9 6

9_5

9_7

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 6's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 6 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X3,12,4,13 X5,14,6,15 X7,16,8,17 X9,18,10,1 X15,6,16,7 X17,8,18,9 X13,10,14,11 X11,2,12,3
Gauss code	-1, 9, -2, 1, -3, 6, -4, 7, -5, 8, -9, 2, -8, 3, -6, 4, -7, 5
Dowker-Thistlethwaite code	4 12 14 16 18 2 10 6 8
Conway Notation	[522]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 10, width is 3, Braid index is 3		[{11, 2}, {1, 9}, {8, 10}, {9, 11}, {10, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 8}, {7, 1}]

[edit Notes on presentations of 9 6]

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["9 6"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3,12,4,13 X5,14,6,15 X7,16,8,17 X9,18,10,1 X15,6,16,7 X17,8,18,9 X13,10,14,11 X11,2,12,3

In[5]:=

GaussCode[K]

Out[5]=

-1, 9, -2, 1, -3, 6, -4, 7, -5, 8, -9, 2, -8, 3, -6, 4, -7, 5

In[6]:=

DTCode[K]

Out[6]=

4 12 14 16 18 2 10 6 8

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[522]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 2 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-16][5] Hyperbolic Volume 7.2036 A-Polynomial See Data:9 6/A-polynomial

[edit Notes for 9 6'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 3} 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 3} 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 -6

[edit Notes for 9 6'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 2 t^3-4 t^2+5 t-5+5 t^{-1} -4 t^{-2} +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 2 z^6+8 z^4+7 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	{ 27, -6 }
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 q^{-3} - q^{-4} +3 q^{-5} -3 q^{-6} +4 q^{-7} -5 q^{-8} +4 q^{-9} -3 q^{-10} +2 q^{-11} - q^{-12} }
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^{10}-3 z^2 a^{10}-a^{10}+z^6 a^8+4 z^4 a^8+3 z^2 a^8-a^8+z^6 a^6+5 z^4 a^6+7 z^2 a^6+3 a^6}
Kauffman polynomial (db, data sources)	$z^{3}a^{15}-za^{15}+2z^{4}a^{14}-2z^{2}a^{14}+2z^{5}a^{13}-z^{3}a^{13}+2z^{6}a^{12}-2z^{4}a^{12}+z^{2}a^{12}+2z^{7}a^{11}-5z^{5}a^{11}+6z^{3}a^{11}-2za^{11}+z^{8}a^{10}-2z^{6}a^{10}+2z^{4}a^{10}-3z^{2}a^{10}+a^{10}+3z^{7}a^{9}-10z^{5}a^{9}+8z^{3}a^{9}-za^{9}+z^{8}a^{8}-3z^{6}a^{8}+z^{4}a^{8}+z^{2}a^{8}-a^{8}+z^{7}a^{7}-3z^{5}a^{7}+2za^{7}+z^{6}a^{6}-5z^{4}a^{6}+7z^{2}a^{6}-3a^{6}$
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^{36}-2 q^{26}-q^{22}+q^{20}+2 q^{18}+q^{16}+2 q^{14}+q^{10}}
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^{196}-q^{194}+2 q^{192}-2 q^{190}+q^{186}-2 q^{184}+4 q^{182}-4 q^{180}+4 q^{178}-2 q^{176}-q^{174}+3 q^{172}-4 q^{170}+4 q^{168}-5 q^{166}+3 q^{164}-3 q^{162}-q^{160}+3 q^{158}-4 q^{156}+5 q^{154}-4 q^{152}+2 q^{150}-4 q^{146}+4 q^{144}-2 q^{142}-q^{140}+6 q^{138}-5 q^{136}+2 q^{134}+3 q^{132}-6 q^{130}+9 q^{128}-10 q^{126}+3 q^{124}-5 q^{120}+10 q^{118}-12 q^{116}+6 q^{114}-3 q^{112}-2 q^{110}+3 q^{108}-9 q^{106}+6 q^{104}-4 q^{102}+4 q^{98}-6 q^{96}+4 q^{94}+3 q^{92}-5 q^{90}+7 q^{88}-6 q^{86}+2 q^{84}+5 q^{82}-7 q^{80}+11 q^{78}-7 q^{76}+5 q^{74}+2 q^{72}-4 q^{70}+7 q^{68}-5 q^{66}+5 q^{64}+2 q^{58}-q^{56}+2 q^{54}+q^{50}}

Further Quantum Invariants

Further quantum knot invariants for 9_6.

