10 148: Difference between revisions
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<!-- WARNING! WARNING! WARNING! |
<!-- WARNING! WARNING! WARNING! |
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<!-- This page was generated from the splice base [[Rolfsen_Splice_Base]]. Please do not edit! |
<!-- This page was generated from the splice base [[Rolfsen_Splice_Base]]. Please do not edit! |
Latest revision as of 23:30, 27 May 2009
|
|
(KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 148's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
Planar diagram presentation | X4251 X8493 X5,12,6,13 X13,18,14,19 X9,16,10,17 X17,10,18,11 X15,20,16,1 X19,14,20,15 X11,6,12,7 X2837 |
Gauss code | 1, -10, 2, -1, -3, 9, 10, -2, -5, 6, -9, 3, -4, 8, -7, 5, -6, 4, -8, 7 |
Dowker-Thistlethwaite code | 4 8 -12 2 -16 -6 -18 -20 -10 -14 |
Conway Notation | [(3,2)(3,2-)] |
Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||
Length is 10, width is 3, Braid index is 3 |
[{3, 5}, {2, 4}, {1, 3}, {13, 6}, {5, 12}, {10, 13}, {11, 7}, {6, 8}, {4, 10}, {7, 9}, {8, 11}, {12, 2}, {9, 1}] |
[edit Notes on presentations of 10 148]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 148"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X4251 X8493 X5,12,6,13 X13,18,14,19 X9,16,10,17 X17,10,18,11 X15,20,16,1 X19,14,20,15 X11,6,12,7 X2837 |
In[5]:=
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GaussCode[K]
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Out[5]=
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1, -10, 2, -1, -3, 9, 10, -2, -5, 6, -9, 3, -4, 8, -7, 5, -6, 4, -8, 7 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 8 -12 2 -16 -6 -18 -20 -10 -14 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$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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[(3,2)(3,2-)] |
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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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{BR}(3,\{-1,-1,-1,-1,-2,1,1,-2,1,-2\})} |
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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{ 3, 10, 3 } |
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[{3, 5}, {2, 4}, {1, 3}, {13, 6}, {5, 12}, {10, 13}, {11, 7}, {6, 8}, {4, 10}, {7, 9}, {8, 11}, {12, 2}, {9, 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
|
Polynomial invariants
Alexander polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^3-3 t^2+7 t-9+7 t^{-1} -3 t^{-2} + t^{-3} } |
Conway polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6+3 z^4+4 z^2+1} |
2nd Alexander ideal (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}} |
Determinant and Signature | { 31, -2 } |
Jones polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1+3 q^{-1} -4 q^{-2} +6 q^{-3} -5 q^{-4} +5 q^{-5} -4 q^{-6} +2 q^{-7} - q^{-8} } |
HOMFLY-PT polynomial (db, data sources) | |
Kauffman polynomial (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^5 a^9-3 z^3 a^9+2 z a^9+2 z^6 a^8-5 z^4 a^8+2 z^2 a^8+2 z^7 a^7-4 z^5 a^7+z^3 a^7-z a^7+z^8 a^6-z^6 a^6+2 z^4 a^6-6 z^2 a^6+3 a^6+3 z^7 a^5-7 z^5 a^5+9 z^3 a^5-5 z a^5+z^8 a^4-3 z^6 a^4+10 z^4 a^4-11 z^2 a^4+5 a^4+z^7 a^3-2 z^5 a^3+6 z^3 a^3-3 z a^3+3 z^4 a^2-3 z^2 a^2+a^2+z^3 a-z a} |
The A2 invariant | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{24}-2 q^{20}-q^{18}+q^{16}+3 q^{12}+q^{10}+2 q^8+q^6-q^4+q^2-1} |
The G2 invariant | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{128}-q^{126}+3 q^{124}-4 q^{122}+3 q^{120}-q^{118}-3 q^{116}+9 q^{114}-13 q^{112}+15 q^{110}-11 q^{108}-q^{106}+12 q^{104}-22 q^{102}+25 q^{100}-18 q^{98}+3 q^{96}+10 q^{94}-23 q^{92}+20 q^{90}-11 q^{88}-7 q^{86}+17 q^{84}-21 q^{82}+10 q^{80}+5 q^{78}-21 q^{76}+28 q^{74}-25 q^{72}+12 q^{70}+4 q^{68}-20 q^{66}+31 q^{64}-29 q^{62}+22 q^{60}-5 q^{58}-8 q^{56}+22 q^{54}-24 q^{52}+20 q^{50}-5 q^{48}-6 q^{46}+19 q^{44}-16 q^{42}+7 q^{40}+12 q^{38}-21 q^{36}+25 q^{34}-14 q^{32}-2 q^{30}+17 q^{28}-24 q^{26}+24 q^{24}-13 q^{22}+2 q^{20}+7 q^{18}-13 q^{16}+10 q^{14}-7 q^{12}+3 q^{10}-2 q^6-q^2+1- q^{-2} + q^{-4} } |
A1 Invariants.
