10 149

10_148

10_150

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 149's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 149 at Knotilus!

[edit 10 149 Quick Notes]

[edit 10 149 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₁₄₂₅ X₃₈₄₉ X_12,6,13,5 X_13,18,14,19 X_9,16,10,17 X_17,10,18,11 X_15,20,16,1 X_19,14,20,15 X_6,12,7,11 X₇₂₈₃
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)(21,2-)]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 10, width is 3,

Braid index is 3

[{11, 5}, {4, 9}, {8, 10}, {9, 11}, {6, 1}, {5, 8}, {7, 2}, {1, 3}, {10, 6}, {2, 4}, {3, 7}]

[edit Notes on presentations of 10 149]

Computer Talk

The above data is available with the Mathematica package 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.

Read more at http://katlas.org/wiki/KnotTheory.

In[3]:= K = Knot["10 149"];

In[4]:= PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= X₁₄₂₅ X₃₈₄₉ X_12,6,13,5 X_13,18,14,19 X_9,16,10,17 X_17,10,18,11 X_15,20,16,1 X_19,14,20,15 X_6,12,7,11 X₇₂₈₃

In[5]:= GaussCode[K]

Out[5]= -1, 10, -2, 1, 3, -9, -10, 2, -5, 6, 9, -3, -4, 8, -7, 5, -6, 4, -8, 7

In[6]:= DTCode[K]

Out[6]= 4 8 -12 2 16 -6 18 20 10 14

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:= ConwayNotation[K]

Out[8]= [(3,2)(21,2-)]

In[9]:= br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]= 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,-2,1,-2,-2\})}

In[10]:= {First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[10]= { 3, 10, 3 }

In[11]:= Show[BraidPlot[br]]

Out[11]= -Graphics-

In[12]:= Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]= -Graphics-

In[13]:= ap = ArcPresentation[K]

Out[13]= ArcPresentation[{11, 5}, {4, 9}, {8, 10}, {9, 11}, {6, 1}, {5, 8}, {7, 2}, {1, 3}, {10, 6}, {2, 4}, {3, 7}]

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Chiral
Unknotting number	2
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-13][3]
Hyperbolic Volume	11.4427
A-Polynomial	See Data:10 149/A-polynomial

[edit Notes for 10 149's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	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}
Topological 4 genus	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}
Concordance genus	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 3}
Rasmussen s-Invariant	-4

[edit Notes for 10 149's 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+5 t^2-9 t+11-9 t^{-1} +5 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-z^4+2 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	{ 41, -4 }
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 2 q^{-2} -3 q^{-3} +6 q^{-4} -7 q^{-5} +7 q^{-6} -7 q^{-7} +5 q^{-8} -3 q^{-9} + q^{-10} }
HOMFLY-PT 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^4 a^8+2 z^2 a^8+a^8-z^6 a^6-4 z^4 a^6-6 z^2 a^6-4 a^6+2 z^4 a^4+6 z^2 a^4+4 a^4}
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^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-4 z^3 a^{11}+z a^{11}+4 z^6 a^{10}-5 z^4 a^{10}+z^2 a^{10}+3 z^7 a^9-2 z^5 a^9-z^3 a^9+z a^9+z^8 a^8+3 z^6 a^8-4 z^4 a^8+a^8+4 z^7 a^7-6 z^5 a^7+5 z^3 a^7-3 z a^7+z^8 a^6-z^6 a^6+5 z^4 a^6-9 z^2 a^6+4 a^6+z^7 a^5-z^5 a^5+2 z^3 a^5-3 z a^5+3 z^4 a^4-7 z^2 a^4+4 a^4}
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^{30}-q^{28}+q^{26}-2 q^{22}-3 q^{18}+q^{16}+q^{12}+3 q^{10}+2 q^6}
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^{162}-2 q^{160}+4 q^{158}-6 q^{156}+4 q^{154}-2 q^{152}-4 q^{150}+13 q^{148}-20 q^{146}+24 q^{144}-21 q^{142}+7 q^{140}+11 q^{138}-31 q^{136}+48 q^{134}-46 q^{132}+31 q^{130}-3 q^{128}-28 q^{126}+47 q^{124}-46 q^{122}+28 q^{120}+2 q^{118}-28 q^{116}+38 q^{114}-23 q^{112}-6 q^{110}+39 q^{108}-57 q^{106}+50 q^{104}-22 q^{102}-21 q^{100}+56 q^{98}-74 q^{96}+70 q^{94}-45 q^{92}+4 q^{90}+32 q^{88}-58 q^{86}+59 q^{84}-45 q^{82}+12 q^{80}+17 q^{78}-36 q^{76}+33 q^{74}-16 q^{72}-13 q^{70}+38 q^{68}-47 q^{66}+29 q^{64}+q^{62}-35 q^{60}+59 q^{58}-55 q^{56}+36 q^{54}-4 q^{52}-22 q^{50}+39 q^{48}-37 q^{46}+28 q^{44}-7 q^{42}-4 q^{40}+12 q^{38}-11 q^{36}+9 q^{34}-q^{32}+q^{30}+q^{28}}

