=18.1818%><table cellpadding=0 cellspacing=0> |
<td width=18.1818%><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=9.09091%>-6</td ><td width=9.09091%>-5</td ><td width=9.09091%>-4</td ><td width=9.09091%>-3</td ><td width=9.09091%>-2</td ><td width=9.09091%>-1</td ><td width=9.09091%>0</td ><td width=18.1818%>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 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 bgcolor=yellow>2</td><td>2</td></tr> |
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<tr align=center><td>-1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow>1</td><td>1</td></tr> |
<tr align=center><td>-1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow>1</td><td>1</td></tr> |
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<tr align=center><td>-11</td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> |
<tr align=center><td>-11</td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> |
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<tr align=center><td>-13</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>-13</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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</table> |
</table> | |
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coloured_jones_2 = <math>q^2+ q^{-1} -2 q^{-3} +2 q^{-9} - q^{-10} +2 q^{-12} -2 q^{-13} - q^{-14} +2 q^{-15} - q^{-16} - q^{-17} + q^{-18} </math> | |
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coloured_jones_3 = <math>2 q^4-2 q^2-3 q+6+2 q^{-1} -4 q^{-2} -6 q^{-3} +5 q^{-4} +4 q^{-5} -4 q^{-6} -5 q^{-7} +5 q^{-8} +5 q^{-9} -4 q^{-10} -3 q^{-11} +4 q^{-12} +4 q^{-13} -4 q^{-14} -3 q^{-15} +2 q^{-16} +3 q^{-17} -3 q^{-18} -2 q^{-19} + q^{-20} + q^{-21} - q^{-22} + q^{-24} - q^{-26} + q^{-27} +2 q^{-28} - q^{-29} -2 q^{-30} +2 q^{-32} - q^{-34} - q^{-35} + q^{-36} </math> | |
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{{Display Coloured Jones|J2=<math>q^2+ q^{-1} -2 q^{-3} +2 q^{-9} - q^{-10} +2 q^{-12} -2 q^{-13} - q^{-14} +2 q^{-15} - q^{-16} - q^{-17} + q^{-18} </math>|J3=<math>2 q^4-2 q^2-3 q+6+2 q^{-1} -4 q^{-2} -6 q^{-3} +5 q^{-4} +4 q^{-5} -4 q^{-6} -5 q^{-7} +5 q^{-8} +5 q^{-9} -4 q^{-10} -3 q^{-11} +4 q^{-12} +4 q^{-13} -4 q^{-14} -3 q^{-15} +2 q^{-16} +3 q^{-17} -3 q^{-18} -2 q^{-19} + q^{-20} + q^{-21} - q^{-22} + q^{-24} - q^{-26} + q^{-27} +2 q^{-28} - q^{-29} -2 q^{-30} +2 q^{-32} - q^{-34} - q^{-35} + q^{-36} </math>|J4=<math>q^8+2 q^6-2 q^5-4 q^4+q^3+q^2+8 q-2-9 q^{-1} - q^{-2} +13 q^{-4} - q^{-5} -10 q^{-6} -3 q^{-7} - q^{-8} +15 q^{-9} + q^{-10} -9 q^{-11} -3 q^{-12} - q^{-13} +12 q^{-14} + q^{-15} -8 q^{-16} -3 q^{-17} -3 q^{-18} +10 q^{-19} +2 q^{-20} -7 q^{-21} -3 q^{-22} -4 q^{-23} +8 q^{-24} +5 q^{-25} -4 q^{-26} -3 q^{-27} -5 q^{-28} +5 q^{-29} +7 q^{-30} - q^{-31} -2 q^{-32} -6 q^{-33} + q^{-34} +6 q^{-35} + q^{-36} - q^{-37} -4 q^{-38} -2 q^{-39} +3 q^{-40} + q^{-42} - q^{-43} -2 q^{-44} +3 q^{-45} -2 q^{-46} +4 q^{-50} -2 q^{-51} - q^{-52} - q^{-53} - q^{-54} +3 q^{-55} - q^{-58} - q^{-59} + q^{-60} </math>|J5=<math>2 q^{12}-2 q^9-2 q^8-3 q^7+2 q^6+6 q^5+6 q^4-2 q^3-8 q^2-11 q-1+11 q^{-1} +16 q^{-2} +3 q^{-3} -11 q^{-4} -17 q^{-5} -7 q^{-6} +10 q^{-7} +18 q^{-8} +10 q^{-9} -9 q^{-10} -17 q^{-11} -9 q^{-12} +8 q^{-13} +16 q^{-14} +10 q^{-15} -9 q^{-16} -16 q^{-17} -8 