10 136: Difference between revisions
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{{Rolfsen Knot Page| |
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n = 10 | |
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k = 136 | |
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KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,4,-3,1,-2,-10,5,3,-4,2,6,-9,10,-5,7,-8,9,-6,8,-7/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=10|k=136|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,4,-3,1,-2,-10,5,3,-4,2,6,-9,10,-5,7,-8,9,-6,8,-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:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr> |
<tr><td>[[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr> |
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<tr><td>[[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr> |
<tr><td>[[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart2.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr> |
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<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]]</td></tr> |
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart0.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.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:BraidPart2.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]]</td></tr> |
<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]]</td></tr> |
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</table> |
</table> | |
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braid_crossings = 10 | |
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braid_width = 5 | |
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[[Invariants from Braid Theory|Length]] is 10, width is 5. |
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braid_index = 4 | |
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same_alexander = [[8_21]], | |
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[[Invariants from Braid Theory|Braid index]] is 4. |
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same_jones = [[K11n92]], | |
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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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{[[8_21]], ...} |
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Same [[The Jones Polynomial|Jones Polynomial]] (up to mirroring, <math>q\leftrightarrow q^{-1}</math>): |
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{[[K11n92]], ...} |
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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%>-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=8.33333%>1</td ><td width=8.33333%>2</td ><td width=8.33333%>3</td ><td width=16.6667%>χ</td></tr> |
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<tr align=center><td>9</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td>-1</td></tr> |
<tr align=center><td>9</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td>-1</td></tr> |
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<tr align=center><td>7</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow> </td><td>1</td></tr> |
<tr align=center><td>7</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow> </td><td>1</td></tr> |
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<tr align=center><td>-5</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>-5</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>-7</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>-7</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^{13}-q^{12}-2 q^{11}+3 q^{10}-4 q^8+3 q^7+2 q^6-3 q^5+q^4+q^3-q+3 q^{-1} -3 q^{-2} -2 q^{-3} +6 q^{-4} -3 q^{-5} -3 q^{-6} +5 q^{-7} - q^{-8} -2 q^{-9} + q^{-10} </math> | |