Further Quantum Invariants

Further quantum knot invariants for 10_149.

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^{21}-2 q^{19}+2 q^{17}-2 q^{15}-q^9+3 q^7-q^5+2 q^3}
2	$q^{58}-2q^{56}-q^{54}+6q^{52}-5q^{50}-6q^{48}+13q^{46}-2q^{44}-12q^{42}+11q^{40}+3q^{38}-9q^{36}+2q^{34}+5q^{32}-2q^{30}-8q^{28}+5q^{26}+6q^{24}-13q^{22}+2q^{20}+12q^{18}-10q^{16}-2q^{14}+10q^{12}-2q^{10}-3q^{8}+3q^{6}+q^{4}$
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^{111}-2 q^{109}-q^{107}+3 q^{105}+3 q^{103}-5 q^{101}-9 q^{99}+11 q^{97}+17 q^{95}-12 q^{93}-31 q^{91}+7 q^{89}+48 q^{87}+4 q^{85}-60 q^{83}-21 q^{81}+58 q^{79}+39 q^{77}-51 q^{75}-47 q^{73}+36 q^{71}+49 q^{69}-16 q^{67}-43 q^{65}-q^{63}+34 q^{61}+18 q^{59}-26 q^{57}-30 q^{55}+14 q^{53}+44 q^{51}-6 q^{49}-53 q^{47}-7 q^{45}+61 q^{43}+17 q^{41}-58 q^{39}-35 q^{37}+49 q^{35}+44 q^{33}-36 q^{31}-47 q^{29}+15 q^{27}+44 q^{25}+2 q^{23}-30 q^{21}-10 q^{19}+17 q^{17}+12 q^{15}-3 q^{13}-8 q^{11}+q^9+2 q^7+2 q^5}
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^{265}-2 q^{263}-q^{261}+3 q^{259}-3 q^{251}-3 q^{249}+8 q^{247}+9 q^{245}-5 q^{243}-16 q^{241}-19 q^{239}+35 q^{235}+61 q^{233}+19 q^{231}-78 q^{229}-130 q^{227}-77 q^{225}+90 q^{223}+254 q^{221}+237 q^{219}-50 q^{217}-407 q^{215}-487 q^{213}-132 q^{211}+489 q^{209}+839 q^{207}+497 q^{205}-424 q^{203}-1181 q^{201}-1008 q^{199}+121 q^{197}+1362 q^{195}+1573 q^{193}+408 q^{191}-1289 q^{189}-2030 q^{187}-1032 q^{185}+935 q^{183}+2205 q^{181}+1615 q^{179}-379 q^{177}-2080 q^{175}-1982 q^{173}-195 q^{171}+1672 q^{169}+2053 q^{167}+692 q^{165}-1136 q^{163}-1863 q^{161}-989 q^{159}+595 q^{157}+1498 q^{155}+1088 q^{153}-133 q^{151}-1092 q^{149}-1058 q^{147}-191 q^{145}+721 q^{143}+962 q^{141}+420 q^{139}-438 q^{137}-905 q^{135}-587 q^{133}+246 q^{131}+892 q^{129}+771 q^{127}-103 q^{125}-963 q^{123}-979 q^{121}-44 q^{119}+1061 q^{117}+1266 q^{115}+240 q^{113}-1147 q^{111}-1559 q^{109}-535 q^{107}+1113 q^{105}+1869 q^{103}+911 q^{101}-944 q^{99}-2033 q^{97}-1342 q^{95}+562 q^{93}+2024 q^{91}+1742 q^{89}-52 q^{87}-1759 q^{85}-1965 q^{83}-540 q^{81}+1255 q^{79}+1947 q^{77}+1046 q^{75}-620 q^{73}-1646 q^{71}-1313 q^{69}-29 q^{67}+1126 q^{65}+1306 q^{63}+502 q^{61}-548 q^{59}-1030 q^{57}-714 q^{55}+48 q^{53}+627 q^{51}+663 q^{49}+249 q^{47}-245 q^{45}-458 q^{43}-317 q^{41}-9 q^{39}+211 q^{37}+249 q^{35}+117 q^{33}-54 q^{31}-123 q^{29}-95 q^{27}-29 q^{25}+35 q^{23}+58 q^{21}+30 q^{19}+q^{17}-10 q^{15}-17 q^{13}-8 q^{11}+q^9+4 q^7+2 q^5+2 q^3}