q^{-18} +8 q^{-19} +14 q^{-20} +8 q^{-21} -10 q^{-22} -15 q^{-23} -5 q^{-24} +7 q^{-25} +13 q^{-26} +7 q^{-27} -7 q^{-28} -12 q^{-29} -4 q^{-30} +4 q^{-31} +10 q^{-32} +7 q^{-33} -2 q^{-34} -8 q^{-35} -5 q^{-36} - q^{-37} +6 q^{-38} +6 q^{-39} +2 q^{-40} -4 q^{-41} -5 q^{-42} -4 q^{-43} +3 q^{-45} +5 q^{-46} +2 q^{-47} -2 q^{-48} -4 q^{-49} -4 q^{-50} -3 q^{-51} +4 q^{-52} +6 q^{-53} +3 q^{-54} -4 q^{-56} -6 q^{-57} - q^{-58} +4 q^{-59} +4 q^{-60} +4 q^{-61} - q^{-62} -4 q^{-63} -3 q^{-64} - q^{-65} + q^{-66} +3 q^{-67} + q^{-68} - q^{-69} - q^{-70} - q^{-74} - q^{-75} +2 q^{-77} +2 q^{-78} - q^{-80} - q^{-81} -2 q^{-82} +2 q^{-84} + q^{-85} - q^{-88} - q^{-89} + q^{-90} </math>|J6=<math>q^{18}+2 q^{16}-2 q^{14}-4 q^{13}-4 q^{12}-q^{11}+3 q^{10}+12 q^9+7 q^8+2 q^7-10 q^6-15 q^5-17 q^4-2 q^3+24 q^2+22 q+19-7 q^{-1} -23 q^{-2} -37 q^{-3} -16 q^{-4} +24 q^{-5} +30 q^{-6} +32 q^{-7} +2 q^{-8} -19 q^{-9} -44 q^{-10} -25 q^{-11} +18 q^{-12} +27 q^{-13} +35 q^{-14} +6 q^{-15} -14 q^{-16} -42 q^{-17} -27 q^{-18} +16 q^{-19} +25 q^{-20} +33 q^{-21} +4 q^{-22} -13 q^{-23} -41 q^{-24} -24 q^{-25} +17 q^{-26} +25 q^{-27} +32 q^{-28} +3 q^{-29} -13 q^{-30} -39 q^{-31} -20 q^{-32} +15 q^{-33} +22 q^{-34} +30 q^{-35} +5 q^{-36} -10 q^{-37} -36 q^{-38} -18 q^{-39} +10 q^{-40} +15 q^{-41} +27 q^{-42} +8 q^{-43} -4 q^{-44} -32 q^{-45} -17 q^{-46} +2 q^{-47} +8 q^{-48} +23 q^{-49} +11 q^{-50} +5 q^{-51} -24 q^{-52} -15 q^{-53} -6 q^{-54} - q^{-55} +17 q^{-56} +13 q^{-57} +14 q^{-58} -13 q^{-59} -9 q^{-60} -10 q^{-61} -10 q^{-62} +8 q^{-63} +9 q^{-64} +16 q^{-65} -3 q^{-66} + q^{-67} -7 q^{-68} -12 q^{-69} -2 q^{-70} +9 q^{-72} +9 q^{-74} + q^{-75} -6 q^{-76} -2 q^{-77} -6 q^{-78} -2 q^{-79} -5 q^{-80} +7 q^{-81} +5 q^{-82} +2 q^{-83} +5 q^{-84} -2 q^{-85} -4 q^{-86} -8 q^{-87} - q^{-88} +2 q^{-90} +8 q^{-91} +3 q^{-92} + q^{-93} -3 q^{-94} -3 q^{-95} -4 q^{-96} -3 q^{-97} +5 q^{-98} +2 q^{-100} + q^{-101} - q^{-103} -2 q^{-104} +4 q^{-105} -3 q^{-106} - q^{-107} - q^{-108} +5 q^{-112} - q^{-113} - q^{-115} - q^{-116} -2 q^{-117} - q^{-118} +3 q^{-119} + q^{-121} - q^{-124} - q^{-125} + q^{-126} </math>|J7=<math>2 q^{24}-4 q^{19}-4 q^{18}-5 q^{17}+2 q^{16}+6 q^{15}+8 q^{14}+12 q^{13}+8 q^{12}-4 q^{11}-20 q^{10}-29 q^9-15 q^8+6 q^7+19 q^6+37 q^5+40 q^4+11 q^3-27 q^2-58 q-47-17 q^{-1} +14 q^{-2} +56 q^{-3} +69 q^{-4} +38 q^{-5} -12 q^{-6} -63 q^{-7} -71 q^{-8} -40 q^{-9} -4 q^{-10} +53 q^{-11} +76 q^{-12} +52 q^{-13} +5 q^{-14} -53 q^{-15} -69 q^{-16} -47 q^{-17} -15 q^{-18} +46 q^{-19} +70 q^{-20} +49 q^{-21} +10 q^{-22} -47 q^{-23} -65 q^{-24} -44 q^{-25} -12 q^{-26} +46 q^{-27} +67 q^{-28} +44 q^{-29} +9 q^{-30} -48 q^{-31} -65 q^{-32} -43 q^{-33} -6 q^{-34} +48 q^{-35} +66 q^{-36} +40 q^{-37} +6 q^{-38} -46 q^{-39} -63 q^{-40} -41 q^{-41} -4 q^{-42} +44 q^{-43} +60 q^{-44} +35 q^{-45} +7 q^{-46} -38 q^{-47} -57 q^{-48} -39 q^{-49} -6 q^{-50} +35 q^{-51} +49 q^{-52} +33 q^{-53} +13 q^{-54} -25 q^{-55} -48 q^{-56} -37 q^{-57} -13 q^{-58} +21 q^{-59} +39 q^{-60} +32 q^{-61} +22 q^{-62} -8 q^{-63} -35 q^{-64} -34 q^{-65} -24 q^{-66} +4 q^{-67} +25 q^{-68} +29 q^{-69} +28 q^{-70} +10 q^{-71} -17 q^{-72} -26 q^{-73} -29 q^{-74} -14 q^{-75} +5 q^{-76} +16 q^{-77} +28 q^{-78} +22 q^{-79} +3 q^{-80} -8 q^{-81} -22 q^{-82} -24 q^{-83} -14 q^{-84} -2 q^{-85} +14 q^{-86} +22 q^{-87} +17 