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^{25}+q^{23}-q^{21}+q^{19}-q^{17}-q^{15}+q^{13}+2 q^9+q^5}
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^{68}-q^{66}-q^{64}+2 q^{62}-q^{60}+3 q^{56}-3 q^{54}-q^{52}+3 q^{50}-2 q^{48}-2 q^{46}+q^{44}+q^{42}-q^{40}-q^{38}+2 q^{36}-q^{34}-3 q^{32}+2 q^{30}-3 q^{26}+2 q^{24}+3 q^{22}-2 q^{20}+q^{18}+3 q^{16}+q^{10}}
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^{129}+q^{127}+q^{125}-2 q^{121}-q^{119}+2 q^{117}+q^{115}-q^{111}-q^{109}-q^{107}+2 q^{105}+3 q^{103}-3 q^{101}-5 q^{99}+2 q^{97}+7 q^{95}+q^{93}-5 q^{91}-2 q^{89}+4 q^{87}+3 q^{85}-4 q^{81}-2 q^{79}+4 q^{77}+q^{75}-4 q^{73}-5 q^{71}+4 q^{69}+3 q^{67}-3 q^{65}-5 q^{63}+4 q^{61}+5 q^{59}-q^{57}-6 q^{55}-2 q^{53}+5 q^{51}+3 q^{49}-5 q^{47}-6 q^{45}+q^{43}+6 q^{41}+2 q^{39}-6 q^{37}-2 q^{35}+4 q^{33}+5 q^{31}-q^{29}-2 q^{27}+q^{25}+3 q^{23}+q^{21}+q^{15}}
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^{208}-q^{206}-q^{204}+4 q^{198}-q^{196}-2 q^{194}-3 q^{192}-3 q^{190}+8 q^{188}+3 q^{186}+q^{184}-6 q^{182}-10 q^{180}+6 q^{178}+8 q^{176}+9 q^{174}-6 q^{172}-20 q^{170}-3 q^{168}+10 q^{166}+22 q^{164}+5 q^{162}-24 q^{160}-17 q^{158}+q^{156}+25 q^{154}+16 q^{152}-13 q^{150}-17 q^{148}-12 q^{146}+11 q^{144}+17 q^{142}+2 q^{140}-5 q^{138}-12 q^{136}-3 q^{134}+5 q^{132}+8 q^{130}+7 q^{128}-5 q^{126}-11 q^{124}-q^{122}+12 q^{120}+10 q^{118}-3 q^{116}-15 q^{114}-4 q^{112}+15 q^{110}+10 q^{108}-6 q^{106}-20 q^{104}-8 q^{102}+16 q^{100}+14 q^{98}-19 q^{94}-14 q^{92}+9 q^{90}+14 q^{88}+10 q^{86}-9 q^{84}-14 q^{82}-3 q^{80}+2 q^{78}+15 q^{76}+6 q^{74}-6 q^{72}-9 q^{70}-13 q^{68}+5 q^{66}+10 q^{64}+7 q^{62}-16 q^{58}-6 q^{56}+3 q^{54}+10 q^{52}+9 q^{50}-6 q^{48}-6 q^{46}-4 q^{44}+3 q^{42}+8 q^{40}+q^{38}-2 q^{34}+3 q^{30}+q^{28}+q^{26}+q^{20}}
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^{305}+q^{303}+q^{301}-2 q^{295}-2 q^{293}+q^{291}+4 q^{289}+2 q^{287}+q^{285}-4 q^{283}-8 q^{281}-3 q^{279}+5 q^{277}+11 q^{275}+7 q^{273}-3 q^{271}-14 q^{269}-14 q^{267}+3 q^{265}+21 q^{263}+19 q^{261}-2 q^{259}-25 q^{257}-31 q^{255}-5 q^{253}+35 q^{251}+44 q^{249}+11 q^{247}-40 q^{245}-59 q^{243}-24 q^{241}+40 q^{239}+78 q^{237}+42 q^{235}-35 q^{233}-86 q^{231}-58 q^{229}+20 q^{227}+82 q^{225}+73 q^{223}-q^{221}-69 q^{219}-75 q^{217}-20 q^{215}+44 q^{213}+63 q^{211}+33 q^{209}-17 q^{207}-47 q^{205}-38 q^{203}-2 q^{201}+25 q^{199}+31 q^{197}+19 q^{195}-7 q^{193}-23 q^{191}-24 q^{189}-6 q^{187}+17 q^{185}+26 q^{183}+13 q^{181}-14 q^{179}-31 q^{177}-15 q^{175}+19 q^{173}+32 q^{171}+15 q^{169}-21 q^{167}-42 q^{165}-16 q^{163}+33 q^{161}+48 q^{159}+22 q^{157}-27 q^{155}-58 q^{153}-30 q^{151}+30 q^{149}+62 q^{147}+39 q^{145}-18 q^{143}-62 q^{141}-51 q^{139}+6 q^{137}+55 q^{135}+56 q^{133}+9 q^{131}-45 q^{129}-58 q^{127}-26 q^{125}+25 q^{123}+49 q^{121}+35 q^{119}-7 q^{117}-35 q^{115}-34 q^{113}-12 q^{111}+13 q^{109}+26 q^{107}+23 q^{105}+6 q^{103}-11 q^{101}-18 q^{99}-21 q^{97}-8 q^{95}+10 q^{93}+22 q^{91}+18 q^{89}+5 q^{87}-14 q^{85}-28 q^{83}-17 q^{81}+3 q^{79}+19 q^{77}+22 q^{75}+9 q^{73}-11 q^{71}-20 q^{69}-14 q^{67}+13 q^{63}+14 q^{61}+5 q^{59}-3 q^{57}-9 q^{55}-6 q^{53}+q^{51}+6 q^{49}+4 q^{47}+3 q^{45}-2 q^{41}+2 q^{37}+q^{35}+q^{33}+q^{31}+q^{25}}
