9 20

9_19

9_21

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 20's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 20 at Knotilus!

From knotilus

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X5,14,6,15 X7,16,8,17 X11,1,12,18 X15,6,16,7 X17,13,18,12 X13,8,14,9 X9,2,10,3
Gauss code	-1, 9, -2, 1, -3, 6, -4, 8, -9, 2, -5, 7, -8, 3, -6, 4, -7, 5
Dowker-Thistlethwaite code	4 10 14 16 2 18 8 6 12
Conway Notation	[31212]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 9, width is 4, Braid index is 4		[{12, 2}, {1, 10}, {8, 11}, {10, 12}, {9, 3}, {2, 4}, {3, 5}, {4, 8}, {6, 9}, {5, 7}, {11, 6}, {7, 1}]

[edit Notes on presentations of 9 20]

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 20"];

In[4]:=

PD[K]

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

Out[4]=

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

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 14 16 2 18 8 6 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[31212]

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]=

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

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, 9, 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[{12, 2}, {1, 10}, {8, 11}, {10, 12}, {9, 3}, {2, 4}, {3, 5}, {4, 8}, {6, 9}, {5, 7}, {11, 6}, {7, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-12][1] Hyperbolic Volume 9.6443 A-Polynomial See Data:9 20/A-polynomial

[edit Notes for 9 20's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $2$ Topological 4 genus $2$ Concordance genus $3$ Rasmussen s-Invariant -4

[edit Notes for 9 20's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+5t^{2}-9t+11-9t^{-1}+5t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 41, -4 }
Jones polynomial	$1-2q^{-1}+4q^{-2}-5q^{-3}+7q^{-4}-7q^{-5}+6q^{-6}-5q^{-7}+3q^{-8}-q^{-9}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{8}-a^{8}+2z^{4}a^{6}+5z^{2}a^{6}+2a^{6}-z^{6}a^{4}-4z^{4}a^{4}-5z^{2}a^{4}-2a^{4}+z^{4}a^{2}+3z^{2}a^{2}+2a^{2}$
Kauffman polynomial (db, data sources)	$z^{3}a^{11}+3z^{4}a^{10}-z^{2}a^{10}+5z^{5}a^{9}-5z^{3}a^{9}+2za^{9}+5z^{6}a^{8}-6z^{4}a^{8}+3z^{2}a^{8}-a^{8}+3z^{7}a^{7}-7z^{3}a^{7}+2za^{7}+z^{8}a^{6}+5z^{6}a^{6}-16z^{4}a^{6}+10z^{2}a^{6}-2a^{6}+5z^{7}a^{5}-12z^{5}a^{5}+5z^{3}a^{5}+z^{8}a^{4}+z^{6}a^{4}-11z^{4}a^{4}+11z^{2}a^{4}-2a^{4}+2z^{7}a^{3}-7z^{5}a^{3}+6z^{3}a^{3}+z^{6}a^{2}-4z^{4}a^{2}+5z^{2}a^{2}-2a^{2}$
The A2 invariant	$-q^{28}+q^{24}-q^{22}+q^{20}-q^{18}+q^{14}-q^{12}+2q^{10}-q^{8}+q^{6}+q^{4}+1$
The G2 invariant	$q^{148}-2q^{146}+3q^{144}-4q^{142}+2q^{140}-q^{138}-2q^{136}+9q^{134}-12q^{132}+15q^{130}-13q^{128}+5q^{126}+2q^{124}-13q^{122}+23q^{120}-27q^{118}+23q^{116}-13q^{114}-q^{112}+15q^{110}-23q^{108}+25q^{106}-21q^{104}+5q^{102}+7q^{100}-17q^{98}+16q^{96}-5q^{94}-9q^{92}+23q^{90}-24q^{88}+14q^{86}+3q^{84}-26q^{82}+44q^{80}-44q^{78}+30q^{76}-4q^{74}-21q^{72}+43q^{70}-45q^{68}+33q^{66}-17q^{64}-6q^{62}+22q^{60}-28q^{58}+22q^{56}-6q^{54}-9q^{52}+19q^{50}-19q^{48}+7q^{46}+8q^{44}-23q^{42}+31q^{40}-28q^{38}+12q^{36}+10q^{34}-26q^{32}+36q^{30}-30q^{28}+17q^{26}-q^{24}-13q^{22}+21q^{20}-19q^{18}+14q^{16}-3q^{14}-2q^{12}+5q^{10}-5q^{8}+4q^{6}-q^{4}+q^{2}$

Further Quantum Invariants

Further quantum knot invariants for 9_20.