Weight | Invariant |
---|---|
1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{17}+q^{15}-2 q^{13}+q^{11}+q^7+2 q^5-q^3+2 q- q^{-1} } |
2 | |
3 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{93}+q^{91}+q^{89}-q^{87}-3 q^{85}+q^{83}+7 q^{81}+q^{79}-11 q^{77}-7 q^{75}+12 q^{73}+18 q^{71}-8 q^{69}-24 q^{67}-2 q^{65}+26 q^{63}+14 q^{61}-25 q^{59}-21 q^{57}+17 q^{55}+26 q^{53}-11 q^{51}-26 q^{49}+3 q^{47}+24 q^{45}-q^{43}-18 q^{41}-6 q^{39}+13 q^{37}+9 q^{35}-9 q^{33}-17 q^{31}+3 q^{29}+24 q^{27}+5 q^{25}-25 q^{23}-13 q^{21}+29 q^{19}+21 q^{17}-21 q^{15}-24 q^{13}+12 q^{11}+24 q^9-q^7-18 q^5-3 q^3+8 q+7 q^{-1} -2 q^{-3} -4 q^{-5} - q^{-7} + q^{-11} } |
5 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{225}+q^{223}+q^{221}-q^{219}-q^{213}+q^{211}+4 q^{209}-q^{207}-6 q^{205}-5 q^{203}-3 q^{201}+7 q^{199}+19 q^{197}+18 q^{195}-7 q^{193}-34 q^{191}-43 q^{189}-21 q^{187}+34 q^{185}+82 q^{183}+80 q^{181}+3 q^{179}-100 q^{177}-154 q^{175}-100 q^{173}+51 q^{171}+208 q^{169}+238 q^{167}+73 q^{165}-179 q^{163}-345 q^{161}-273 q^{159}+37 q^{157}+376 q^{155}+471 q^{153}+190 q^{151}-277 q^{149}-584 q^{147}-454 q^{145}+64 q^{143}+584 q^{141}+659 q^{139}+189 q^{137}-452 q^{135}-754 q^{133}-435 q^{131}+258 q^{129}+735 q^{127}+587 q^{125}-53 q^{123}-619 q^{121}-641 q^{119}-108 q^{117}+477 q^{115}+596 q^{113}+203 q^{111}-328 q^{109}-510 q^{107}-222 q^{105}+215 q^{103}+398 q^{101}+212 q^{99}-134 q^{97}-312 q^{95}-182 q^{93}+88 q^{91}+240 q^{89}+173 q^{87}-44 q^{85}-217 q^{83}-197 q^{81}-5 q^{79}+201 q^{77}+257 q^{75}+91 q^{73}-190 q^{71}-356 q^{69}-219 q^{67}+148 q^{65}+448 q^{63}+388 q^{61}-43 q^{59}-516 q^{57}-583 q^{55}-110 q^{53}+505 q^{51}+733 q^{49}+329 q^{47}-397 q^{45}-807 q^{43}-535 q^{41}+200 q^{39}+754 q^{37}+673 q^{35}+51 q^{33}-576 q^{31}-695 q^{29}-268 q^{27}+327 q^{25}+595 q^{23}+381 q^{21}-72 q^{19}-393 q^{17}-381 q^{15}-104 q^{13}+193 q^{11}+278 q^9+162 q^7-23 q^5-150 q^3-143 q-48 q^{-1} +48 q^{-3} +78 q^{-5} +54 q^{-7} +7 q^{-9} -27 q^{-11} -33 q^{-13} -15 q^{-15} +4 q^{-17} +8 q^{-19} +8 q^{-21} +3 q^{-23} -2 q^{-25} -2 q^{-27} } |