6	$q^{420}-q^{418}-q^{416}+2q^{410}+2q^{406}-3q^{404}-3q^{402}-q^{398}+4q^{396}+4q^{394}+8q^{392}-5q^{390}-8q^{388}-7q^{386}-8q^{384}+3q^{382}+11q^{380}+21q^{378}-8q^{374}-15q^{372}-19q^{370}-2q^{368}+18q^{366}+33q^{364}+2q^{362}-13q^{360}-25q^{358}-26q^{356}+5q^{354}+37q^{352}+46q^{350}-5q^{348}-42q^{346}-58q^{344}-38q^{342}+32q^{340}+96q^{338}+96q^{336}-3q^{334}-97q^{332}-141q^{330}-96q^{328}+43q^{326}+172q^{324}+194q^{322}+59q^{320}-113q^{318}-223q^{316}-201q^{314}-26q^{312}+176q^{310}+263q^{308}+163q^{306}-27q^{304}-195q^{302}-248q^{300}-132q^{298}+61q^{296}+204q^{294}+197q^{292}+88q^{290}-59q^{288}-167q^{286}-155q^{284}-58q^{282}+60q^{280}+114q^{278}+108q^{276}+48q^{274}-36q^{272}-85q^{270}-84q^{268}-36q^{266}+14q^{264}+60q^{262}+69q^{260}+36q^{258}-19q^{256}-65q^{254}-56q^{252}-15q^{250}+38q^{248}+65q^{246}+45q^{244}-19q^{242}-77q^{240}-65q^{238}+q^{236}+71q^{234}+92q^{232}+48q^{230}-49q^{228}-124q^{226}-98q^{224}+6q^{222}+109q^{220}+140q^{218}+79q^{216}-55q^{214}-163q^{212}-152q^{210}-27q^{208}+113q^{206}+179q^{204}+137q^{202}-10q^{200}-157q^{198}-196q^{196}-98q^{194}+56q^{192}+175q^{190}+196q^{188}+79q^{186}-87q^{184}-194q^{182}-169q^{180}-52q^{178}+95q^{176}+196q^{174}+158q^{172}+30q^{170}-112q^{168}-168q^{166}-135q^{164}-28q^{162}+103q^{160}+149q^{158}+108q^{156}+6q^{154}-71q^{152}-115q^{150}-92q^{148}-13q^{146}+49q^{144}+79q^{142}+53q^{140}+32q^{138}-15q^{136}-46q^{134}-42q^{132}-33q^{130}-9q^{128}+36q^{124}+40q^{122}+31q^{120}+14q^{118}-15q^{116}-35q^{114}-57q^{112}-24q^{110}+4q^{108}+34q^{106}+48q^{104}+36q^{102}+9q^{100}-37q^{98}-41q^{96}-36q^{94}-11q^{92}+17q^{90}+36q^{88}+34q^{86}+5q^{84}-8q^{82}-24q^{80}-22q^{78}-12q^{76}+7q^{74}+18q^{72}+10q^{70}+9q^{68}-q^{66}-7q^{64}-9q^{62}-2q^{60}+4q^{58}+2q^{56}+5q^{54}+3q^{52}+q^{50}-2q^{48}+2q^{44}+q^{40}+q^{38}+q^{36}+q^{30}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{36}-2q^{26}-q^{22}+q^{20}+2q^{18}+q^{16}+2q^{14}+q^{10}$
1,1	$q^{100}-2q^{98}+4q^{96}-6q^{94}+7q^{92}-10q^{90}+12q^{88}-10q^{86}+11q^{84}-10q^{82}+10q^{80}-10q^{78}+7q^{76}-10q^{74}+8q^{72}-4q^{70}-q^{68}+10q^{66}-16q^{64}+26q^{62}-31q^{60}+38q^{58}-40q^{56}+36q^{54}-35q^{52}+24q^{50}-22q^{48}+2q^{46}+2q^{44}-18q^{42}+20q^{40}-24q^{38}+28q^{36}-18q^{34}+22q^{32}-8q^{30}+12q^{28}-2q^{26}+4q^{24}+q^{20}$
2,0	$q^{90}-q^{84}+q^{80}+q^{78}+q^{76}-q^{74}+q^{70}-2q^{66}+q^{62}-2q^{60}-2q^{58}-q^{56}-3q^{52}-2q^{50}-q^{48}-3q^{46}-q^{44}+q^{42}+q^{40}+4q^{36}+4q^{34}+2q^{32}+q^{30}+3q^{28}+2q^{26}+q^{20}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{82}-q^{80}+2q^{76}-2q^{74}-q^{72}+3q^{70}-q^{68}-2q^{66}+3q^{64}-2q^{60}+q^{58}-q^{56}-q^{54}-2q^{52}+q^{50}-4q^{46}-q^{44}-q^{42}-4q^{40}-q^{38}+3q^{36}+4q^{32}+4q^{30}+2q^{28}+3q^{26}+2q^{24}+q^{20}$
1,0,0	$-q^{47}-q^{43}+q^{41}-q^{39}-2q^{35}-q^{33}-q^{31}+2q^{27}+q^{25}+3q^{23}+q^{21}+2q^{19}+q^{15}$
1,0,1	$q^{132}-2q^{130}+3q^{128}-q^{126}-3q^{124}+6q^{122}-8q^{120}+4q^{118}+2q^{116}-5q^{114}+9q^{112}-5q^{110}+2q^{108}+2q^{106}-5q^{104}-q^{102}+4q^{100}-11q^{98}+9q^{96}-9q^{92}+17q^{90}-16q^{88}+15q^{86}-6q^{84}+3q^{82}+7q^{80}-6q^{78}+11q^{76}-7q^{74}+9q^{72}-12q^{70}+7q^{68}-17q^{66}-5q^{64}-10q^{62}-18q^{60}+2q^{58}-22q^{56}+11q^{54}-8q^{52}+9q^{50}+12q^{48}+2q^{46}+19q^{44}+3q^{42}+10q^{40}+6q^{38}+2q^{36}+4q^{34}+q^{30}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{104}-q^{100}+q^{98}+2q^{96}-q^{94}-q^{92}+2q^{90}+q^{88}-q^{86}+2q^{82}-q^{80}-2q^{78}+q^{76}-q^{74}-3q^{72}-5q^{66}-3q^{64}-2q^{62}-4q^{60}-6q^{58}-3q^{56}-q^{54}-q^{52}+q^{50}+4q^{48}+5q^{46}+5q^{44}+7q^{42}+4q^{40}+4q^{38}+3q^{36}+2q^{34}+q^{30}$
1,0,0,0	$-q^{58}-q^{54}-q^{48}-2q^{44}-q^{42}-2q^{40}+2q^{34}+2q^{32}+2q^{30}+3q^{28}+q^{26}+2q^{24}+q^{20}$