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^{366}-2 q^{364}-q^{362}+3 q^{360}-3 q^{354}+5 q^{352}-2 q^{350}-7 q^{348}+11 q^{346}+2 q^{344}-6 q^{342}-18 q^{340}+3 q^{338}+3 q^{336}+6 q^{334}+52 q^{332}+29 q^{330}-27 q^{328}-106 q^{326}-77 q^{324}-49 q^{322}+52 q^{320}+260 q^{318}+273 q^{316}+92 q^{314}-295 q^{312}-505 q^{310}-564 q^{308}-210 q^{306}+614 q^{304}+1189 q^{302}+1124 q^{300}+131 q^{298}-1116 q^{296}-2211 q^{294}-2045 q^{292}-176 q^{290}+2251 q^{288}+3812 q^{286}+2998 q^{284}+186 q^{282}-3777 q^{280}-5987 q^{278}-4367 q^{276}+433 q^{274}+5897 q^{272}+8181 q^{270}+5703 q^{268}-1382 q^{266}-8446 q^{264}-10636 q^{262}-6118 q^{260}+2935 q^{258}+10738 q^{256}+12548 q^{254}+5828 q^{252}-4946 q^{250}-12943 q^{248}-12903 q^{246}-4535 q^{244}+6735 q^{242}+14172 q^{240}+12093 q^{238}+2490 q^{236}-8445 q^{234}-13670 q^{232}-10081 q^{230}-446 q^{228}+9229 q^{226}+12125 q^{224}+7406 q^{222}-1548 q^{220}-8668 q^{218}-9805 q^{216}-4859 q^{214}+2751 q^{212}+7553 q^{210}+7320 q^{208}+2586 q^{206}-3049 q^{204}-6218 q^{202}-5220 q^{200}-1048 q^{198}+3168 q^{196}+5101 q^{194}+3559 q^{192}-3430 q^{188}-4384 q^{186}-2400 q^{184}+1196 q^{182}+4070 q^{180}+4003 q^{178}+1157 q^{176}-2854 q^{174}-4952 q^{172}-3683 q^{170}+646 q^{168}+5007 q^{166}+5952 q^{164}+2647 q^{162}-3199 q^{160}-7211 q^{158}-6588 q^{156}-735 q^{154}+6290 q^{152}+9322 q^{150}+6044 q^{148}-1983 q^{146}-9142 q^{144}-10766 q^{142}-4600 q^{140}+5071 q^{138}+11688 q^{136}+10782 q^{134}+2503 q^{132}-7581 q^{130}-13317 q^{128}-9992 q^{126}-241 q^{124}+9619 q^{122}+13372 q^{120}+8614 q^{118}-1344 q^{116}-10625 q^{114}-12652 q^{112}-7014 q^{110}+2585 q^{108}+10140 q^{106}+11276 q^{104}+5779 q^{102}-3062 q^{100}-9058 q^{98}-9496 q^{96}-4483 q^{94}+2537 q^{92}+7492 q^{90}+7793 q^{88}+3493 q^{86}-1894 q^{84}-5681 q^{82}-5884 q^{80}-2989 q^{78}+1166 q^{76}+4066 q^{74}+4208 q^{72}+2353 q^{70}-475 q^{68}-2517 q^{66}-2981 q^{64}-1757 q^{62}+78 q^{60}+1384 q^{58}+1842 q^{56}+1258 q^{54}+220 q^{52}-730 q^{50}-1022 q^{48}-772 q^{46}-272 q^{44}+260 q^{42}+509 q^{40}+462 q^{38}+180 q^{36}-51 q^{34}-191 q^{32}-217 q^{30}-125 q^{28}-11 q^{26}+71 q^{24}+74 q^{22}+55 q^{20}+23 q^{18}-9 q^{16}-24 q^{14}-19 q^{12}-7 q^{10}-q^8+3 q^6+3 q^4+3 q^2+1}