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coloured_jones_3 = <math>-q^{25}+3 q^{23}+3 q^{22}-7 q^{21}-6 q^{20}+8 q^{19}+13 q^{18}-10 q^{17}-21 q^{16}+11 q^{15}+25 q^{14}-8 q^{13}-29 q^{12}+8 q^{11}+29 q^{10}-6 q^9-30 q^8+8 q^7+26 q^6-5 q^5-26 q^4+6 q^3+21 q^2-2 q-19+2 q^{-1} +14 q^{-2} -10 q^{-4} + q^{-5} +5 q^{-6} -2 q^{-7} -2 q^{-8} +5 q^{-9} -7 q^{-11} +7 q^{-13} +2 q^{-14} -7 q^{-15} -2 q^{-16} +4 q^{-17} +2 q^{-18} - q^{-19} -2 q^{-20} + q^{-21} </math> | |
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{{Display Coloured Jones|J2=<math>q^{13}-q^{12}-2 q^{11}+3 q^{10}-4 q^8+3 q^7+2 q^6-3 q^5+q^4+q^3-q+3 q^{-1} -3 q^{-2} -2 q^{-3} +6 q^{-4} -3 q^{-5} -3 q^{-6} +5 q^{-7} - q^{-8} -2 q^{-9} + q^{-10} </math>|J3=<math>-q^{25}+3 q^{23}+3 q^{22}-7 q^{21}-6 q^{20}+8 q^{19}+13 q^{18}-10 q^{17}-21 q^{16}+11 q^{15}+25 q^{14}-8 q^{13}-29 q^{12}+8 q^{11}+29 q^{10}-6 q^9-30 q^8+8 q^7+26 q^6-5 q^5-26 q^4+6 q^3+21 q^2-2 q-19+2 q^{-1} +14 q^{-2} -10 q^{-4} + q^{-5} +5 q^{-6} -2 q^{-7} -2 q^{-8} +5 q^{-9} -7 q^{-11} +7 q^{-13} +2 q^{-14} -7 q^{-15} -2 q^{-16} +4 q^{-17} +2 q^{-18} - q^{-19} -2 q^{-20} + q^{-21} </math>|J4=<math>q^{42}-q^{41}-2 q^{39}-2 q^{38}+5 q^{37}+q^{36}+8 q^{35}-7 q^{34}-15 q^{33}+q^{32}+2 q^{31}+33 q^{30}+2 q^{29}-31 q^{28}-22 q^{27}-13 q^{26}+62 q^{25}+26 q^{24}-33 q^{23}-42 q^{22}-36 q^{21}+71 q^{20}+45 q^{19}-22 q^{18}-48 q^{17}-51 q^{16}+67 q^{15}+51 q^{14}-16 q^{13}-45 q^{12}-54 q^{11}+57 q^{10}+52 q^9-9 q^8-39 q^7-55 q^6+41 q^5+53 q^4+4 q^3-29 q^2-58 q+17+50 q^{-1} +20 q^{-2} -13 q^{-3} -54 q^{-4} -9 q^{-5} +36 q^{-6} +27 q^{-7} +6 q^{-8} -37 q^{-9} -23 q^{-10} +18 q^{-11} +17 q^{-12} +13 q^{-13} -14 q^{-14} -17 q^{-15} +11 q^{-16} +4 q^{-18} -4 q^{-19} -7 q^{-20} +16 q^{-21} -3 q^{-22} -3 q^{-23} -7 q^{-24} -6 q^{-25} +15 q^{-26} + q^{-27} -5 q^{-29} -6 q^{-30} +5 q^{-31} + q^{-32} +2 q^{-33} - q^{-34} -2 q^{-35} + q^{-36} </math>|J5=<math>-q^{62}+q^{60}+3 q^{59}+2 q^{58}-7 q^{56}-11 q^{55}-4 q^{54}+10 q^{53}+22 q^{52}+19 q^{51}-4 q^{50}-37 q^{49}-44 q^{48}-12 q^{47}+45 q^{46}+73 q^{45}+45 q^{44}-37 q^{43}-106 q^{42}-86 q^{41}+19 q^{40}+124 q^{39}+128 q^{38}+12 q^{37}-130 q^{36}-165 q^{35}-43 q^{34}+126 q^{33}+184 q^{32}+69 q^{31}-113 q^{30}-191 q^{29}-87 q^{28}+103 q^{27}+188 q^{26}+95 q^{25}-93 q^{24}-184 q^{23}-94 q^{22}+86 q^{21}+176 q^{20}+94 q^{19}-81 q^{18}-171 q^{17}-90 q^{16}+75 q^{15}+159 q^{14}+90 q^{13}-64 q^{12}-149 q^{11}-89 q^{10}+49 q^9+131 q^8+92 q^7-28 q^6-110 q^5-92 q^4+4 q^3+86 q^2+87 q+21-52 q^{-1} -80 q^{-2} -44 q^{-3} +22 q^{-4} +60 q^{-5} +54 q^{-6} +19 q^{-7} -37 q^{-8} -61 q^{-9} -40 q^{-10} +3 q^{-11} +48 q^{-12} +61 q^{-13} +24 q^{-14} -30 q^{-15} -58 q^{-16} -46 q^{-17} +4 q^{-18} +51 q^{-19} +51 q^{-20} +14 q^{-21} -31 q^{-22} -45 q^{-23} -23 q^{-24} +13 q^{-25} +32 q^{-26} +20 q^{-27} -5 q^{-28} -16 q^{-29} -10 q^{-30} +11 q^{-32} +4 q^{-33} -10 q^{-34} -6 q^{-35} +2 q^{-36} +8 q^{-37} +10 q^{-38} +2 q^{-39} -12 q^{-40} -10 q^{-41} -2 q^{-42} +6 q^{-43} +8 q^{-44} +4 q^{-45} -7 q^{-47} -4 q^{-48} + q^{-49} +2 q^{-50} + q^{-51} +2 q^{-52} - q^{-53} -2 q^{-54} + q^{-55} </math>|J6=<math>q^{87}-q^{86}-2 q^{83}-2 q^{82}-q^{81}+5 q^{80}+4 q^{79}+10 q^{78}+5 q^{77}-6 q^{76}-22 q^{75}-25 q^{74}-14 q^{73}+4 q^{72}+54 q^{71}+61 q^{70}+41 q^{69}-30 q^{68}-86 q^{67}-123 q^{66}-88 q^{65}+64 q^{64}+167 q^{63}+213 q^{62}+90 q^{61}-80 q^{60}-290 q^{59}-322 q^{58}-83 q^{57}+187 q^{56}+422 q^{55}+346 q^{54}+98 q^{53}-358 q^{52}-560 q^{51}-339 q^{50}+53 q^{49}+506 q^{48}+562 q^{47}+341 q^{46}-284 q^{45}-649 q^{44}-522 q^{43}-113 q^{42}+461 q^{41}+633 q^{40}+489 q^{39}-193 q^{38}-625 q^{37}-574 q^{36}-196 q^{35}+403 q^{34}+620 q^{33}+528 q^{32}-159 q^{31}-588 q^{30}-566 q^{29}-214 q^{28}+378 q^{27}+599 q^{26}+523 q^{25}-151 q^{24}-557 q^{23}-548 q^{22}-221 q^{21}+347 q^{20}+571 q^{19}+516 q^{18}-119 q^{17}-497 q^{16}-527 q^{15}-251 q^{14}+273 q^{13}+511 q^{12}+516 q^{11}-37 q^{10}-384 q^9-487 q^8-307 q^7+141 q^6+402 q^5+503 q^4+81 q^3-210 q^2-398 q-352-28 q^{-1} +231 q^{-2} +427 q^{-3} +177 q^{-4} -4 q^{-5} -233 q^{-6} -315 q^{-7} -155 