9 18

9_17

9_19

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 18's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 18 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X3,12,4,13 X5,14,6,15 X9,18,10,1 X17,6,18,7 X7,16,8,17 X15,8,16,9 X13,10,14,11 X11,2,12,3
Gauss code	-1, 9, -2, 1, -3, 5, -6, 7, -4, 8, -9, 2, -8, 3, -7, 6, -5, 4
Dowker-Thistlethwaite code	4 12 14 16 18 2 10 8 6
Conway Notation	[3222]

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

[edit Notes on presentations of 9 18]

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 18"];

In[4]:=

PD[K]

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

Out[4]=

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

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 12 14 16 18 2 10 8 6

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[3222]

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]=

${\textrm {BR}}(4,\{-1,-1,-1,-2,1,-2,-2,-2,-3,2,-3\})$

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]=

{ 4, 11, 4 }

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-14][3] Hyperbolic Volume 10.0577 A-Polynomial See Data:9 18/A-polynomial

[edit Notes for 9 18's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $2$ Topological 4 genus $2$ Concordance genus $2$ Rasmussen s-Invariant -4

[edit Notes for 9 18's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$4t^{2}-10t+13-10t^{-1}+4t^{-2}$
Conway polynomial	$4z^{4}+6z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 41, -4 }
Jones polynomial	$q^{-2}-2q^{-3}+5q^{-4}-6q^{-5}+7q^{-6}-7q^{-7}+6q^{-8}-4q^{-9}+2q^{-10}-q^{-11}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{10}-a^{10}+z^{4}a^{8}+z^{2}a^{8}+2z^{4}a^{6}+4z^{2}a^{6}+a^{6}+z^{4}a^{4}+2z^{2}a^{4}+a^{4}$
Kauffman polynomial (db, data sources)	$z^{5}a^{13}-3z^{3}a^{13}+2za^{13}+2z^{6}a^{12}-5z^{4}a^{12}+3z^{2}a^{12}+2z^{7}a^{11}-3z^{5}a^{11}+z^{8}a^{10}+z^{6}a^{10}-2z^{4}a^{10}-2z^{2}a^{10}+a^{10}+4z^{7}a^{9}-5z^{5}a^{9}+z^{3}a^{9}+z^{8}a^{8}+2z^{6}a^{8}-2z^{4}a^{8}+2z^{7}a^{7}+z^{5}a^{7}-4z^{3}a^{7}+2za^{7}+3z^{6}a^{6}-4z^{4}a^{6}+3z^{2}a^{6}-a^{6}+2z^{5}a^{5}-2z^{3}a^{5}+z^{4}a^{4}-2z^{2}a^{4}+a^{4}$
The A2 invariant	$-q^{34}-2q^{28}+q^{26}+q^{20}-q^{18}+2q^{16}+q^{12}+2q^{10}-q^{8}+q^{6}$
The G2 invariant	$q^{176}-q^{174}+3q^{172}-4q^{170}+3q^{168}-2q^{166}-2q^{164}+10q^{162}-15q^{160}+18q^{158}-15q^{156}+5q^{154}+9q^{152}-26q^{150}+36q^{148}-37q^{146}+23q^{144}-2q^{142}-24q^{140}+37q^{138}-38q^{136}+27q^{134}-6q^{132}-16q^{130}+24q^{128}-23q^{126}+5q^{124}+16q^{122}-29q^{120}+31q^{118}-16q^{116}-7q^{114}+34q^{112}-51q^{110}+54q^{108}-40q^{106}+9q^{104}+23q^{102}-48q^{100}+58q^{98}-47q^{96}+23q^{94}+4q^{92}-26q^{90}+32q^{88}-25q^{86}+4q^{84}+16q^{82}-23q^{80}+20q^{78}-2q^{76}-18q^{74}+35q^{72}-36q^{70}+29q^{68}-12q^{66}-11q^{64}+29q^{62}-34q^{60}+33q^{58}-18q^{56}+6q^{54}+6q^{52}-14q^{50}+16q^{48}-13q^{46}+9q^{44}-2q^{42}-q^{40}+3q^{38}-3q^{36}+3q^{34}-q^{32}+q^{30}$

Further Quantum Invariants

Further quantum knot invariants for 9_18.