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^{30}-q^{28}+q^{26}-2 q^{22}-3 q^{18}+q^{16}+q^{12}+3 q^{10}+2 q^6}
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^{84}-4 q^{82}+10 q^{80}-20 q^{78}+36 q^{76}-60 q^{74}+88 q^{72}-120 q^{70}+152 q^{68}-174 q^{66}+178 q^{64}-158 q^{62}+114 q^{60}-40 q^{58}-52 q^{56}+148 q^{54}-240 q^{52}+308 q^{50}-348 q^{48}+356 q^{46}-324 q^{44}+270 q^{42}-186 q^{40}+92 q^{38}-q^{36}-88 q^{34}+146 q^{32}-192 q^{30}+193 q^{28}-180 q^{26}+146 q^{24}-104 q^{22}+69 q^{20}-32 q^{18}+20 q^{16}+2 q^{12}+2 q^{10}}
2,0	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^{76}-q^{74}-q^{72}+2 q^{70}-q^{68}-2 q^{66}-q^{64}+4 q^{62}+q^{60}-6 q^{58}+2 q^{56}+5 q^{54}-q^{52}-q^{50}+5 q^{48}+3 q^{46}-4 q^{44}-2 q^{42}-5 q^{38}-6 q^{36}+3 q^{34}-q^{32}-6 q^{30}+3 q^{28}+4 q^{26}-q^{24}-2 q^{22}+5 q^{20}+5 q^{18}-q^{16}+4 q^{12}+q^{10}}

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^{68}-2 q^{66}+4 q^{62}-6 q^{60}+8 q^{56}-9 q^{54}+10 q^{50}-7 q^{48}+6 q^{44}-q^{40}+3 q^{36}-4 q^{34}-11 q^{32}+2 q^{30}-2 q^{28}-12 q^{26}+9 q^{24}+5 q^{22}-3 q^{20}+9 q^{18}+4 q^{16}-q^{14}+3 q^{12}}
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^{39}-q^{37}+2 q^{35}-q^{33}+q^{31}-2 q^{29}-q^{27}-3 q^{25}-2 q^{23}+3 q^{17}+q^{15}+4 q^{13}+2 q^9}
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^{110}-4 q^{108}+8 q^{106}-8 q^{104}-2 q^{102}+23 q^{100}-44 q^{98}+45 q^{96}-13 q^{94}-50 q^{92}+111 q^{90}-130 q^{88}+81 q^{86}+27 q^{84}-142 q^{82}+211 q^{80}-188 q^{78}+83 q^{76}+54 q^{74}-158 q^{72}+179 q^{70}-123 q^{68}+30 q^{66}+22 q^{64}-16 q^{62}-40 q^{60}+85 q^{58}-54 q^{56}-24 q^{54}+156 q^{52}-216 q^{50}+211 q^{48}-112 q^{46}-39 q^{44}+133 q^{42}-206 q^{40}+141 q^{38}-82 q^{36}-20 q^{34}+64 q^{32}-61 q^{30}+52 q^{28}+2 q^{26}+4 q^{24}+18 q^{22}+q^{20}+3 q^{18}+2 q^{16}}

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^{86}-q^{84}-q^{82}+4 q^{80}-q^{78}-7 q^{76}+2 q^{74}+4 q^{72}-6 q^{70}-4 q^{68}+7 q^{66}+3 q^{64}-5 q^{62}+4 q^{60}+12 q^{58}-q^{56}-q^{54}+13 q^{52}-q^{50}-11 q^{48}-q^{46}-4 q^{44}-19 q^{42}-12 q^{40}-q^{38}-3 q^{36}-6 q^{34}+6 q^{32}+13 q^{30}+4 q^{28}+6 q^{26}+9 q^{24}+4 q^{22}+3 q^{18}}

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^{48}-q^{46}+2 q^{44}+q^{38}-2 q^{36}-q^{34}-4 q^{32}-2 q^{30}-3 q^{28}+3 q^{22}+3 q^{20}+2 q^{18}+4 q^{16}+2 q^{12}}