q^{-8} +31 q^{-9} +253 q^{-10} +166 q^{-11} +142 q^{-12} -33 q^{-13} -166 q^{-14} -150 q^{-15} -93 q^{-16} +48 q^{-17} +35 q^{-18} +132 q^{-19} +77 q^{-20} +4 q^{-21} -21 q^{-22} -63 q^{-23} -47 q^{-24} -92 q^{-25} +8 q^{-26} +33 q^{-27} +52 q^{-28} +86 q^{-29} +42 q^{-30} -2 q^{-31} -90 q^{-32} -65 q^{-33} -51 q^{-34} -10 q^{-35} +73 q^{-36} +69 q^{-37} +50 q^{-38} -16 q^{-39} -34 q^{-40} -53 q^{-41} -45 q^{-42} +25 q^{-43} +22 q^{-44} +30 q^{-45} +7 q^{-46} - q^{-47} -18 q^{-48} -22 q^{-49} +24 q^{-50} +7 q^{-52} -6 q^{-53} -6 q^{-54} -14 q^{-55} -11 q^{-56} +28 q^{-57} +5 q^{-58} +9 q^{-59} -2 q^{-60} -5 q^{-61} -15 q^{-62} -12 q^{-63} +11 q^{-64} +2 q^{-65} +8 q^{-66} +3 q^{-67} +3 q^{-68} -7 q^{-69} -6 q^{-70} +3 q^{-71} -2 q^{-72} +2 q^{-73} + q^{-74} +2 q^{-75} - q^{-76} -2 q^{-77} + q^{-78} </math>|J7=<math>-q^{115}+q^{113}+q^{112}+2 q^{111}+2 q^{110}-q^{108}-9 q^{107}-10 q^{106}-6 q^{105}-2 q^{104}+12 q^{103}+23 q^{102}+32 q^{101}+29 q^{100}-5 q^{99}-45 q^{98}-64 q^{97}-82 q^{96}-48 q^{95}+22 q^{94}+109 q^{93}+184 q^{92}+154 q^{91}+46 q^{90}-99 q^{89}-272 q^{88}-336 q^{87}-235 q^{86}+346 q^{84}+539 q^{83}+497 q^{82}+225 q^{81}-284 q^{80}-708 q^{79}-840 q^{78}-577 q^{77}+103 q^{76}+791 q^{75}+1145 q^{74}+994 q^{73}+211 q^{72}-728 q^{71}-1362 q^{70}-1414 q^{69}-601 q^{68}+552 q^{67}+1476 q^{66}+1729 q^{65}+965 q^{64}-298 q^{63}-1434 q^{62}-1937 q^{61}-1286 q^{60}+46 q^{59}+1340 q^{58}+2021 q^{57}+1470 q^{56}+169 q^{55}-1195 q^{54}-2016 q^{53}-1585 q^{52}-312 q^{51}+1087 q^{50}+1972 q^{49}+1607 q^{48}+387 q^{47}-988 q^{46}-1922 q^{45}-1614 q^{44}-415 q^{43}+951 q^{42}+1875 q^{41}+1580 q^{40}+430 q^{39}-907 q^{38}-1850 q^{37}-1583 q^{36}-422 q^{35}+907 q^{34}+1811 q^{33}+1544 q^{32}+440 q^{31}-852 q^{30}-1783 q^{29}-1560 q^{28}-452 q^{27}+830 q^{26}+1720 q^{25}+1518 q^{24}+504 q^{23}-719 q^{22}-1652 q^{21}-1534 q^{20}-564 q^{19}+630 q^{18}+1538 q^{17}+1490 q^{16}+666 q^{15}-447 q^{14}-1400 q^{13}-1488 q^{12}-772 q^{11}+265 q^{10}+1210 q^9+1418 q^8+897 q^7-11 q^6-978 q^5-1338 q^4-1001 q^3-232 q^2+689 q+1174+1068 q^{-1} +489 q^{-2} -368 q^{-3} -962 q^{-4} -1049 q^{-5} -692 q^{-6} +30 q^{-7} +672 q^{-8} +952 q^{-9} +815 q^{-10} +263 q^{-11} -345 q^{-12} -745 q^{-13} -825 q^{-14} -492 q^{-15} +34 q^{-16} +478 q^{-17} +709 q^{-18} +591 q^{-19} +220 q^{-20} -184 q^{-21} -508 q^{-22} -556 q^{-23} -359 q^{-24} -59 q^{-25} +255 q^{-26} +417 q^{-27} +370 q^{-28} +200 q^{-29} -35 q^{-30} -223 q^{-31} -265 q^{-32} -228 q^{-33} -100 q^{-34} +46 q^{-35} +117 q^{-36} +150 q^{-37} +124 q^{-38} +60 q^{-39} +21 q^{-40} -26 q^{-41} -73 q^{-42} -75 q^{-43} -92 q^{-44} -71 q^{-45} -18 q^{-46} +20 q^{-47} +91 q^{-48} +123 q^{-49} +85 q^{-50} +37 q^{-51} -40 q^{-52} -104 q^{-53} -99 q^{-54} -88 q^{-55} -18 q^{-56} +66 q^{-57} +82 q^{-58} +86 q^{-59} +38 q^{-60} -23 q^{-61} -35 q^{-62} -59 q^{-63} -50 q^{-64} +5 q^{-65} +18 q^{-66} +36 q^{-67} +23 q^{-68} -14 q^{-69} - q^{-70} -12 q^{-71} -16 q^{-72} +12 q^{-73} +11 q^{-74} +17 q^{-75} +10 q^{-76} -24 q^{-77} -12 q^{-78} -13 q^{-79} -12 q^{-80} +13 q^{-81} +10 q^{-82} +15 q^{-83} +16 q^{-84} -4 q^{-85} -8 q^{-86} -12 q^{-87} -14 q^{-88} +4 q^{-89} +3 q^{-91} +10 q^{-92} +3 q^{-93} +2 q^{-94} -4 q^{-95} -6 q^{-96} + q^{-97} -2 q^{-99} +2 q^{-100} + q^{-101} +2 q^{-102} - q^{-103} -2 q^{-104} + q^{-105} </math>}} |
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coloured_jones_4 = <math>q^{42}-q^{41}-2 q^{39}-2 q^{38}+5 q^{37}+q^{36}+8 q^{35}-7 q^{34}-15 q^{33}+q^{32}+2 q^{31}+33 q^{30}+2 q^{29}-31 q^{28}-22 q^{27}-13 q^{26}+62 q^{25}+26 q^{24}-33 q^{23}-42 q^{22}-36 q^{21}+71 q^{20}+45 q^{19}-22 q^{18}-48 q^{17}-51 q^{16}+67 q^{15}+51 q^{14}-16 q^{13}-45 q^{12}-54 q^{11}+57 q^{10}+52 q^9-9 q^8-39 q^7-55 q^6+41 q^5+53 q^4+4 q^3-29 q^2-58 q+17+50 q^{-1} +20 q^{-2} -13 q^{-3} -54 q^{-4} -9 q^{-5} +36 q^{-6} +27 q^{-7} +6 q^{-8} -37 q^{-9} -23 q^{-10} +18 q^{-11} +17 q^{-12} +13 q^{-13} -14 q^{-14} -17 q^{-15} +11 q^{-16} +4 q^{-18} -4 q^{-19} -7 q^{-20} +16 q^{-21} -3 q^{-22} -3 q^{-23} -7 q^{-24} -6 q^{-25} +15 q^{-26} + q^{-27} -5 q^{-29} -6 q^{-30} +5 q^{-31} + q^{-32} +2 q^{-33} - q^{-34} -2 q^{-35} + q^{-36} </math> | |