q^{-88} +12 q^{-89} -4 q^{-90} -17 q^{-91} -18 q^{-92} -16 q^{-93} -5 q^{-94} +7 q^{-95} +13 q^{-96} +18 q^{-97} +13 q^{-98} -5 q^{-100} -12 q^{-101} -14 q^{-102} -7 q^{-103} -2 q^{-104} +6 q^{-105} +10 q^{-106} +7 q^{-107} +8 q^{-108} +3 q^{-109} -5 q^{-110} -5 q^{-111} -6 q^{-112} -7 q^{-113} -3 q^{-114} -2 q^{-115} +4 q^{-116} +7 q^{-117} +4 q^{-118} +4 q^{-119} +4 q^{-120} -2 q^{-121} -4 q^{-122} -8 q^{-123} -6 q^{-124} - q^{-126} +3 q^{-127} +8 q^{-128} +5 q^{-129} +3 q^{-130} - q^{-131} -5 q^{-132} -2 q^{-133} -4 q^{-134} -3 q^{-135} +2 q^{-136} + q^{-137} +3 q^{-138} +2 q^{-139} -2 q^{-140} +2 q^{-141} - q^{-143} +2 q^{-144} -2 q^{-145} - q^{-146} -3 q^{-148} + q^{-151} +4 q^{-152} + q^{-155} -2 q^{-156} - q^{-157} -2 q^{-158} - q^{-159} +2 q^{-160} + q^{-161} + q^{-163} - q^{-166} - q^{-167} + q^{-168} </math>}} |
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coloured_jones_4 = <math>q^8+2 q^6-2 q^5-4 q^4+q^3+q^2+8 q-2-9 q^{-1} - q^{-2} +13 q^{-4} - q^{-5} -10 q^{-6} -3 q^{-7} - q^{-8} +15 q^{-9} + q^{-10} -9 q^{-11} -3 q^{-12} - q^{-13} +12 q^{-14} + q^{-15} -8 q^{-16} -3 q^{-17} -3 q^{-18} +10 q^{-19} +2 q^{-20} -7 q^{-21} -3 q^{-22} -4 q^{-23} +8 q^{-24} +5 q^{-25} -4 q^{-26} -3 q^{-27} -5 q^{-28} +5 q^{-29} +7 q^{-30} - q^{-31} -2 q^{-32} -6 q^{-33} + q^{-34} +6 q^{-35} + q^{-36} - q^{-37} -4 q^{-38} -2 q^{-39} +3 q^{-40} + q^{-42} - q^{-43} -2 q^{-44} +3 q^{-45} -2 q^{-46} +4 q^{-50} -2 q^{-51} - q^{-52} - q^{-53} - q^{-54} +3 q^{-55} - q^{-58} - q^{-59} + q^{-60} </math> | |
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coloured_jones_5 = <math>2 q^{12}-2 q^9-2 q^8-3 q^7+2 q^6+6 q^5+6 q^4-2 q^3-8 q^2-11 q-1+11 q^{-1} +16 q^{-2} +3 q^{-3} -11 q^{-4} -17 q^{-5} -7 q^{-6} +10 q^{-7} +18 q^{-8} +10 q^{-9} -9 q^{-10} -17 q^{-11} -9 q^{-12} +8 q^{-13} +16 q^{-14} +10 q^{-15} -9 q^{-16} -16 q^{-17} -8 q^{-18} +8 q^{-19} +14 q^{-20} +8 q^{-21} -10 q^{-22} -15 q^{-23} -5 q^{-24} +7 q^{-25} +13 q^{-26} +7 q^{-27} -7 q^{-28} -12 q^{-29} -4 q^{-30} +4 q^{-31} +10 q^{-32} +7 q^{-33} -2 q^{-34} -8 q^{-35} -5 q^{-36} - q^{-37} +6 q^{-38} +6 q^{-39} +2 q^{-40} -4 q^{-41} -5 q^{-42} -4 q^{-43} +3 q^{-45} +5 q^{-46} +2 q^{-47} -2 q^{-48} -4 q^{-49} -4 q^{-50} -3 q^{-51} +4 q^{-52} +6 q^{-53} +3 q^{-54} -4 q^{-56} -6 q^{-57} - q^{-58} +4 q^{-59} +4 q^{-60} +4 q^{-61} - q^{-62} -4 q^{-63} -3 q^{-64} - q^{-65} + q^{-66} +3 q^{-67} + q^{-68} - q^{-69} - q^{-70} - q^{-74} - q^{-75} +2 q^{-77} +2 q^{-78} - q^{-80} - q^{-81} -2 q^{-82} +2 q^{-84} + q^{-85} - q^{-88} - q^{-89} + q^{-90} </math> | |
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{{Computer Talk Header}} |
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coloured_jones_6 = <math>q^{18}+2 q^{16}-2 q^{14}-4 q^{13}-4 q^{12}-q^{11}+3 q^{10}+12 q^9+7 q^8+2 q^7-10 q^6-15 q^5-17 q^4-2 q^3+24 q^2+22 q+19-7 q^{-1} -23 q^{-2} -37 q^{-3} -16 q^{-4} +24 q^{-5} +30 q^{-6} +32 q^{-7} +2 q^{-8} -19 q^{-9} -44 q^{-10} -25 q^{-11} +18 q^{-12} +27 q^{-13} +35 q^{-14} +6 q^{-15} -14 q^{-16} -42 q^{-17} -27 q^{-18} +16 q^{-19} +25 q^{-20} +33 q^{-21} +4 q^{-22} -13 q^{-23} -41 q^{-24} -24 q^{-25} +17 q^{-26} +25 q^{-27} +32 q^{-28} +3 q^{-29} -13 q^{-30} -39 q^{-31} -20 q^{-32} +15 q^{-33} +22 q^{-34} +30 q^{-35} +5 q^{-36} -10 q^{-37} -36 q^{-38} -18 q^{-39} +10 q^{-40} +15 q^{-41} +27 q^{-42} +8 