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^{68}-2 q^{66}+4 q^{64}-6 q^{62}+8 q^{60}-10 q^{58}+10 q^{56}-9 q^{54}+6 q^{52}-2 q^{50}-3 q^{48}+8 q^{46}-12 q^{44}+16 q^{42}-19 q^{40}+18 q^{38}-17 q^{36}+12 q^{34}-9 q^{32}+2 q^{30}+2 q^{28}-6 q^{26}+9 q^{24}-9 q^{22}+11 q^{20}-7 q^{18}+8 q^{16}-3 q^{14}+3 q^{12}}

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^{110}-2 q^{106}-2 q^{104}+2 q^{102}+5 q^{100}-7 q^{96}-5 q^{94}+6 q^{92}+9 q^{90}-2 q^{88}-11 q^{86}-3 q^{84}+10 q^{82}+7 q^{80}-6 q^{78}-7 q^{76}+4 q^{74}+8 q^{72}-q^{70}-7 q^{68}+8 q^{64}+2 q^{62}-6 q^{60}-5 q^{58}+4 q^{56}+4 q^{54}-6 q^{52}-10 q^{50}+q^{48}+8 q^{46}-q^{44}-11 q^{42}-6 q^{40}+9 q^{38}+10 q^{36}-2 q^{34}-8 q^{32}+q^{30}+8 q^{28}+6 q^{26}-2 q^{24}-2 q^{22}+q^{20}+3 q^{18}}

D4 Invariants.

Weight

Invariant

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^{94}-2 q^{92}+2 q^{90}-3 q^{88}+5 q^{86}-7 q^{84}+6 q^{82}-7 q^{80}+9 q^{78}-8 q^{76}+5 q^{74}-4 q^{72}+4 q^{70}+2 q^{68}-5 q^{66}+6 q^{64}-7 q^{62}+14 q^{60}-12 q^{58}+14 q^{56}-13 q^{54}+15 q^{52}-12 q^{50}+7 q^{48}-14 q^{46}-6 q^{42}-6 q^{40}-2 q^{38}-7 q^{36}+9 q^{34}-4 q^{32}+12 q^{30}-3 q^{28}+12 q^{26}-q^{24}+7 q^{22}-2 q^{20}+3 q^{18}}

G2 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^{162}-2 q^{160}+4 q^{158}-6 q^{156}+4 q^{154}-2 q^{152}-4 q^{150}+13 q^{148}-20 q^{146}+24 q^{144}-21 q^{142}+7 q^{140}+11 q^{138}-31 q^{136}+48 q^{134}-46 q^{132}+31 q^{130}-3 q^{128}-28 q^{126}+47 q^{124}-46 q^{122}+28 q^{120}+2 q^{118}-28 q^{116}+38 q^{114}-23 q^{112}-6 q^{110}+39 q^{108}-57 q^{106}+50 q^{104}-22 q^{102}-21 q^{100}+56 q^{98}-74 q^{96}+70 q^{94}-45 q^{92}+4 q^{90}+32 q^{88}-58 q^{86}+59 q^{84}-45 q^{82}+12 q^{80}+17 q^{78}-36 q^{76}+33 q^{74}-16 q^{72}-13 q^{70}+38 q^{68}-47 q^{66}+29 q^{64}+q^{62}-35 q^{60}+59 q^{58}-55 q^{56}+36 q^{54}-4 q^{52}-22 q^{50}+39 q^{48}-37 q^{46}+28 q^{44}-7 q^{42}-4 q^{40}+12 q^{38}-11 q^{36}+9 q^{34}-q^{32}+q^{30}+q^{28}}

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

In[3]:= K = Knot["10 149"];

In[4]:= Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[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 -t^3+5 t^2-9 t+11-9 t^{-1} +5 t^{-2} - t^{-3} }

In[5]:= Conway[K][z]

Out[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 -z^6-z^4+2 z^2+1}

In[6]:= Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]= 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]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 41, -4 }

In[8]:= Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]= 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^{-2} -3 q^{-3} +6 q^{-4} -7 q^{-5} +7 q^{-6} -7 q^{-7} +5 q^{-8} -3 q^{-9} + q^{-10} }

In[9]:= HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]= 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^4 a^8+2 z^2 a^8+a^8-z^6 a^6-4 z^4 a^6-6 z^2 a^6-4 a^6+2 z^4 a^4+6 z^2 a^4+4 a^4}