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coloured_jones_5 = <math>-q^{62}+q^{60}+3 q^{59}+2 q^{58}-7 q^{56}-11 q^{55}-4 q^{54}+10 q^{53}+22 q^{52}+19 q^{51}-4 q^{50}-37 q^{49}-44 q^{48}-12 q^{47}+45 q^{46}+73 q^{45}+45 q^{44}-37 q^{43}-106 q^{42}-86 q^{41}+19 q^{40}+124 q^{39}+128 q^{38}+12 q^{37}-130 q^{36}-165 q^{35}-43 q^{34}+126 q^{33}+184 q^{32}+69 q^{31}-113 q^{30}-191 q^{29}-87 q^{28}+103 q^{27}+188 q^{26}+95 q^{25}-93 q^{24}-184 q^{23}-94 q^{22}+86 q^{21}+176 q^{20}+94 q^{19}-81 q^{18}-171 q^{17}-90 q^{16}+75 q^{15}+159 q^{14}+90 q^{13}-64 q^{12}-149 q^{11}-89 q^{10}+49 q^9+131 q^8+92 q^7-28 q^6-110 q^5-92 q^4+4 q^3+86 q^2+87 q+21-52 q^{-1} -80 q^{-2} -44 q^{-3} +22 q^{-4} +60 q^{-5} +54 q^{-6} +19 q^{-7} -37 q^{-8} -61 q^{-9} -40 q^{-10} +3 q^{-11} +48 q^{-12} +61 q^{-13} +24 q^{-14} -30 q^{-15} -58 q^{-16} -46 q^{-17} +4 q^{-18} +51 q^{-19} +51 q^{-20} +14 q^{-21} -31 q^{-22} -45 q^{-23} -23 q^{-24} +13 q^{-25} +32 q^{-26} +20 q^{-27} -5 q^{-28} -16 q^{-29} -10 q^{-30} +11 q^{-32} +4 q^{-33} -10 q^{-34} -6 q^{-35} +2 q^{-36} +8 q^{-37} +10 q^{-38} +2 q^{-39} -12 q^{-40} -10 q^{-41} -2 q^{-42} +6 q^{-43} +8 q^{-44} +4 q^{-45} -7 q^{-47} -4 q^{-48} + q^{-49} +2 q^{-50} + q^{-51} +2 q^{-52} - q^{-53} -2 q^{-54} + q^{-55} </math> | |
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coloured_jones_6 = <math>q^{87}-q^{86}-2 q^{83}-2 q^{82}-q^{81}+5 q^{80}+4 q^{79}+10 q^{78}+5 q^{77}-6 q^{76}-22 q^{75}-25 q^{74}-14 q^{73}+4 q^{72}+54 q^{71}+61 q^{70}+41 q^{69}-30 q^{68}-86 q^{67}-123 q^{66}-88 q^{65}+64 q^{64}+167 q^{63}+213 q^{62}+90 q^{61}-80 q^{60}-290 q^{59}-322 q^{58}-83 q^{57}+187 q^{56}+422 q^{55}+346 q^{54}+98 q^{53}-358 q^{52}-560 q^{51}-339 q^{50}+53 q^{49}+506 q^{48}+562 q^{47}+341 q^{46}-284 q^{45}-649 q^{44}-522 q^{43}-113 q^{42}+461 q^{41}+633 q^{40}+489 q^{39}-193 q^{38}-625 q^{37}-574 q^{36}-196 q^{35}+403 q^{34}+620 q^{33}+528 q^{32}-159 q^{31}-588 q^{30}-566 q^{29}-214 q^{28}+378 q^{27}+599 q^{26}+523 q^{25}-151 q^{24}-557 q^{23}-548 q^{22}-221 q^{21}+347 q^{20}+571 q^{19}+516 q^{18}-119 q^{17}-497 q^{16}-527 q^{15}-251 q^{14}+273 q^{13}+511 q^{12}+516 q^{11}-37 q^{10}-384 q^9-487 q^8-307 q^7+141 q^6+402 q^5+503 q^4+81 q^3-210 q^2-398 q-352-28 q^{-1} +231 q^{-2} +427 q^{-3} +177 q^{-4} -4 q^{-5} -233 q^{-6} -315 q^{-7} -155 q^{-8} +31 q^{-9} +253 q^{-10} +166 q^{-11} +142 q^{-12} -33 q^{-13} -166 q^{-14} -150 q^{-15} -93 q^{-16} +48 q^{-17} +35 q^{-18} +132 q^{-19} +77 q^{-20} +4 q^{-21} -21 q^{-22} -63 q^{-23} -47 q^{-24} -92 q^{-25} +8 q^{-26} +33 q^{-27} +52 q^{-28} +86 q^{-29} +42 q^{-30} -2 q^{-31} -90 q^{-32} -65 q^{-33} -51 q^{-34} -10 q^{-35} +73 q^{-36} +69 q^{-37} +50 q^{-38} -16 q^{-39} -34 q^{-40} -53 q^{-41} -45 q^{-42} +25 q^{-43} +22 q^{-44} +30 q^{-45} +7 q^{-46} - q^{-47} -18 q^{-48} -22 q^{-49} +24 q^{-50} +7 q^{-52} -6 q^{-53} -6 q^{-54} -14 q^{-55} -11 q^{-56} +28 q^{-57} +5 q^{-58} +9 q^{-59} -2 q^{-60} -5 q^{-61} -15 q^{-62} -12 q^{-63} +11 q^{-64} +2 q^{-65} +8 q^{-66} +3 q^{-67} +3 q^{-68} -7 q^{-69} -6 q^{-70} +3 q^{-71} -2 q^{-72} +2 q^{-73} + q^{-74} +2 q^{-75} - q^{-76} -2 q^{-77} + q^{-78} </math> | |