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^{82}+q^{80}-2 q^{78}+2 q^{76}-2 q^{74}+3 q^{72}-3 q^{70}+3 q^{68}-2 q^{66}+q^{64}-2 q^{60}+3 q^{58}-5 q^{56}+5 q^{54}-6 q^{52}+5 q^{50}-6 q^{48}+4 q^{46}-3 q^{44}+q^{42}-q^{38}+3 q^{36}-2 q^{34}+4 q^{32}-2 q^{30}+4 q^{28}-q^{26}+2 q^{24}+q^{20}}
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^{132}-q^{128}-q^{126}+q^{124}+2 q^{122}-2 q^{118}-2 q^{116}+q^{114}+3 q^{112}+q^{110}-2 q^{108}-2 q^{106}+q^{104}+3 q^{102}-3 q^{98}-q^{96}+2 q^{94}+q^{92}-3 q^{90}-2 q^{88}+q^{86}+2 q^{84}-q^{82}-q^{80}+q^{76}-q^{74}-3 q^{72}-2 q^{70}+q^{68}+q^{66}-3 q^{64}-4 q^{62}+4 q^{58}+2 q^{56}-q^{54}-2 q^{52}+3 q^{50}+4 q^{48}+2 q^{46}-q^{44}+q^{42}+2 q^{40}+2 q^{38}+q^{30}}