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^{23}+q^{21}-2 q^{19}+2 q^{17}-q^{15}+q^{11}-q^9+3 q^7-q^5+q^3}
2	$q^{64}-q^{62}-q^{60}+4q^{58}-2q^{56}-6q^{54}+8q^{52}+q^{50}-10q^{48}+8q^{46}+4q^{44}-11q^{42}+2q^{40}+5q^{38}-4q^{36}-3q^{34}+3q^{32}+5q^{30}-8q^{28}-q^{26}+11q^{24}-8q^{22}-4q^{20}+11q^{18}-3q^{16}-3q^{14}+5q^{12}-q^{8}+q^{6}$
3	$-q^{123}+q^{121}+q^{119}-q^{117}-3q^{115}+2q^{113}+7q^{111}-q^{109}-13q^{107}-3q^{105}+18q^{103}+11q^{101}-22q^{99}-21q^{97}+21q^{95}+31q^{93}-14q^{91}-39q^{89}+9q^{87}+42q^{85}+q^{83}-42q^{81}-10q^{79}+38q^{77}+15q^{75}-30q^{73}-18q^{71}+19q^{69}+21q^{67}-8q^{65}-23q^{63}-7q^{61}+22q^{59}+18q^{57}-20q^{55}-33q^{53}+17q^{51}+41q^{49}-9q^{47}-45q^{45}+2q^{43}+41q^{41}+6q^{39}-35q^{37}-11q^{35}+27q^{33}+10q^{31}-14q^{29}-9q^{27}+10q^{25}+7q^{23}-4q^{21}-3q^{19}+3q^{17}+2q^{15}-q^{11}+q^{9}$
4	$q^{200}-q^{198}-q^{196}+q^{194}+3q^{190}-4q^{188}-5q^{186}+3q^{184}+4q^{182}+15q^{180}-7q^{178}-23q^{176}-10q^{174}+7q^{172}+50q^{170}+14q^{168}-39q^{166}-55q^{164}-30q^{162}+83q^{160}+77q^{158}-4q^{156}-94q^{154}-117q^{152}+55q^{150}+132q^{148}+88q^{146}-68q^{144}-192q^{142}-29q^{140}+122q^{138}+164q^{136}+6q^{134}-197q^{132}-104q^{130}+64q^{128}+181q^{126}+66q^{124}-146q^{122}-124q^{120}+9q^{118}+142q^{116}+87q^{114}-72q^{112}-111q^{110}-38q^{108}+83q^{106}+93q^{104}+10q^{102}-86q^{100}-88q^{98}+4q^{96}+96q^{94}+110q^{92}-43q^{90}-137q^{88}-92q^{86}+73q^{84}+195q^{82}+30q^{80}-137q^{78}-171q^{76}+6q^{74}+209q^{72}+94q^{70}-69q^{68}-172q^{66}-64q^{64}+138q^{62}+99q^{60}+6q^{58}-106q^{56}-76q^{54}+56q^{52}+52q^{50}+30q^{48}-37q^{46}-43q^{44}+15q^{42}+14q^{40}+19q^{38}-9q^{36}-15q^{34}+7q^{32}+2q^{30}+7q^{28}-2q^{26}-4q^{24}+3q^{22}+2q^{18}-q^{14}+q^{12}$
5	$-q^{295}+q^{293}+q^{291}-q^{289}-q^{283}+2q^{281}+4q^{279}-3q^{277}-7q^{275}-4q^{273}-q^{271}+11q^{269}+20q^{267}+8q^{265}-20q^{263}-39q^{261}-29q^{259}+14q^{257}+67q^{255}+74q^{253}+13q^{251}-86q^{249}-137q^{247}-76q^{245}+73q^{243}+199q^{241}+182q^{239}-5q^{237}-239q^{235}-308q^{233}-121q^{231}+209q^{229}+422q^{227}+305q^{225}-100q^{223}-487q^{221}-502q^{219}-79q^{217}+462q^{215}+665q^{213}+314q^{211}-353q^{209}-771q^{207}-532q^{205}+180q^{203}+772q^{201}+718q^{199}+25q^{197}-709q^{195}-827q^{193}-204q^{191}+583q^{189}+846q^{187}+349q^{185}-443q^{183}-803q^{181}-430q^{179}+309q^{177}+714q^{175}+452q^{173}-186q^{171}-605q^{169}-448q^{167}+91q^{165}+500q^{163}+427q^{161}-7q^{159}-389q^{157}-415q^{155}-89q^{153}+291q^{151}+419q^{149}+199q^{147}-177q^{145}-426q^{143}-345q^{141}+44q^{139}+437q^{137}+497q^{135}+130q^{133}-414q^{131}-656q^{129}-324q^{127}+342q^{125}+771q^{123}+538q^{121}-213q^{119}-828q^{117}-719q^{115}+35q^{113}+775q^{111}+850q^{109}+160q^{107}-656q^{105}-871q^{103}-332q^{101}+455q^{99}+800q^{97}+446q^{95}-249q^{93}-656q^{91}-466q^{89}+70q^{87}+461q^{85}+418q^{83}+59q^{81}-290q^{79}-326q^{77}-96q^{75}+139q^{73}+214q^{71}+107q^{69}-55q^{67}-129q^{65}-76q^{63}+13q^{61}+62q^{59}+49q^{57}-27q^{53}-23q^{51}-3q^{49}+15q^{47}+11q^{45}-q^{43}-6q^{41}-q^{37}+4q^{35}+5q^{33}-2q^{31}-2q^{29}+2q^{27}+2q^{21}-q^{17}+q^{15}$