6 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{312}-q^{310}-q^{308}+q^{306}-2 q^{300}+3 q^{298}-4 q^{294}+3 q^{292}+4 q^{290}+4 q^{288}-5 q^{286}-q^{284}-9 q^{282}-18 q^{280}-2 q^{278}+17 q^{276}+36 q^{274}+24 q^{272}+23 q^{270}-24 q^{268}-81 q^{266}-87 q^{264}-47 q^{262}+47 q^{260}+115 q^{258}+195 q^{256}+142 q^{254}-27 q^{252}-218 q^{250}-336 q^{248}-285 q^{246}-92 q^{244}+298 q^{242}+561 q^{240}+567 q^{238}+236 q^{236}-296 q^{234}-790 q^{232}-996 q^{230}-545 q^{228}+252 q^{226}+1081 q^{224}+1421 q^{222}+1031 q^{220}-71 q^{218}-1389 q^{216}-1985 q^{214}-1561 q^{212}-115 q^{210}+1569 q^{208}+2619 q^{206}+2168 q^{204}+301 q^{202}-1848 q^{200}-3159 q^{198}-2641 q^{196}-518 q^{194}+2188 q^{192}+3668 q^{190}+2940 q^{188}+438 q^{186}-2487 q^{184}-3955 q^{182}-3058 q^{180}-123 q^{178}+2840 q^{176}+4015 q^{174}+2701 q^{172}-322 q^{170}-3053 q^{168}-3818 q^{166}-2022 q^{164}+898 q^{162}+3084 q^{160}+3170 q^{158}+1206 q^{156}-1361 q^{154}-2857 q^{152}-2277 q^{150}-351 q^{148}+1604 q^{146}+2265 q^{144}+1365 q^{142}-303 q^{140}-1563 q^{138}-1537 q^{136}-543 q^{134}+689 q^{132}+1242 q^{130}+882 q^{128}-34 q^{126}-822 q^{124}-902 q^{122}-366 q^{120}+416 q^{118}+821 q^{116}+667 q^{114}+20 q^{112}-669 q^{110}-933 q^{108}-573 q^{106}+272 q^{104}+984 q^{102}+1182 q^{100}+553 q^{98}-563 q^{96}-1538 q^{94}-1584 q^{92}-480 q^{90}+1074 q^{88}+2227 q^{86}+1977 q^{84}+342 q^{82}-1865 q^{80}-3036 q^{78}-2226 q^{76}+108 q^{74}+2696 q^{72}+3684 q^{70}+2333 q^{68}-781 q^{66}-3493 q^{64}-4023 q^{62}-2045 q^{60}+1366 q^{58}+3933 q^{56}+4082 q^{54}+1550 q^{52}-1814 q^{50}-3942 q^{48}-3658 q^{46}-1135 q^{44}+1901 q^{42}+3640 q^{40}+3001 q^{38}+757 q^{36}-1674 q^{34}-2929 q^{32}-2368 q^{30}-556 q^{28}+1334 q^{26}+2124 q^{24}+1705 q^{22}+458 q^{20}-832 q^{18}-1437 q^{16}-1168 q^{14}-341 q^{12}+420 q^{10}+829 q^8+745 q^6+300 q^4-182 q^2-431-401 q^{-2} -216 q^{-4} +24 q^{-6} +187 q^{-8} +209 q^{-10} +120 q^{-12} +16 q^{-14} -54 q^{-16} -83 q^{-18} -63 q^{-20} -17 q^{-22} +13 q^{-24} +21 q^{-26} +18 q^{-28} +11 q^{-30} +2 q^{-32} -6 q^{-34} -3 q^{-36} - q^{-38} - q^{-40} - q^{-42} + q^{-46} } |
A2 Invariants.