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^{114}-q^{112}+q^{110}-q^{108}+2 q^{106}-2 q^{104}+q^{102}-2 q^{100}+3 q^{98}-2 q^{96}+q^{94}-2 q^{92}+2 q^{90}+q^{84}-2 q^{82}+3 q^{80}-4 q^{78}+3 q^{76}-6 q^{74}+3 q^{72}-5 q^{70}+3 q^{68}-5 q^{66}+2 q^{64}-4 q^{62}-3 q^{58}-2 q^{56}-2 q^{52}+2 q^{50}+6 q^{46}+q^{44}+6 q^{42}+q^{40}+5 q^{38}+q^{36}+2 q^{34}+q^{30}}

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^{196}-q^{194}+2 q^{192}-2 q^{190}+q^{186}-2 q^{184}+4 q^{182}-4 q^{180}+4 q^{178}-2 q^{176}-q^{174}+3 q^{172}-4 q^{170}+4 q^{168}-5 q^{166}+3 q^{164}-3 q^{162}-q^{160}+3 q^{158}-4 q^{156}+5 q^{154}-4 q^{152}+2 q^{150}-4 q^{146}+4 q^{144}-2 q^{142}-q^{140}+6 q^{138}-5 q^{136}+2 q^{134}+3 q^{132}-6 q^{130}+9 q^{128}-10 q^{126}+3 q^{124}-5 q^{120}+10 q^{118}-12 q^{116}+6 q^{114}-3 q^{112}-2 q^{110}+3 q^{108}-9 q^{106}+6 q^{104}-4 q^{102}+4 q^{98}-6 q^{96}+4 q^{94}+3 q^{92}-5 q^{90}+7 q^{88}-6 q^{86}+2 q^{84}+5 q^{82}-7 q^{80}+11 q^{78}-7 q^{76}+5 q^{74}+2 q^{72}-4 q^{70}+7 q^{68}-5 q^{66}+5 q^{64}+2 q^{58}-q^{56}+2 q^{54}+q^{50}}

.