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coloured_jones_7 = <math>-q^{115}+q^{113}+q^{112}+2 q^{111}+2 q^{110}-q^{108}-9 q^{107}-10 q^{106}-6 q^{105}-2 q^{104}+12 q^{103}+23 q^{102}+32 q^{101}+29 q^{100}-5 q^{99}-45 q^{98}-64 q^{97}-82 q^{96}-48 q^{95}+22 q^{94}+109 q^{93}+184 q^{92}+154 q^{91}+46 q^{90}-99 q^{89}-272 q^{88}-336 q^{87}-235 q^{86}+346 q^{84}+539 q^{83}+497 q^{82}+225 q^{81}-284 q^{80}-708 q^{79}-840 q^{78}-577 q^{77}+103 q^{76}+791 q^{75}+1145 q^{74}+994 q^{73}+211 q^{72}-728 q^{71}-1362 q^{70}-1414 q^{69}-601 q^{68}+552 q^{67}+1476 q^{66}+1729 q^{65}+965 q^{64}-298 q^{63}-1434 q^{62}-1937 q^{61}-1286 q^{60}+46 q^{59}+1340 q^{58}+2021 q^{57}+1470 q^{56}+169 q^{55}-1195 q^{54}-2016 q^{53}-1585 q^{52}-312 q^{51}+1087 q^{50}+1972 q^{49}+1607 q^{48}+387 q^{47}-988 q^{46}-1922 q^{45}-1614 q^{44}-415 q^{43}+951 q^{42}+1875 q^{41}+1580 q^{40}+430 q^{39}-907 q^{38}-1850 q^{37}-1583 q^{36}-422 q^{35}+907 q^{34}+1811 q^{33}+1544 q^{32}+440 q^{31}-852 q^{30}-1783 q^{29}-1560 q^{28}-452 q^{27}+830 q^{26}+1720 q^{25}+1518 q^{24}+504 q^{23}-719 q^{22}-1652 q^{21}-1534 q^{20}-564 q^{19}+630 q^{18}+1538 q^{17}+1490 q^{16}+666 q^{15}-447 q^{14}-1400 q^{13}-1488 q^{12}-772 q^{11}+265 q^{10}+1210 q^9+1418 q^8+897 q^7-11 q^6-978 q^5-1338 q^4-1001 q^3-232 q^2+689 q+1174+1068 q^{-1} +489 q^{-2} -368 q^{-3} -962 q^{-4} -1049 q^{-5} -692 q^{-6} +30 q^{-7} +672 q^{-8} +952 q^{-9} +815 q^{-10} +263 q^{-11} -345 q^{-12} -745 q^{-13} -825 q^{-14} -492 q^{-15} +34 q^{-16} +478 q^{-17} +709 q^{-18} +591 q^{-19} +220 q^{-20} -184 q^{-21} -508 q^{-22} -556 q^{-23} -359 q^{-24} -59 q^{-25} +255 q^{-26} +417 q^{-27} +370 q^{-28} +200 q^{-29} -35 q^{-30} -223 q^{-31} -265 q^{-32} -228 q^{-33} -100 q^{-34} +46 q^{-35} +117 q^{-36} +150 q^{-37} +124 q^{-38} +60 q^{-39} +21 q^{-40} -26 q^{-41} -73 q^{-42} -75 q^{-43} -92 q^{-44} -71 q^{-45} -18 q^{-46} +20 q^{-47} +91 q^{-48} +123 q^{-49} +85 q^{-50} +37 q^{-51} -40 q^{-52} -104 q^{-53} -99 q^{-54} -88 q^{-55} -18 q^{-56} +66 q^{-57} +82 q^{-58} +86 q^{-59} +38 q^{-60} -23 q^{-61} -35 q^{-62} -59 q^{-63} -50 q^{-64} +5 q^{-65} +18 q^{-66} +36 q^{-67} +23 q^{-68} -14 q^{-69} - q^{-70} -12 q^{-71} -16 q^{-72} +12 q^{-73} +11 q^{-74} +17 q^{-75} +10 q^{-76} -24 q^{-77} -12 q^{-78} -13 q^{-79} -12 q^{-80} +13 q^{-81} +10 q^{-82} +15 q^{-83} +16 q^{-84} -4 q^{-85} -8 q^{-86} -12 q^{-87} -14 q^{-88} +4 q^{-89} +3 q^{-91} +10 q^{-92} +3 q^{-93} +2 q^{-94} -4 q^{-95} -6 q^{-96} + q^{-97} -2 q^{-99} +2 q^{-100} + q^{-101} +2 q^{-102} - q^{-103} -2 q^{-104} + q^{-105} </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[10, 136]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[1, 4, 2, 5], X[5, 10, 6, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], |
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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[10, 136]]</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[1, 4, 2, 5], X[5, 10, 6, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], |
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X[14, 8, 15, 7], X[18, 12, 19, 11], X[20, 15, 1, 16], |
X[14, 8, 15, 7], X[18, 12, 19, 11], X[20, 15, 1, 16], |
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X[16, 19, 17, 20], X[12, 18, 13, 17], X[6, 14, 7, 13]]</nowiki></ |
X[16, 19, 17, 20], X[12, 18, 13, 17], X[6, 14, 7, 13]]</nowiki></code></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[10, 136]]</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[10, 136]]</nowiki></code></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, -10, 5, 3, -4, 2, 6, -9, 10, -5, 7, -8, 9, |
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-6, 8, -7]</nowiki></ |
-6, 8, -7]</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[10, 136]]</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[10, 136]]</nowiki></code></td></tr> |
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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[10, 136]]</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, -18, -6, -20, -12, -16]</nowiki></code></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: blue; border: 0px; padding: 0em"><nowiki>Out[6]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{5, 10}</nowiki></pre></td></tr> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[5]:=</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>br = BR[Knot[10, 136]]</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">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[5, {1, -2, 1, -2, -3, 2, 2, 4, -3, 4}]</nowiki></code></td></tr> |