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^{34}-2 q^{28}+q^{26}+q^{20}-q^{18}+2 q^{16}+q^{12}+2 q^{10}-q^8+q^6}
1,1	$q^{92}-2q^{90}+6q^{88}-12q^{86}+23q^{84}-40q^{82}+60q^{80}-84q^{78}+110q^{76}-132q^{74}+142q^{72}-136q^{70}+114q^{68}-76q^{66}+18q^{64}+52q^{62}-119q^{60}+184q^{58}-236q^{56}+268q^{54}-280q^{52}+260q^{50}-224q^{48}+162q^{46}-97q^{44}+24q^{42}+38q^{40}-92q^{38}+123q^{36}-134q^{34}+136q^{32}-120q^{30}+103q^{28}-72q^{26}+56q^{24}-34q^{22}+23q^{20}-10q^{18}+6q^{16}-2q^{14}+q^{12}$
2,0	$q^{86}+2q^{78}-4q^{74}-q^{72}+4q^{70}+2q^{68}-4q^{66}-q^{64}+5q^{62}+q^{60}-7q^{58}-2q^{56}+2q^{54}-3q^{52}-3q^{50}+q^{48}+q^{46}-3q^{44}+q^{42}+3q^{40}-4q^{38}-q^{36}+7q^{34}+2q^{32}-5q^{30}+3q^{28}+7q^{26}+q^{24}-4q^{22}+2q^{20}+4q^{18}-q^{16}-q^{14}+q^{12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{74}-q^{72}+q^{70}+2q^{68}-4q^{66}+2q^{64}+4q^{62}-8q^{60}+3q^{58}+5q^{56}-10q^{54}+q^{52}+5q^{50}-6q^{48}-2q^{46}+2q^{44}-3q^{40}-2q^{38}+7q^{36}-2q^{34}-5q^{32}+10q^{30}-6q^{26}+9q^{24}+q^{22}-3q^{20}+4q^{18}+q^{16}-q^{14}+q^{12}$
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^{45}-q^{41}-2 q^{37}+q^{35}-q^{33}+q^{31}+q^{27}+2 q^{21}+2 q^{17}+2 q^{13}-q^{11}+q^9}