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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 6"];

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 2 t^3-4 t^2+5 t-5+5 t^{-1} -4 t^{-2} +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 2 z^6+8 z^4+7 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]=

{ 27, -6 }

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 q^{-3} - q^{-4} +3 q^{-5} -3 q^{-6} +4 q^{-7} -5 q^{-8} +4 q^{-9} -3 q^{-10} +2 q^{-11} - q^{-12} }

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^{10}-3 z^2 a^{10}-a^{10}+z^6 a^8+4 z^4 a^8+3 z^2 a^8-a^8+z^6 a^6+5 z^4 a^6+7 z^2 a^6+3 a^6}

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{3}a^{15}-za^{15}+2z^{4}a^{14}-2z^{2}a^{14}+2z^{5}a^{13}-z^{3}a^{13}+2z^{6}a^{12}-2z^{4}a^{12}+z^{2}a^{12}+2z^{7}a^{11}-5z^{5}a^{11}+6z^{3}a^{11}-2za^{11}+z^{8}a^{10}-2z^{6}a^{10}+2z^{4}a^{10}-3z^{2}a^{10}+a^{10}+3z^{7}a^{9}-10z^{5}a^{9}+8z^{3}a^{9}-za^{9}+z^{8}a^{8}-3z^{6}a^{8}+z^{4}a^{8}+z^{2}a^{8}-a^{8}+z^{7}a^{7}-3z^{5}a^{7}+2za^{7}+z^{6}a^{6}-5z^{4}a^{6}+7z^{2}a^{6}-3a^{6}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 6"];

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 2 t^3-4 t^2+5 t-5+5 t^{-1} -4 t^{-2} +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 q^{-3} - q^{-4} +3 q^{-5} -3 q^{-6} +4 q^{-7} -5 q^{-8} +4 q^{-9} -3 q^{-10} +2 q^{-11} - q^{-12} } }

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]=

{}

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:

(7, -18)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $28$ 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 -144} 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 392} 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{2834}{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{382}{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 -4032} 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 -6912} 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 -1184} 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 -784} ${\frac {10976}{3}}$ 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 10368} 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{79352}{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{10696}{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{1559017}{30}} 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{13882}{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{797474}{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{5687}{18}} 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{64777}{30}}

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=} -6 is the signature of 9 6. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -5                   1 1 -7                 1 1 0 -9               2     2 -11             1 1     0 -13           3 2       1 -15         2 1         -1 -17       2 3           -1 -19     1 2             1 -21   1 2               -1 -23   1                 1 -25 1                   -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=-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 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 r=-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 {\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=-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}\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=-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}^{2}\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}^{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=-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}\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=-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}^{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=-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}\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=-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}_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=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}} 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}}