q^{-43} -4 q^{-44} -32 q^{-45} -17 q^{-46} +2 q^{-47} +8 q^{-48} +23 q^{-49} +11 q^{-50} +5 q^{-51} -24 q^{-52} -15 q^{-53} -6 q^{-54} - q^{-55} +17 q^{-56} +13 q^{-57} +14 q^{-58} -13 q^{-59} -9 q^{-60} -10 q^{-61} -10 q^{-62} +8 q^{-63} +9 q^{-64} +16 q^{-65} -3 q^{-66} + q^{-67} -7 q^{-68} -12 q^{-69} -2 q^{-70} +9 q^{-72} +9 q^{-74} + q^{-75} -6 q^{-76} -2 q^{-77} -6 q^{-78} -2 q^{-79} -5 q^{-80} +7 q^{-81} +5 q^{-82} +2 q^{-83} +5 q^{-84} -2 q^{-85} -4 q^{-86} -8 q^{-87} - q^{-88} +2 q^{-90} +8 q^{-91} +3 q^{-92} + q^{-93} -3 q^{-94} -3 q^{-95} -4 q^{-96} -3 q^{-97} +5 q^{-98} +2 q^{-100} + q^{-101} - q^{-103} -2 q^{-104} +4 q^{-105} -3 q^{-106} - q^{-107} - q^{-108} +5 q^{-112} - q^{-113} - q^{-115} - q^{-116} -2 q^{-117} - q^{-118} +3 q^{-119} + q^{-121} - q^{-124} - q^{-125} + q^{-126} </math> | |
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coloured_jones_7 = <math>2 q^{24}-4 q^{19}-4 q^{18}-5 q^{17}+2 q^{16}+6 q^{15}+8 q^{14}+12 q^{13}+8 q^{12}-4 q^{11}-20 q^{10}-29 q^9-15 q^8+6 q^7+19 q^6+37 q^5+40 q^4+11 q^3-27 q^2-58 q-47-17 q^{-1} +14 q^{-2} +56 q^{-3} +69 q^{-4} +38 q^{-5} -12 q^{-6} -63 q^{-7} -71 q^{-8} -40 q^{-9} -4 q^{-10} +53 q^{-11} +76 q^{-12} +52 q^{-13} +5 q^{-14} -53 q^{-15} -69 q^{-16} -47 q^{-17} -15 q^{-18} +46 q^{-19} +70 q^{-20} +49 q^{-21} +10 q^{-22} -47 q^{-23} -65 q^{-24} -44 q^{-25} -12 q^{-26} +46 q^{-27} +67 q^{-28} +44 q^{-29} +9 q^{-30} -48 q^{-31} -65 q^{-32} -43 q^{-33} -6 q^{-34} +48 q^{-35} +66 q^{-36} +40 q^{-37} +6 q^{-38} -46 q^{-39} -63 q^{-40} -41 q^{-41} -4 q^{-42} +44 q^{-43} +60 q^{-44} +35 q^{-45} +7 q^{-46} -38 q^{-47} -57 q^{-48} -39 q^{-49} -6 q^{-50} +35 q^{-51} +49 q^{-52} +33 q^{-53} +13 q^{-54} -25 q^{-55} -48 q^{-56} -37 q^{-57} -13 q^{-58} +21 q^{-59} +39 q^{-60} +32 q^{-61} +22 q^{-62} -8 q^{-63} -35 q^{-64} -34 q^{-65} -24 q^{-66} +4 q^{-67} +25 q^{-68} +29 q^{-69} +28 q^{-70} +10 q^{-71} -17 q^{-72} -26 q^{-73} -29 q^{-74} -14 q^{-75} +5 q^{-76} +16 q^{-77} +28 q^{-78} +22 q^{-79} +3 q^{-80} -8 q^{-81} -22 q^{-82} -24 q^{-83} -14 q^{-84} -2 q^{-85} +14 q^{-86} +22 q^{-87} +17 q^{-88} +12 q^{-89} -4 q^{-90} -17 q^{-91} -18 q^{-92} -16 q^{-93} -5 q^{-94} +7 q^{-95} +13 q^{-96} +18 q^{-97} +13 q^{-98} -5 q^{-100} -12 q^{-101} -14 q^{-102} -7 q^{-103} -2 q^{-104} +6 q^{-105} +10 q^{-106} +7 q^{-107} +8 q^{-108} +3 q^{-109} -5 q^{-110} -5 q^{-111} -6 q^{-112} -7 q^{-113} -3 q^{-114} -2 q^{-115} +4 q^{-116} +7 q^{-117} +4 q^{-118} +4 q^{-119} +4 q^{-120} -2 q^{-121} -4 q^{-122} -8 q^{-123} -6 q^{-124} - q^{-126} +3 q^{-127} +8 q^{-128} +5 q^{-129} +3 q^{-130} - q^{-131} -5 q^{-132} -2 q^{-133} -4 q^{-134} -3 q^{-135} +2 q^{-136} + q^{-137} +3 q^{-138} +2 q^{-139} -2 q^{-140} +2 q^{-141} - q^{-143} +2 q^{-144} -2 q^{-145} - q^{-146} -3 q^{-148} + q^{-151} +4 q^{-152} + q^{-155} -2 q^{-156} - q^{-157} -2 q^{-158} - q^{-159} +2 q^{-160} + q^{-161} + q^{-163} - q^{-166} - q^{-167} + q^{-168} </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, 46]]</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, 46]]</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[7, 12, 8, 13], X[10, 3, 11, 4], X[2, 11, 3, 12], |
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X[5, 14, 6, 15], X[13, 6, 14, 7], X[15, 18, 16, 1], X[9, 17, 10, 16], |
X[5, 14, 6, 15], X[13, 6, 14, 7], X[15, 18, 16, 1], X[9, 17, 10, 16], |