In[10]:= Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]= 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^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-4 z^3 a^{11}+z a^{11}+4 z^6 a^{10}-5 z^4 a^{10}+z^2 a^{10}+3 z^7 a^9-2 z^5 a^9-z^3 a^9+z a^9+z^8 a^8+3 z^6 a^8-4 z^4 a^8+a^8+4 z^7 a^7-6 z^5 a^7+5 z^3 a^7-3 z a^7+z^8 a^6-z^6 a^6+5 z^4 a^6-9 z^2 a^6+4 a^6+z^7 a^5-z^5 a^5+2 z^3 a^5-3 z a^5+3 z^4 a^4-7 z^2 a^4+4 a^4}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_20, K11n26,}

Same Jones Polynomial (up to mirroring, 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\leftrightarrow q^{-1}} ): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

In[3]:= K = Knot["10 149"];

In[4]:= {A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]= { 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+5 t^2-9 t+11-9 t^{-1} +5 t^{-2} - t^{-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 2 q^{-2} -3 q^{-3} +6 q^{-4} -7 q^{-5} +7 q^{-6} -7 q^{-7} +5 q^{-8} -3 q^{-9} + q^{-10} } }

In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]= {9_20, K11n26,}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]= {}

Vassiliev invariants

V₂ and V₃:

(2, -2)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

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 8}

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 -16}

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 32}

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 \frac{172}{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 \frac{68}{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 -128}

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 -\frac{736}{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 -\frac{256}{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 -80}

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 \frac{256}{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 128}

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 \frac{1376}{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 \frac{544}{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 \frac{14791}{15}}

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 -\frac{548}{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 \frac{39724}{45}}

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 \frac{185}{9}}

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 \frac{1831}{15}}

V_2,1 through V_6,9 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 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^rq^j} are shown, along with their alternating sums 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 \chi} (fixed 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} , 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 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} , 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=} -4 is the signature of 10 149. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-3

2

-5

2

1

-1

-7

4

1

3

-9

3

2

-1

-11

4

0

-13

3

0

-15

2

4

-2

-17

1

3

2

-19

2

-2

-21

1

Integral Khovanov Homology

(db, data source)