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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[10, 136]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:10_136_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[10, 136]]&) /@ {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">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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<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[10, 136]][t]</nowiki></pre></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>{5, 10}</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[10, 136]]</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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<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[10, 136]]]</nowiki></code></td></tr> |
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<tr align=left><td></td><td>[[Image:10_136_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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<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[10, 136]]&) /@ { |
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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, NotAvailable, 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[10, 136]][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 4 2 |
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-5 - t + - + 4 t - t |
-5 - t + - + 4 t - t |
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t</nowiki></ |
t</nowiki></code></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[10, 136]][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[10, 136]][z]</nowiki></code></td></tr> |
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1 - 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> 4 |
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1 - 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[10, 136]], KnotSignature[Knot[10, 136]]}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[13]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{15, 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 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> -3 2 2 2 3 4 |
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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[8, 21], Knot[10, 136]}</nowiki></code></td></tr> |
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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[10, 136]], KnotSignature[Knot[10, 136]]}</nowiki></code></td></tr> |
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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>{15, 2}</nowiki></code></td></tr> |
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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[10, 136]][q]</nowiki></code></td></tr> |
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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> -3 2 2 2 3 4 |
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-2 + q - -- + - + 3 q - 2 q + 2 q - q |
-2 + q - -- + - + 3 q - 2 q + 2 q - q |
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2 q |
2 q |
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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[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 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[10, 136]][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[10, 136], Knot[11, NonAlternating, 92]}</nowiki></code></td></tr> |
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q - q + q + 2 q + q + q - q - q</nowiki></pre></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[10, 136]][a, z]</nowiki></pre></td></tr> |
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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[10, 136]][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> -10 -2 4 6 8 10 12 14 |