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^{-6} - q^{-7} +4 q^{-9} -3 q^{-10} -3 q^{-11} +9 q^{-12} -4 q^{-13} -8 q^{-14} +12 q^{-15} -2 q^{-16} -13 q^{-17} +14 q^{-18} + q^{-19} -16 q^{-20} +14 q^{-21} +3 q^{-22} -16 q^{-23} +11 q^{-24} +3 q^{-25} -11 q^{-26} +7 q^{-27} + q^{-28} -5 q^{-29} +4 q^{-30} -2 q^{-32} + q^{-33} }
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^{-9} - q^{-10} + q^{-12} +3 q^{-13} -3 q^{-14} -3 q^{-15} +2 q^{-16} +9 q^{-17} -4 q^{-18} -9 q^{-19} -2 q^{-20} +17 q^{-21} -14 q^{-23} -9 q^{-24} +18 q^{-25} +8 q^{-26} -12 q^{-27} -16 q^{-28} +14 q^{-29} +13 q^{-30} -6 q^{-31} -17 q^{-32} +5 q^{-33} +15 q^{-34} -16 q^{-36} -4 q^{-37} +16 q^{-38} +5 q^{-39} -13 q^{-40} -10 q^{-41} +14 q^{-42} +9 q^{-43} -10 q^{-44} -9 q^{-45} +8 q^{-46} +6 q^{-47} -4 q^{-48} -3 q^{-49} +3 q^{-50} - q^{-51} -2 q^{-52} +3 q^{-53} +2 q^{-54} -4 q^{-55} -2 q^{-56} +3 q^{-57} +3 q^{-58} -3 q^{-59} - q^{-60} +2 q^{-62} - q^{-63} }
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^{-12} - q^{-13} + q^{-15} +3 q^{-17} -4 q^{-18} -2 q^{-19} +3 q^{-20} + q^{-21} +10 q^{-22} -9 q^{-23} -9 q^{-24} + q^{-25} + q^{-26} +25 q^{-27} -8 q^{-28} -16 q^{-29} -8 q^{-30} -9 q^{-31} +41 q^{-32} - q^{-33} -13 q^{-34} -13 q^{-35} -27 q^{-36} +45 q^{-37} +2 q^{-38} - q^{-39} -4 q^{-40} -40 q^{-41} +40 q^{-42} -9 q^{-43} +4 q^{-44} +15 q^{-45} -36 q^{-46} +35 q^{-47} -32 q^{-48} - q^{-49} +34 q^{-50} -22 q^{-51} +37 q^{-52} -56 q^{-53} -13 q^{-54} +48 q^{-55} -6 q^{-56} +42 q^{-57} -75 q^{-58} -24 q^{-59} +60 q^{-60} +7 q^{-61} +44 q^{-62} -88 q^{-63} -34 q^{-64} +66 q^{-65} +19 q^{-66} +45 q^{-67} -91 q^{-68} -42 q^{-69} +57 q^{-70} +26 q^{-71} +52 q^{-72} -76 q^{-73} -48 q^{-74} +34 q^{-75} +21 q^{-76} +56 q^{-77} -47 q^{-78} -39 q^{-79} +10 q^{-80} +3 q^{-81} +49 q^{-82} -18 q^{-83} -22 q^{-84} -2 q^{-85} -10 q^{-86} +32 q^{-87} -4 q^{-88} -7 q^{-89} -3 q^{-90} -12 q^{-91} +16 q^{-92} - q^{-95} -7 q^{-96} +5 q^{-97} + q^{-99} -2 q^{-101} + q^{-102} }
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^{-15} - q^{-16} + q^{-18} +2 q^{-21} -3 q^{-22} -2 q^{-23} +3 q^{-24} +3 q^{-25} + q^{-26} +4 q^{-27} -8 q^{-28} -9 q^{-29} +9 q^{-31} +9 q^{-32} +13 q^{-33} -9 q^{-34} -22 q^{-35} -14 q^{-36} +3 q^{-37} +18 q^{-38} +33 q^{-39} +4 q^{-40} -25 q^{-41} -30 q^{-42} -17 q^{-43} +7 q^{-44} +47 q^{-45} +23 q^{-46} -12 q^{-47} -26 q^{-48} -29 q^{-49} -11 q^{-50} +34 q^{-51} +26 q^{-52} -5 q^{-53} -9 q^{-54} -12 q^{-55} -8 q^{-56} +21 q^{-57} + q^{-58} -27 q^{-59} -10 q^{-60} +16 q^{-61} +34 q^{-62} +35 q^{-63} -23 q^{-64} -78 q^{-65} -42 q^{-66} +29 q^{-67} +88 q^{-68} +82 q^{-69} -24 q^{-70} -127 q^{-71} -99 q^{-72} +18 q^{-73} +132 q^{-74} +139 q^{-75} - q^{-76} -159 q^{-77} -159 q^{-78} -10 q^{-79} +163 q^{-80} +188 q^{-81} +25 q^{-82} -174 q^{-83} -208 q^{-84} -36 q^{-85} +184 q^{-86} +224 q^{-87} +42 q^{-88} -187 q^{-89} -242 q^{-90} -52 q^{-91} +201 q^{-92} +251 q^{-93} +55 q^{-94} -196 q^{-95} -265 q^{-96} -70 q^{-97} +202 q^{-98} +267 q^{-99} +81 q^{-100} -184 q^{-101} -271 q^{-102} -97 