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{96}-q^{92}+2q^{90}+3q^{88}-2q^{86}-q^{84}+5q^{82}+3q^{80}-6q^{78}-4q^{76}+4q^{74}-3q^{72}-10q^{70}+4q^{66}-5q^{64}-3q^{62}+4q^{60}-q^{58}-6q^{56}+q^{54}+4q^{52}-5q^{50}-2q^{48}+9q^{46}+2q^{44}-5q^{42}+4q^{40}+8q^{38}-2q^{34}+4q^{32}+5q^{30}-q^{28}+3q^{24}+q^{22}-q^{20}+q^{18}$
1,0,0,0	$-q^{56}-q^{52}-q^{50}-2q^{46}+q^{44}-q^{42}+q^{38}+q^{34}+q^{30}+2q^{26}+2q^{22}+q^{20}+2q^{16}-q^{14}+q^{12}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{74}+q^{72}-3q^{70}+4q^{68}-6q^{66}+8q^{64}-8q^{62}+8q^{60}-7q^{58}+5q^{56}-2q^{54}-3q^{52}+7q^{50}-12q^{48}+14q^{46}-16q^{44}+16q^{42}-15q^{40}+12q^{38}-7q^{36}+4q^{34}+q^{32}-4q^{30}+8q^{28}-8q^{26}+9q^{24}-7q^{22}+7q^{20}-4q^{18}+3q^{16}-q^{14}+q^{12}$
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^{120}-q^{116}-q^{114}+2 q^{112}+3 q^{110}-q^{108}-5 q^{106}-2 q^{104}+6 q^{102}+6 q^{100}-4 q^{98}-9 q^{96}-q^{94}+9 q^{92}+5 q^{90}-7 q^{88}-8 q^{86}+2 q^{84}+7 q^{82}-7 q^{78}-2 q^{76}+5 q^{74}+2 q^{72}-6 q^{70}-4 q^{68}+4 q^{66}+5 q^{64}-3 q^{62}-6 q^{60}+2 q^{58}+7 q^{56}+q^{54}-8 q^{52}-3 q^{50}+8 q^{48}+8 q^{46}-4 q^{44}-8 q^{42}+9 q^{38}+5 q^{36}-3 q^{34}-5 q^{32}+q^{30}+4 q^{28}+2 q^{26}-q^{24}-q^{22}+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^{102}-q^{100}+2 q^{98}-2 q^{96}+4 q^{94}-5 q^{92}+5 q^{90}-6 q^{88}+7 q^{86}-7 q^{84}+5 q^{82}-5 q^{80}+4 q^{78}-3 q^{76}-3 q^{74}+2 q^{72}-5 q^{70}+7 q^{68}-12 q^{66}+10 q^{64}-12 q^{62}+13 q^{60}-13 q^{58}+9 q^{56}-10 q^{54}+9 q^{52}-4 q^{50}+3 q^{48}-q^{46}+7 q^{42}-4 q^{40}+6 q^{38}-6 q^{36}+9 q^{34}-4 q^{32}+6 q^{30}-4 q^{28}+5 q^{26}-q^{24}+2 q^{22}-q^{20}+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^{176}-q^{174}+3 q^{172}-4 q^{170}+3 q^{168}-2 q^{166}-2 q^{164}+10 q^{162}-15 q^{160}+18 q^{158}-15 q^{156}+5 q^{154}+9 q^{152}-26 q^{150}+36 q^{148}-37 q^{146}+23 q^{144}-2 q^{142}-24 q^{140}+37 q^{138}-38 q^{136}+27 q^{134}-6 q^{132}-16 q^{130}+24 q^{128}-23 q^{126}+5 q^{124}+16 q^{122}-29 q^{120}+31 q^{118}-16 q^{116}-7 q^{114}+34 q^{112}-51 q^{110}+54 q^{108}-40 q^{106}+9 q^{104}+23 q^{102}-48 q^{100}+58 q^{98}-47 q^{96}+23 q^{94}+4 q^{92}-26 q^{90}+32 q^{88}-25 q^{86}+4 q^{84}+16 q^{82}-23 q^{80}+20 q^{78}-2 q^{76}-18 q^{74}+35 q^{72}-36 q^{70}+29 q^{68}-12 q^{66}-11 q^{64}+29 q^{62}-34 q^{60}+33 q^{58}-18 q^{56}+6 q^{54}+6 q^{52}-14 q^{50}+16 q^{48}-13 q^{46}+9 q^{44}-2 q^{42}-q^{40}+3 q^{38}-3 q^{36}+3 q^{34}-q^{32}+q^{30}}

.

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 18"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$4t^{2}-10t+13-10t^{-1}+4t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$4z^{4}+6z^{2}+1$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 41, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{2}a^{10}-a^{10}+z^{4}a^{8}+z^{2}a^{8}+2z^{4}a^{6}+4z^{2}a^{6}+a^{6}+z^{4}a^{4}+2z^{2}a^{4}+a^{4}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a246,}

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

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 18"];

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 4 t^2-10 t+13-10 t^{-1} +4 t^{-2} } , $q^{-2}-2q^{-3}+5q^{-4}-6q^{-5}+7q^{-6}-7q^{-7}+6q^{-8}-4q^{-9}+2q^{-10}-q^{-11}$ }