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q - q + q + 2 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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<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[10, 136]][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[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 |
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-4 3 2 2 2 z 2 2 4 |
-4 3 2 2 2 z 2 2 4 |
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-2 - a + -- + a - 3 z + ---- + a z - z |
-2 - a + -- + a - 3 z + ---- + a z - z |
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2 2 |
2 2 |
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a a</nowiki></ |
a a</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[18]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kauffman[Knot[10, 136]][a, z]</nowiki></pre></td></tr> |
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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[10, 136]][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 2 |
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-4 3 2 2 z 4 z 2 z 4 z 2 2 |
-4 3 2 2 z 4 z 2 z 4 z 2 2 |
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-2 - a - -- - a - --- - --- - 2 a z + 6 z + -- + ---- + 3 a z + |
-2 - a - -- - a - --- - --- - 2 a z + 6 z + -- + ---- + 3 a z + |
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Line 165: | Line 206: | ||
9 a z - 3 z - ---- + a z + -- + ---- + 2 a z + z + -- |
9 a z - 3 z - ---- + a z + -- + ---- + 2 a z + z + -- |
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2 3 a 2 |
2 3 a 2 |
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a a a</nowiki></ |
a a a</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[10, 136]], Vassiliev[3][Knot[10, 136]]}</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[10, 136]], Vassiliev[3][Knot[10, 136]]}</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[10, 136]][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>{0, 1}</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[10, 136]][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>1 3 1 1 1 1 1 2 q |
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- + 2 q + 2 q + ----- + ----- + ----- + ----- + ---- + --- + - + |
- + 2 q + 2 q + ----- + ----- + ----- + ----- + ---- + --- + - + |
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q 7 4 5 3 3 3 3 2 2 q t t |
q 7 4 5 3 3 3 3 2 2 q t t |
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Line 177: | Line 226: | ||
3 5 5 2 7 2 9 3 |
3 5 5 2 7 2 9 3 |
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q t + q t + q t + q t + q t</nowiki></ |
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[10, 136], 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[10, 136], 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> -10 2 -8 5 3 3 6 2 3 3 3 4 |
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q - -- - q + -- - -- - -- + -- - -- - -- + - - q + q + q - |
q - -- - q + -- - -- - -- + -- - -- - -- + - - q + q + q - |
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9 7 6 5 4 3 2 q |
9 7 6 5 4 3 2 q |
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Line 186: | Line 239: | ||
5 6 7 8 10 11 12 13 |
5 6 7 8 10 11 12 13 |
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3 q + 2 q + 3 q - 4 q + 3 q - 2 q - q + q</nowiki></ |
3 q + 2 q + 3 q - 4 q + 3 q - 2 q - q + q</nowiki></code></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]] |
Latest revision as of 17:03, 1 September 2005
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(KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 136's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
Planar diagram presentation | X1425 X5,10,6,11 X3948 X9,3,10,2 X14,8,15,7 X18,12,19,11 X20,15,1,16 X16,19,17,20 X12,18,13,17 X6,14,7,13 |