Weight | Invariant |
---|---|
1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{24}-2 q^{20}-q^{18}+q^{16}+3 q^{12}+q^{10}+2 q^8+q^6-q^4+q^2-1} |
1,1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{68}-2 q^{66}+6 q^{64}-12 q^{62}+21 q^{60}-32 q^{58}+48 q^{56}-60 q^{54}+64 q^{52}-66 q^{50}+54 q^{48}-30 q^{46}-3 q^{44}+38 q^{42}-70 q^{40}+98 q^{38}-118 q^{36}+118 q^{34}-120 q^{32}+96 q^{30}-77 q^{28}+44 q^{26}-8 q^{24}-16 q^{22}+54 q^{20}-60 q^{18}+72 q^{16}-60 q^{14}+53 q^{12}-36 q^{10}+20 q^8-12 q^6+6 q^4-2 q^2-2 q^{-2} + q^{-4} } |
2,0 |
A3 Invariants.
Weight | Invariant |
---|---|
0,1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{54}-q^{52}+q^{50}+2 q^{48}-3 q^{46}+2 q^{44}+2 q^{42}-6 q^{40}+3 q^{38}-8 q^{34}-2 q^{32}-2 q^{30}-4 q^{28}-q^{26}+4 q^{24}+6 q^{22}+5 q^{20}+3 q^{18}+9 q^{16}-q^{14}-3 q^{12}+5 q^{10}-4 q^8-4 q^6+3 q^4-q^2-1+ q^{-2} } |
1,0,0 | |
1,0,1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{88}-2 q^{86}+5 q^{84}-5 q^{82}+2 q^{80}+7 q^{78}-19 q^{76}+28 q^{74}-23 q^{72}+5 q^{70}+22 q^{68}-51 q^{66}+60 q^{64}-51 q^{62}+22 q^{60}+20 q^{58}-47 q^{56}+68 q^{54}-52 q^{52}+31 q^{50}-14 q^{48}-8 q^{46}-19 q^{44}-4 q^{42}-14 q^{40}-33 q^{38}+53 q^{36}-67 q^{34}+76 q^{32}-21 q^{30}+10 q^{28}+56 q^{26}-50 q^{24}+59 q^{22}-26 q^{20}-q^{18}+16 q^{16}-22 q^{14}+7 q^{12}+q^{10}-7 q^8+4 q^6-4 q^2+3-2 q^{-2} + q^{-4} } |
A4 Invariants.
Weight | Invariant |
---|---|
0,1,0,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{68}+q^{64}+3 q^{62}+5 q^{56}+q^{54}-5 q^{52}-2 q^{50}-q^{48}-9 q^{46}-13 q^{44}-6 q^{42}-5 q^{40}-10 q^{38}-q^{36}+10 q^{34}+5 q^{32}+9 q^{30}+18 q^{28}+11 q^{26}+4 q^{24}+6 q^{22}+4 q^{20}-5 q^{18}-7 q^{16}-q^{12}-5 q^{10}-q^8+3 q^6-q^4+1} |
1,0,0,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{38}-3 q^{34}-2 q^{32}-2 q^{30}-2 q^{28}+q^{26}+q^{24}+4 q^{22}+3 q^{20}+4 q^{18}+2 q^{16}+2 q^{14}-2 q^6+q^4-q^2} |
B2 Invariants.