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]=

{K11a246,}

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:

(6, -15)

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 $24$ $-120$ $288$ 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 748} $108$ $-2880$ $-5200$ $-896$ $-664$ $2304$ $7200$ $17952$ $2592$ 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{184471}{5}} ${\frac {17828}{15}}$ ${\frac {207524}{15}}$ ${\frac {889}{3}}$ ${\frac {8791}{5}}$

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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$-4 is the signature of 9 18. 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 -3                   1 1 -5                 2 1 -1 -7               3     3 -9             3 2     -1 -11           4 3       1 -13         3 3         0 -15       3 4           -1 -17     1 3             2 -19   1 3               -2 -21   1                 1 -23 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-5$ $i=-3$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{-4}-2q^{-5}+q^{-6}+6q^{-7}-10q^{-8}+q^{-9}+20q^{-10}-25q^{-11}-3q^{-12}+39q^{-13}-37q^{-14}-10q^{-15}+52q^{-16}-39q^{-17}-16q^{-18}+51q^{-19}-30q^{-20}-19q^{-21}+38q^{-22}-15q^{-23}-15q^{-24}+20q^{-25}-4q^{-26}-8q^{-27}+6q^{-28}-2q^{-30}+q^{-31}$
3	$q^{-6}-2q^{-7}+q^{-8}+2q^{-9}+2q^{-10}-8q^{-11}+13q^{-13}+5q^{-14}-27q^{-15}-5q^{-16}+37q^{-17}+22q^{-18}-65q^{-19}-29q^{-20}+78q^{-21}+57q^{-22}-104q^{-23}-76q^{-24}+114q^{-25}+107q^{-26}-128q^{-27}-126q^{-28}+127q^{-29}+145q^{-30}-124q^{-31}-155q^{-32}+111q^{-33}+160q^{-34}-95q^{-35}-157q^{-36}+74q^{-37}+148q^{-38}-50q^{-39}-134q^{-40}+26q^{-41}+116q^{-42}-7q^{-43}-93q^{-44}-7q^{-45}+68q^{-46}+18q^{-47}-48q^{-48}-17q^{-49}+26q^{-50}+17q^{-51}-15q^{-52}-10q^{-53}+5q^{-54}+7q^{-55}-3q^{-56}-2q^{-57}+2q^{-59}-q^{-60}$
4	$q^{-8}-2q^{-9}+q^{-10}+2q^{-11}-2q^{-12}+4q^{-13}-9q^{-14}+3q^{-15}+11q^{-16}-7q^{-17}+9q^{-18}-31q^{-19}+9q^{-20}+39q^{-21}-12q^{-22}+10q^{-23}-89q^{-24}+15q^{-25}+106q^{-26}+10q^{-27}+14q^{-28}-221q^{-29}-15q^{-30}+218q^{-31}+103q^{-32}+53q^{-33}-423q^{-34}-123q^{-35}+321q^{-36}+266q^{-37}+168q^{-38}-626q^{-39}-300q^{-40}+355q^{-41}+433q^{-42}+333q^{-43}-748q^{-44}-465q^{-45}+310q^{-46}+527q^{-47}+486q^{-48}-762q^{-49}-557q^{-50}+218q^{-51}+529q^{-52}+582q^{-53}-679q^{-54}-567q^{-55}+97q^{-56}+456q^{-57}+621q^{-58}-520q^{-59}-512q^{-60}-36q^{-61}+323q^{-62}+599q^{-63}-308q^{-64}-397q^{-65}-153q^{-66}+155q^{-67}+506q^{-68}-105q^{-69}-239q^{-70}-195q^{-71}+4q^{-72}+343q^{-73}+19q^{-74}-83q^{-75}-151q^{-76}-73q^{-77}+171q^{-78}+42q^{-79}+7q^{-80}-70q^{-81}-67q^{-82}+58q^{-83}+17q^{-84}+23q^{-85}-17q^{-86}-31q^{-87}+15q^{-88}+10q^{-90}-q^{-91}-9q^{-92}+4q^{-93}-q^{-94}+2q^{-95}-2q^{-97}+q^{-98}$
5	${\textrm {NotAvailable}}(q)$
6	${\textrm {NotAvailable}}(q)$
7	${\textrm {NotAvailable}}(q)$

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_17 9_19