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X[17, 9, 18, 8]]</nowiki></pre></td></tr> |
X[17, 9, 18, 8]]</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, 46]]</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, -4, 3, -1, -5, 6, -2, 9, -8, -3, 4, 2, -6, 5, -7, 8, -9, 7]</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, 46]]</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, -14, -12, -16, 2, -6, -18, -8]</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, 46]]</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[4, {-1, 2, -1, 2, -3, -2, 1, -2, -3}]</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>{4, 9}</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, 46]]</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>4</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, 46]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:9_46_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, 46]]&) /@ {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, 2, 1, 3, 4, 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[10]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>alex = Alexander[Knot[9, 46]][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 |
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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, 46]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:9_46_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, 46]]&) /@ {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, 2, 1, 3, 4, 2}</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, 46]][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 |
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5 - - - 2 t |
5 - - - 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, 46]][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 |
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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 |
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1 - 2 z</nowiki></pre></td></tr> |
1 - 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[6, 1], Knot[9, 46], Knot[11, NonAlternating, 67], |
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<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[6, 1], Knot[9, 46], Knot[11, NonAlternating, 67], |
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Knot[11, NonAlternating, 97], Knot[11, NonAlternating, 139]}</nowiki></pre></td></tr> |
Knot[11, NonAlternating, 97], Knot[11, NonAlternating, 139]}</nowiki></pre></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, 46]], KnotSignature[Knot[9, 46]]}</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[13]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{9, 0}</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[14]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Jones[Knot[9, 46]][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> -6 -5 -4 2 -2 1 |
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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, 46]][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> -6 -5 -4 2 -2 1 |
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2 + q - q + q - -- + q - - |
2 + q - q + q - -- + q - - |
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3 q |
3 q |