Weight | Invariant |
---|---|
0,1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{54}+q^{52}-3 q^{50}+4 q^{48}-5 q^{46}+6 q^{44}-6 q^{42}+4 q^{40}-3 q^{38}+2 q^{34}-6 q^{32}+8 q^{30}-10 q^{28}+11 q^{26}-10 q^{24}+10 q^{22}-5 q^{20}+5 q^{18}+q^{16}-q^{14}+5 q^{12}-5 q^{10}+6 q^8-6 q^6+5 q^4-3 q^2+1- q^{-2} } |
1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{88}-q^{84}-q^{82}+2 q^{80}+3 q^{78}-q^{76}-4 q^{74}-q^{72}+5 q^{70}+4 q^{68}-5 q^{66}-6 q^{64}+q^{62}+6 q^{60}-7 q^{56}-4 q^{54}+2 q^{52}+q^{50}-4 q^{48}-4 q^{46}+q^{44}+4 q^{42}-3 q^{38}+q^{36}+7 q^{34}+4 q^{32}-q^{30}-2 q^{28}+6 q^{26}+6 q^{24}-q^{22}-6 q^{20}+q^{18}+6 q^{16}+2 q^{14}-5 q^{12}-5 q^{10}+q^8+4 q^6-2 q^2-1+ q^{-4} } |
D4 Invariants.
Weight | Invariant |
---|---|
1,0,0,0 |
G2 Invariants.
Weight | Invariant |
---|---|
1,0 |
.
KnotTheory`
, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
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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["10 148"];
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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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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^3-3 t^2+7 t-9+7 t^{-1} -3 t^{-2} + t^{-3} } |
In[5]:=
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Conway[K][z]
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Out[5]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6+3 z^4+4 z^2+1} |
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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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}} |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 31, -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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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1+3 q^{-1} -4 q^{-2} +6 q^{-3} -5 q^{-4} +5 q^{-5} -4 q^{-6} +2 q^{-7} - q^{-8} } |
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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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^5 a^9-3 z^3 a^9+2 z a^9+2 z^6 a^8-5 z^4 a^8+2 z^2 a^8+2 z^7 a^7-4 z^5 a^7+z^3 a^7-z a^7+z^8 a^6-z^6 a^6+2 z^4 a^6-6 z^2 a^6+3 a^6+3 z^7 a^5-7 z^5 a^5+9 z^3 a^5-5 z a^5+z^8 a^4-3 z^6 a^4+10 z^4 a^4-11 z^2 a^4+5 a^4+z^7 a^3-2 z^5 a^3+6 z^3 a^3-3 z a^3+3 z^4 a^2-3 z^2 a^2+a^2+z^3 a-z a} |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, ): {}
KnotTheory`
. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
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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["10 148"];
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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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{} |
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: | (4, -7) |