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 \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}}	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 i=-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 i=-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 r=-8}	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 {\mathbb Z}}
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=-7}	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 {\mathbb Z}^{2}\oplus{\mathbb Z}_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 {\mathbb Z}}
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=-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 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=-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 {\mathbb Z}^{4}\oplus{\mathbb Z}_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 {\mathbb Z}^{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 r=-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 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=-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 {\mathbb Z}^{4}\oplus{\mathbb Z}_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 {\mathbb Z}^{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 r=-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 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=-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 {\mathbb Z}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}\oplus{\mathbb Z}_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 {\mathbb Z}^{2}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the 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+1} -dimensional representation of 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 sl(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 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 q^{-3} +2 q^{-4} -6 q^{-5} +2 q^{-6} +14 q^{-7} -18 q^{-8} -6 q^{-9} +36 q^{-10} -28 q^{-11} -21 q^{-12} +55 q^{-13} -29 q^{-14} -34 q^{-15} +61 q^{-16} -22 q^{-17} -37 q^{-18} +50 q^{-19} -10 q^{-20} -29 q^{-21} +27 q^{-22} -14 q^{-24} +8 q^{-25} + q^{-26} -3 q^{-27} + q^{-28} }
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 2 q^{-4} - q^{-6} -9 q^{-7} +7 q^{-8} +15 q^{-9} +4 q^{-10} -36 q^{-11} -13 q^{-12} +47 q^{-13} +46 q^{-14} -65 q^{-15} -75 q^{-16} +58 q^{-17} +126 q^{-18} -60 q^{-19} -159 q^{-20} +35 q^{-21} +201 q^{-22} -16 q^{-23} -227 q^{-24} -11 q^{-25} +248 q^{-26} +34 q^{-27} -257 q^{-28} -55 q^{-29} +252 q^{-30} +78 q^{-31} -241 q^{-32} -90 q^{-33} +210 q^{-34} +105 q^{-35} -176 q^{-36} -103 q^{-37} +127 q^{-38} +101 q^{-39} -86 q^{-40} -84 q^{-41} +48 q^{-42} +62 q^{-43} -22 q^{-44} -40 q^{-45} +7 q^{-46} +24 q^{-47} -3 q^{-48} -11 q^{-49} + q^{-50} +4 q^{-51} + q^{-52} -3 q^{-53} + q^{-54} }
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^{-4} +2 q^{-5} -6 q^{-7} -4 q^{-8} -4 q^{-9} +17 q^{-10} +26 q^{-11} -10 q^{-12} -26 q^{-13} -63 q^{-14} +12 q^{-15} +104 q^{-16} +59 q^{-17} + q^{-18} -206 q^{-19} -119 q^{-20} +140 q^{-21} +222 q^{-22} +211 q^{-23} -310 q^{-24} -385 q^{-25} -17 q^{-26} +337 q^{-27} +604 q^{-28} -226 q^{-29} -631 q^{-30} -354 q^{-31} +271 q^{-32} +1008 q^{-33} +27 q^{-34} -729 q^{-35} -718 q^{-36} +63 q^{-37} +1289 q^{-38} +310 q^{-39} -699 q^{-40} -987 q^{-41} -164 q^{-42} +1416 q^{-43} +532 q^{-44} -592 q^{-45} -1135 q^{-46} -362 q^{-47} +1396 q^{-48} +682 q^{-49} -416 q^{-50} -1144 q^{-51} -538 q^{-52} +1193 q^{-53} +743 q^{-54} -151 q^{-55} -972 q^{-56} -658 q^{-57} +807 q^{-58} +652 q^{-59} +123 q^{-60} -622 q^{-61} -623 q^{-62} +370 q^{-63} +404 q^{-64} +251 q^{-65} -251 q^{-66} -415 q^{-67} +91 q^{-68} +140 q^{-69} +190 q^{-70} -38 q^{-71} -182 q^{-72} +9 q^{-73} +10 q^{-74} +79 q^{-75} +9 q^{-76} -56 q^{-77} +7 q^{-78} -9 q^{-79} +20 q^{-80} +5 q^{-81} -14 q^{-82} +4 q^{-83} -3 q^{-84} +4 q^{-85} + q^{-86} -3 q^{-87} + q^{-88} }