Gauss code | -1, 4, -3, 1, -2, -10, 5, 3, -4, 2, 6, -9, 10, -5, 7, -8, 9, -6, 8, -7 |
Dowker-Thistlethwaite code | 4 8 10 -14 2 -18 -6 -20 -12 -16 |
Conway Notation | [22,22,2-] |
Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||||
Length is 10, width is 5, Braid index is 4 |
[{11, 2}, {1, 7}, {9, 5}, {7, 11}, {8, 10}, {2, 9}, {6, 4}, {5, 8}, {3, 6}, {4, 1}, {10, 3}] |
[edit Notes on presentations of 10 136] The knot 10_136 is the only knot in the Rolfsen Knot Table whose braid index is smaller than the width of its minimum braid.
The next such knot is K11n8.
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 136"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X1425 X5,10,6,11 X3948 X9,3,10,2 X14,8,15,7 X18,12,19,11 X20,15,1,16 X16,19,17,20 X12,18,13,17 X6,14,7,13 |
In[5]:=
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GaussCode[K]
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Out[5]=
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-1, 4, -3, 1, -2, -10, 5, 3, -4, 2, 6, -9, 10, -5, 7, -8, 9, -6, 8, -7 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 8 10 -14 2 -18 -6 -20 -12 -16 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$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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[22,22,2-] |
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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{ 5, 10, 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[{11, 2}, {1, 7}, {9, 5}, {7, 11}, {8, 10}, {2, 9}, {6, 4}, {5, 8}, {3, 6}, {4, 1}, {10, 3}] |
In[14]:=
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Draw[ap]
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Out[14]=
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-Graphics- |
Three dimensional invariants
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Four dimensional invariants
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Polynomial invariants
A1 Invariants.
Weight | Invariant |
---|---|
1 | |
2 | |
3 |
A2 Invariants.
Weight | Invariant |
---|---|
1,0 | |
2,0 |
A3 Invariants.
Weight | Invariant |
---|---|
0,1,0 | |
1,0,0 |
B2 Invariants.
Weight | Invariant |
---|---|
0,1 | |
1,0 |
G2 Invariants.
Weight | Invariant |
---|---|
1,0 |
.
KnotTheory`
, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
|
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["10 136"];
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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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{ 15, 2 } |
In[8]:=
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Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
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In[9]:=
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HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
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In[10]:=
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Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
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"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {8_21,}
Same Jones Polynomial (up to mirroring, ): {K11n92,}
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["10 136"];
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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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{8_21,} |
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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{K11n92,} |
Vassiliev invariants
V2 and V3: | (0, 1) |
V2,1 through V6,9: |
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V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.
Khovanov Homology
The coefficients of the monomials are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where 2 is the signature of 10 136. 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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