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} -2 is the signature of 10 148. 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
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n} |
2 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2+3 q^{-1} +4 q^{-2} -12 q^{-3} +6 q^{-4} +15 q^{-5} -24 q^{-6} +5 q^{-7} +26 q^{-8} -30 q^{-9} + q^{-10} +29 q^{-11} -26 q^{-12} -5 q^{-13} +26 q^{-14} -16 q^{-15} -9 q^{-16} +17 q^{-17} -5 q^{-18} -7 q^{-19} +6 q^{-20} -2 q^{-22} + q^{-23} } |
3 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^4-q^3-q^2-3 q+3+8 q^{-1} -14 q^{-3} -12 q^{-4} +25 q^{-5} +25 q^{-6} -26 q^{-7} -48 q^{-8} +28 q^{-9} +67 q^{-10} -18 q^{-11} -90 q^{-12} +16 q^{-13} +97 q^{-14} + q^{-15} -111 q^{-16} -4 q^{-17} +105 q^{-18} +19 q^{-19} -107 q^{-20} -23 q^{-21} +93 q^{-22} +36 q^{-23} -82 q^{-24} -44 q^{-25} +64 q^{-26} +51 q^{-27} -45 q^{-28} -53 q^{-29} +26 q^{-30} +47 q^{-31} -6 q^{-32} -41 q^{-33} -2 q^{-34} +25 q^{-35} +10 q^{-36} -15 q^{-37} -8 q^{-38} +6 q^{-39} +6 q^{-40} -3 q^{-41} -2 q^{-42} +2 q^{-44} - q^{-45} } |
4 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^8+q^7+3 q^6-2 q^4-9 q^3-6 q^2+14 q+16+14 q^{-1} -23 q^{-2} -53 q^{-3} - q^{-4} +36 q^{-5} +90 q^{-6} +19 q^{-7} -125 q^{-8} -92 q^{-9} -12 q^{-10} +197 q^{-11} +159 q^{-12} -137 q^{-13} -214 q^{-14} -161 q^{-15} +249 q^{-16} +333 q^{-17} -67 q^{-18} -283 q^{-19} -329 q^{-20} +227 q^{-21} +445 q^{-22} +22 q^{-23} -277 q^{-24} -437 q^{-25} +172 q^{-26} +476 q^{-27} +85 q^{-28} -232 q^{-29} -473 q^{-30} +111 q^{-31} +445 q^{-32} +130 q^{-33} -156 q^{-34} -463 q^{-35} +31 q^{-36} +363 q^{-37} +173 q^{-38} -43 q^{-39} -406 q^{-40} -68 q^{-41} +224 q^{-42} +189 q^{-43} +89 q^{-44} -285 q^{-45} -135 q^{-46} +61 q^{-47} +134 q^{-48} +167 q^{-49} -123 q^{-50} -113 q^{-51} -50 q^{-52} +34 q^{-53} +141 q^{-54} -11 q^{-55} -36 q^{-56} -57 q^{-57} -27 q^{-58} +62 q^{-59} +12 q^{-60} +9 q^{-61} -20 q^{-62} -24 q^{-63} +16 q^{-64} + q^{-65} +8 q^{-66} -2 q^{-67} -8 q^{-68} +4 q^{-69} - q^{-70} +2 q^{-71} -2 q^{-73} + q^{-74} } |
5 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 q^{11}+5 q^9+5 q^8-4 q^6-21 q^5-18 q^4+11 q^3+39 q^2+47 q+20-51 q^{-1} -114 q^{-2} -84 q^{-3} +32 q^{-4} +174 q^{-5} +205 q^{-6} +65 q^{-7} -199 q^{-8} -381 q^{-9} -245 q^{-10} +162 q^{-11} +526 q^{-12} +518 q^{-13} +15 q^{-14} -649 q^{-15} -840 q^{-16} -265 q^{-17} +645 q^{-18} +1145 q^{-19} +637 q^{-20} -568 q^{-21} -1394 q^{-22} -1000 q^{-23} +373 q^{-24} +1555 q^{-25} +1363 q^{-26} -164 q^{-27} -1622 q^{-28} -1615 q^{-29} -100 q^{-30} +1622 q^{-31} +1836 q^{-32} +267 q^{-33} -1562 q^{-34} -1915 q^{-35} -467 q^{-36} +1485 q^{-37} +2002 q^{-38} +548 q^{-39} -1396 q^{-40} -1971 q^{-41} -673 q^{-42} +1293 q^{-43} +1982 q^{-44} +721 q^{-45} -1179 q^{-46} -1904 q^{-47} -825 q^{-48} +1023 q^{-49} +1852 q^{-50} +899 q^{-51} -833 q^{-52} -1718 q^{-53} -1008 q^{-54} +586 q^{-55} +1564 q^{-56} +1081 q^{-57} -302 q^{-58} -1325 q^{-59} -1127 