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q</nowiki></pre></td></tr> |
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, 46]}</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, 46]][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> -20 -18 -12 -10 -8 -6 -2 2 |
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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, 46]][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> -20 -18 -12 -10 -8 -6 -2 2 |
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2 + q + q - q - q - q - q + q + 2 q</nowiki></pre></td></tr> |
2 + q + q - q - q - q - q + q + 2 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, 46]][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> 2 4 6 2 2 4 2 |
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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> 2 4 6 2 2 4 2 |
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2 - a - a + a - a z - a z</nowiki></pre></td></tr> |
2 - a - a + a - 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, 46]][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> 2 4 6 3 5 2 2 4 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> 2 4 6 3 5 2 2 4 2 |
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2 + a - a - a - 2 a z - 6 a z - 4 a z + 3 a z + 9 a z + |
2 + a - a - a - 2 a z - 6 a z - 4 a z + 3 a z + 9 a z + |
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Line 152: | Line 101: | ||
3 5 5 5 2 6 4 6 6 6 3 7 5 7 |
3 5 5 5 2 6 4 6 6 6 3 7 5 7 |
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5 a z - 5 a z + a z + 2 a z + a z + a z + a z</nowiki></pre></td></tr> |
5 a z - 5 a z + a z + 2 a z + 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, 46]], Vassiliev[3][Knot[9, 46]]}</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>{-2, 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, 46]][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>1 1 1 1 1 1 1 1 |
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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, 46]][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>1 1 1 1 1 1 1 1 |
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- + 2 q + ------ + ----- + ----- + ----- + ----- + ----- + ---- |
- + 2 q + ------ + ----- + ----- + ----- + ----- + ----- + ---- |
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q 13 6 9 5 9 4 7 3 5 3 3 2 3 |
q 13 6 9 5 9 4 7 3 5 3 3 2 3 |
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q t 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 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, 46], 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> -18 -17 -16 2 -14 2 2 -10 2 2 1 2 |
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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> -18 -17 -16 2 -14 2 2 -10 2 2 1 2 |
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q - q - q + --- - q - --- + --- - q + -- - -- + - + q |
q - q - q + --- - q - --- + --- - q + -- - -- + - + q |
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15 13 12 9 3 q |
15 13 12 9 3 q |
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q q q q q</nowiki></pre></td></tr> |
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 10:34, 30 August 2005
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(KnotPlot image) |