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^{-4} +2 q^{-6} -3 q^{-7} -9 q^{-8} -9 q^{-9} +9 q^{-10} +11 q^{-11} +31 q^{-12} +25 q^{-13} -32 q^{-14} -73 q^{-15} -57 q^{-16} -17 q^{-17} +100 q^{-18} +196 q^{-19} +100 q^{-20} -111 q^{-21} -277 q^{-22} -325 q^{-23} -41 q^{-24} +409 q^{-25} +594 q^{-26} +303 q^{-27} -313 q^{-28} -904 q^{-29} -803 q^{-30} +93 q^{-31} +1076 q^{-32} +1353 q^{-33} +491 q^{-34} -1084 q^{-35} -1958 q^{-36} -1191 q^{-37} +743 q^{-38} +2379 q^{-39} +2157 q^{-40} -183 q^{-41} -2650 q^{-42} -2986 q^{-43} -682 q^{-44} +2585 q^{-45} +3864 q^{-46} +1611 q^{-47} -2368 q^{-48} -4448 q^{-49} -2586 q^{-50} +1894 q^{-51} +4953 q^{-52} +3466 q^{-53} -1410 q^{-54} -5204 q^{-55} -4234 q^{-56} +870 q^{-57} +5365 q^{-58} +4853 q^{-59} -384 q^{-60} -5409 q^{-61} -5339 q^{-62} -65 q^{-63} +5381 q^{-64} +5713 q^{-65} +490 q^{-66} -5288 q^{-67} -5985 q^{-68} -898 q^{-69} +5063 q^{-70} +6180 q^{-71} +1348 q^{-72} -4746 q^{-73} -6226 q^{-74} -1810 q^{-75} +4196 q^{-76} +6146 q^{-77} +2307 q^{-78} -3525 q^{-79} -5816 q^{-80} -2713 q^{-81} +2612 q^{-82} +5272 q^{-83} +3034 q^{-84} -1697 q^{-85} -4455 q^{-86} -3094 q^{-87} +745 q^{-88} +3485 q^{-89} +2936 q^{-90} +4 q^{-91} -2461 q^{-92} -2504 q^{-93} -525 q^{-94} +1518 q^{-95} +1938 q^{-96} +745 q^{-97} -764 q^{-98} -1339 q^{-99} -736 q^{-100} +277 q^{-101} +809 q^{-102} +572 q^{-103} -7 q^{-104} -418 q^{-105} -394 q^{-106} -73 q^{-107} +188 q^{-108} +217 q^{-109} +73 q^{-110} -61 q^{-111} -107 q^{-112} -56 q^{-113} +24 q^{-114} +50 q^{-115} +20 q^{-116} -9 q^{-117} -10 q^{-118} -14 q^{-119} -2 q^{-120} +15 q^{-121} + q^{-122} -6 q^{-123} + q^{-124} -3 q^{-126} +4 q^{-127} + q^{-128} -3 q^{-129} + q^{-130} }
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^{-3} +2 q^{-4} -4 q^{-7} -6 q^{-8} -12 q^{-9} -4 q^{-10} +17 q^{-11} +32 q^{-12} +32 q^{-13} +15 q^{-14} -9 q^{-15} -94 q^{-16} -118 q^{-17} -75 q^{-18} +58 q^{-19} +172 q^{-20} +246 q^{-21} +273 q^{-22} -47 q^{-23} -367 q^{-24} -607 q^{-25} -442 q^{-26} -78 q^{-27} +538 q^{-28} +1223 q^{-29} +991 q^{-30} +217 q^{-31} -1065 q^{-32} -1748 q^{-33} -1913 q^{-34} -686 q^{-35} +1687 q^{-36} +3033 q^{-37} +3045 q^{-38} +790 q^{-39} -1890 q^{-40} -4813 q^{-41} -4841 q^{-42} -1208 q^{-43} +3236 q^{-44} +6832 q^{-45} +6177 q^{-46} +2410 q^{-47} -5114 q^{-48} -9796 q^{-49} -8228 q^{-50} -1777 q^{-51} +7270 q^{-52} +12173 q^{-53} +11251 q^{-54} +383 q^{-55} -10932 q^{-56} -15783 q^{-57} -11376 q^{-58} +1632 q^{-59} +14200 q^{-60} +20532 q^{-61} +10341 q^{-62} -6174 q^{-63} -19536 q^{-64} -21236 q^{-65} -8119 q^{-66} +10875 q^{-67} +26268 q^{-68} +20425 q^{-69} +2105 q^{-70} -18630 q^{-71} -27853 q^{-72} -17790 q^{-73} +4709 q^{-74} +27892 q^{-75} +27584 q^{-76} +10132 q^{-77} -15352 q^{-78} -30885 q^{-79} -24815 q^{-80} -1144 q^{-81} +27269 q^{-82} +31596 q^{-83} +15978 q^{-84} -12047 q^{-85} -31830 q^{-86} -29176 q^{-87} -5473 q^{-88} +26000 q^{-89} +33694 q^{-90} +19989 q^{-91} -9201 q^{-92} -31763 q^{-93} -32050 q^{-94} -9069 q^{-95} +24016 q^{-96} +34648 q^{-97} +23419 q^{-98} -5642 q^{-99} -30221 q^{-100} -33983 q^{-101} -13285 q^{-102} +19844 q^{-103} +33650 q^{-104} +26588 q^{-105} -7 q^{-106} -25487 q^{-107} -33750 q^{-108} -18087 q^{-109} +12234 q^{-110} +28704 q^{-111} +27725 q^{-112} +7113 q^{-113} -16533 q^{-114} -29031 q^{-115} -20983 q^{-116} +2559 q^{-117} +19069 q^{-118} +24136 q^{-119} +12338 q^{-120} -5598 q^{-121} -19428 q^{-122} -18904 q^{-123} -4878 q^{-124} +7799 q^{-125} +15768 q^{-126} +12298 q^{-127} +2399 q^{-128} -8654 q^{-129} -12184 q^{-130} -6688 q^{-131} -4 q^{-132} +6715 q^{-133} +7780 q^{-134} +4589 q^{-135} -1590 q^{-136} -5099 q^{-137} -4210 q^{-138} -2288 q^{-139} +1251 q^{-140} +2980 q^{-141} +2969 q^{-142} +620 q^{-143} -1136 q^{-144} -1398 q^{-145} -1474 q^{-146} -310 q^{-147} +553 q^{-148} +1100 q^{-149} +454 q^{-150} -41 q^{-151} -151 q^{-152} -481 q^{-153} -245 q^{-154} -22 q^{-155} +276 q^{-156} +100 q^{-157} +18 q^{-158} +59 q^{-159} -94 q^{-160} -64 q^{-161} -35 q^{-162} +68 q^{-163} - q^{-164} -10 q^{-165} +30 q^{-166} -15 q^{-167} -8 q^{-168} -12 q^{-169} +22 q^{-170} -4 q^{-171} -10 q^{-172} +9 q^{-173} -3 q^{-174} -3 q^{-176} +4 q^{-177} + q^{-178} -3 q^{-179} + q^{-180} }

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).

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Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

Modifying This Page

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