q^{-103} +166 q^{-104} +258 q^{-105} +111 q^{-106} -134 q^{-107} -241 q^{-108} -116 q^{-109} +102 q^{-110} +203 q^{-111} +117 q^{-112} -66 q^{-113} -167 q^{-114} -107 q^{-115} +40 q^{-116} +125 q^{-117} +89 q^{-118} -15 q^{-119} -90 q^{-120} -71 q^{-121} +2 q^{-122} +61 q^{-123} +54 q^{-124} +4 q^{-125} -39 q^{-126} -38 q^{-127} -7 q^{-128} +21 q^{-129} +28 q^{-130} +10 q^{-131} -16 q^{-132} -17 q^{-133} -5 q^{-134} +3 q^{-135} +11 q^{-136} +10 q^{-137} -5 q^{-138} -7 q^{-139} - q^{-140} -3 q^{-141} +3 q^{-142} +5 q^{-143} - q^{-144} -2 q^{-145} - q^{-147} +2 q^{-149} - q^{-150} }
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^{-18} - q^{-19} + q^{-21} - q^{-24} +3 q^{-25} -3 q^{-26} -2 q^{-27} +4 q^{-28} +2 q^{-29} +2 q^{-30} -4 q^{-31} +5 q^{-32} -9 q^{-33} -9 q^{-34} +6 q^{-35} +8 q^{-36} +12 q^{-37} -3 q^{-38} +13 q^{-39} -20 q^{-40} -28 q^{-41} -4 q^{-42} +6 q^{-43} +28 q^{-44} +10 q^{-45} +42 q^{-46} -18 q^{-47} -47 q^{-48} -32 q^{-49} -19 q^{-50} +23 q^{-51} +14 q^{-52} +88 q^{-53} +9 q^{-54} -35 q^{-55} -46 q^{-56} -49 q^{-57} -5 q^{-58} -19 q^{-59} +110 q^{-60} +29 q^{-61} -6 q^{-62} -29 q^{-63} -40 q^{-64} -9 q^{-65} -55 q^{-66} +101 q^{-67} +5 q^{-68} -15 q^{-69} -33 q^{-70} -9 q^{-71} +38 q^{-72} -34 q^{-73} +127 q^{-74} -25 q^{-75} -77 q^{-76} -112 q^{-77} -32 q^{-78} +82 q^{-79} +43 q^{-80} +229 q^{-81} +16 q^{-82} -123 q^{-83} -243 q^{-84} -139 q^{-85} +49 q^{-86} +99 q^{-87} +371 q^{-88} +144 q^{-89} -85 q^{-90} -344 q^{-91} -286 q^{-92} -68 q^{-93} +74 q^{-94} +478 q^{-95} +310 q^{-96} +32 q^{-97} -365 q^{-98} -405 q^{-99} -222 q^{-100} -24 q^{-101} +517 q^{-102} +457 q^{-103} +179 q^{-104} -323 q^{-105} -471 q^{-106} -362 q^{-107} -149 q^{-108} +506 q^{-109} +565 q^{-110} +313 q^{-111} -262 q^{-112} -502 q^{-113} -465 q^{-114} -253 q^{-115} +480 q^{-116} +640 q^{-117} +410 q^{-118} -218 q^{-119} -523 q^{-120} -535 q^{-121} -319 q^{-122} +468 q^{-123} +698 q^{-124} +474 q^{-125} -198 q^{-126} -550 q^{-127} -588 q^{-128} -360 q^{-129} +459 q^{-130} +744 q^{-131} +529 q^{-132} -165 q^{-133} -559 q^{-134} -634 q^{-135} -410 q^{-136} +411 q^{-137} +743 q^{-138} +578 q^{-139} -81 q^{-140} -499 q^{-141} -628 q^{-142} -464 q^{-143} +293 q^{-144} +646 q^{-145} +566 q^{-146} +27 q^{-147} -352 q^{-148} -519 q^{-149} -457 q^{-150} +150 q^{-151} +453 q^{-152} +450 q^{-153} +80 q^{-154} -184 q^{-155} -329 q^{-156} -357 q^{-157} +63 q^{-158} +251 q^{-159} +275 q^{-160} +58 q^{-161} -74 q^{-162} -157 q^{-163} -222 q^{-164} +41 q^{-165} +122 q^{-166} +136 q^{-167} +13 q^{-168} -30 q^{-169} -63 q^{-170} -123 q^{-171} +41 q^{-172} +58 q^{-173} +66 q^{-174} -7 q^{-175} -14 q^{-176} -26 q^{-177} -72 q^{-178} +32 q^{-179} +26 q^{-180} +35 q^{-181} -6 q^{-182} -2 q^{-183} -11 q^{-184} -41 q^{-185} +17 q^{-186} +6 q^{-187} +18 q^{-188} -2 q^{-189} +5 q^{-190} -3 q^{-191} -20 q^{-192} +7 q^{-193} -2 q^{-194} +7 q^{-195} - q^{-196} +4 q^{-197} -7 q^{-199} +3 q^{-200} -2 q^{-201} +2 q^{-202} + q^{-204} -2 q^{-206} + q^{-207} }

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.

9_5 9_7