q^{-60} + q^{-61} +1031 q^{-62} +1103 q^{-63} +264 q^{-64} -685 q^{-65} -979 q^{-66} -476 q^{-67} +338 q^{-68} +784 q^{-69} +566 q^{-70} -44 q^{-71} -509 q^{-72} -551 q^{-73} -182 q^{-74} +266 q^{-75} +436 q^{-76} +263 q^{-77} -42 q^{-78} -270 q^{-79} -277 q^{-80} -73 q^{-81} +126 q^{-82} +191 q^{-83} +124 q^{-84} -18 q^{-85} -112 q^{-86} -103 q^{-87} -31 q^{-88} +40 q^{-89} +70 q^{-90} +36 q^{-91} -9 q^{-92} -26 q^{-93} -29 q^{-94} -8 q^{-95} +15 q^{-96} +14 q^{-97} + q^{-99} -4 q^{-100} -7 q^{-101} +3 q^{-102} +4 q^{-103} -2 q^{-104} + q^{-106} -2 q^{-107} +2 q^{-109} - q^{-110} } |
6 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{20}-q^{19}-q^{18}-2 q^{15}-3 q^{14}+9 q^{13}+8 q^{12}+6 q^{11}+3 q^{10}-8 q^9-32 q^8-49 q^7-11 q^6+37 q^5+76 q^4+107 q^3+81 q^2-54 q-212-251 q^{-1} -148 q^{-2} +46 q^{-3} +356 q^{-4} +563 q^{-5} +391 q^{-6} -128 q^{-7} -660 q^{-8} -909 q^{-9} -781 q^{-10} +87 q^{-11} +1168 q^{-12} +1681 q^{-13} +1119 q^{-14} -241 q^{-15} -1699 q^{-16} -2671 q^{-17} -1725 q^{-18} +607 q^{-19} +2936 q^{-20} +3550 q^{-21} +2003 q^{-22} -1060 q^{-23} -4410 q^{-24} -4761 q^{-25} -1916 q^{-26} +2652 q^{-27} +5678 q^{-28} +5367 q^{-29} +1472 q^{-30} -4559 q^{-31} -7328 q^{-32} -5327 q^{-33} +674 q^{-34} +6208 q^{-35} +8079 q^{-36} +4586 q^{-37} -3208 q^{-38} -8316 q^{-39} -7915 q^{-40} -1660 q^{-41} +5398 q^{-42} +9250 q^{-43} +6793 q^{-44} -1573 q^{-45} -8066 q^{-46} -9068 q^{-47} -3214 q^{-48} +4294 q^{-49} +9296 q^{-50} +7768 q^{-51} -457 q^{-52} -7437 q^{-53} -9266 q^{-54} -3926 q^{-55} +3449 q^{-56} +8936 q^{-57} +8032 q^{-58} +232 q^{-59} -6790 q^{-60} -9112 q^{-61} -4331 q^{-62} +2667 q^{-63} +8400 q^{-64} +8112 q^{-65} +1020 q^{-66} -5874 q^{-67} -8752 q^{-68} -4884 q^{-69} +1435 q^{-70} +7406 q^{-71} +8086 q^{-72} +2280 q^{-73} -4206 q^{-74} -7852 q^{-75} -5545 q^{-76} -520 q^{-77} +5480 q^{-78} +7506 q^{-79} +3776 q^{-80} -1639 q^{-81} -5888 q^{-82} -5631 q^{-83} -2706 q^{-84} +2560 q^{-85} +5710 q^{-86} +4541 q^{-87} +1092 q^{-88} -2865 q^{-89} -4317 q^{-90} -3881 q^{-91} -403 q^{-92} +2775 q^{-93} +3644 q^{-94} +2560 q^{-95} +60 q^{-96} -1815 q^{-97} -3153 q^{-98} -1883 q^{-99} +69 q^{-100} +1521 q^{-101} +2042 q^{-102} +1371 q^{-103} +334 q^{-104} -1286 q^{-105} -1432 q^{-106} -981 q^{-107} -163 q^{-108} +596 q^{-109} +947 q^{-110} +930 q^{-111} +32 q^{-112} -330 q^{-113} -591 q^{-114} -503 q^{-115} -233 q^{-116} +150 q^{-117} +479 q^{-118} +238 q^{-119} +164 q^{-120} -59 q^{-121} -172 q^{-122} -239 q^{-123} -113 q^{-124} +89 q^{-125} +45 q^{-126} +113 q^{-127} +59 q^{-128} +19 q^{-129} -70 q^{-130} -60 q^{-131} +9 q^{-132} -23 q^{-133} +19 q^{-134} +19 q^{-135} +25 q^{-136} -13 q^{-137} -13 q^{-138} +10 q^{-139} -11 q^{-140} +9 q^{-143} -4 q^{-144} -5 q^{-145} +6 q^{-146} -2 q^{-147} - q^{-149} +2 q^{-150} -2 q^{-152} + q^{-153} } |
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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