See the full Rolfsen Knot Table. Visit 9 46's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
9_46 is also known as the pretzel knot P(3,3,-3). |
Knot presentations
Planar diagram presentation | X4251 X7,12,8,13 X10,3,11,4 X2,11,3,12 X5,14,6,15 X13,6,14,7 X15,18,16,1 X9,17,10,16 X17,9,18,8 |
Gauss code | 1, -4, 3, -1, -5, 6, -2, 9, -8, -3, 4, 2, -6, 5, -7, 8, -9, 7 |
Dowker-Thistlethwaite code | 4 10 -14 -12 -16 2 -6 -18 -8 |
Conway Notation | [3,3,21-] |
Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||
Length is 9, width is 4, Braid index is 4 |
[{9, 5}, {3, 8}, {4, 6}, {5, 2}, {1, 4}, {7, 3}, {6, 9}, {2, 7}, {8, 1}] |
[edit Notes on presentations of 9 46]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["9 46"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X4251 X7,12,8,13 X10,3,11,4 X2,11,3,12 X5,14,6,15 X13,6,14,7 X15,18,16,1 X9,17,10,16 X17,9,18,8 |
In[5]:=
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GaussCode[K]
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Out[5]=
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1, -4, 3, -1, -5, 6, -2, 9, -8, -3, 4, 2, -6, 5, -7, 8, -9, 7 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 10 -14 -12 -16 2 -6 -18 -8 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$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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[3,3,21-] |
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[{9, 5}, {3, 8}, {4, 6}, {5, 2}, {1, 4}, {7, 3}, {6, 9}, {2, 7}, {8, 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 |
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1 | |
2 | |
3 | |
4 | |
5 |
A2 Invariants.
Weight | Invariant |
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1,0 | |
1,1 | |
2,0 |
A3 Invariants.
Weight | Invariant |
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0,1,0 | |
1,0,0 |
A4 Invariants.
Weight | Invariant |
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0,1,0,0 | |
1,0,0,0 |
B2 Invariants.
Weight | Invariant |
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0,1 | |
1,0 |
D4 Invariants.
Weight | Invariant |
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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]:=
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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 46"];
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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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{ 9, 0 } |
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: {6_1, K11n67, K11n97, K11n139,}
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]:=
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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 46"];
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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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{6_1, K11n67, K11n97, K11n139,} |
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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{} |
Vassiliev invariants
V2 and V3: | (-2, 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 0 is the signature of 9 46. 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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