A1 Invariants.

Weight	Invariant
1	$-q^{19}+2q^{17}-2q^{15}+q^{13}-q^{11}+2q^{7}-q^{5}+2q^{3}-q+q^{-1}$
2	$q^{52}-2q^{50}+5q^{46}-6q^{44}-q^{42}+8q^{40}-8q^{38}-q^{36}+9q^{34}-4q^{32}-4q^{30}+4q^{28}+2q^{26}-5q^{24}-3q^{22}+6q^{20}-q^{18}-7q^{16}+8q^{14}+3q^{12}-8q^{10}+5q^{8}+5q^{6}-6q^{4}+q^{2}+4-2q^{-2}-q^{-4}+q^{-6}$
3	$-q^{99}+2q^{97}-3q^{93}+5q^{89}+2q^{87}-10q^{85}+13q^{81}-2q^{79}-18q^{77}+2q^{75}+25q^{73}-2q^{71}-28q^{69}-2q^{67}+30q^{65}+7q^{63}-26q^{61}-13q^{59}+16q^{57}+19q^{55}-6q^{53}-20q^{51}-5q^{49}+21q^{47}+13q^{45}-16q^{43}-22q^{41}+13q^{39}+23q^{37}-9q^{35}-27q^{33}+2q^{31}+28q^{29}+4q^{27}-27q^{25}-11q^{23}+26q^{21}+17q^{19}-19q^{17}-20q^{15}+13q^{13}+23q^{11}-3q^{9}-20q^{7}-2q^{5}+14q^{3}+7q-8q^{-1}-7q^{-3}+3q^{-5}+5q^{-7}-2q^{-11}-q^{-13}+q^{-15}$
4	$q^{160}-2q^{158}+3q^{154}-2q^{152}+q^{150}-6q^{148}+3q^{146}+9q^{144}-8q^{142}+2q^{140}-10q^{138}+11q^{136}+18q^{134}-25q^{132}-10q^{130}-9q^{128}+40q^{126}+37q^{124}-53q^{122}-46q^{120}-15q^{118}+79q^{116}+79q^{114}-60q^{112}-93q^{110}-51q^{108}+86q^{106}+122q^{104}-15q^{102}-92q^{100}-97q^{98}+30q^{96}+111q^{94}+48q^{92}-28q^{90}-96q^{88}-45q^{86}+39q^{84}+78q^{82}+50q^{80}-51q^{78}-87q^{76}-28q^{74}+73q^{72}+85q^{70}-5q^{68}-93q^{66}-68q^{64}+60q^{62}+92q^{60}+24q^{58}-90q^{56}-92q^{54}+44q^{52}+92q^{50}+58q^{48}-70q^{46}-114q^{44}+3q^{42}+76q^{40}+98q^{38}-20q^{36}-112q^{34}-52q^{32}+26q^{30}+114q^{28}+46q^{26}-63q^{24}-76q^{22}-40q^{20}+70q^{18}+73q^{16}+6q^{14}-44q^{12}-68q^{10}+7q^{8}+44q^{6}+35q^{4}+4q^{2}-40-19q^{-2}+3q^{-4}+19q^{-6}+18q^{-8}-8q^{-10}-9q^{-12}-7q^{-14}+q^{-16}+7q^{-18}+q^{-20}-2q^{-24}-q^{-26}+q^{-28}$
5	$-q^{235}+2q^{233}-3q^{229}+2q^{227}+q^{225}+q^{221}-2q^{219}-4q^{217}+2q^{215}+6q^{213}-q^{211}-8q^{209}-5q^{207}+10q^{205}+15q^{203}+8q^{201}-18q^{199}-46q^{197}-15q^{195}+50q^{193}+81q^{191}+32q^{189}-77q^{187}-147q^{185}-73q^{183}+114q^{181}+236q^{179}+133q^{177}-137q^{175}-326q^{173}-232q^{171}+119q^{169}+420q^{167}+350q^{165}-68q^{163}-460q^{161}-464q^{159}-46q^{157}+443q^{155}+556q^{153}+176q^{151}-353q^{149}-573q^{147}-306q^{145}+198q^{143}+514q^{141}+400q^{139}-23q^{137}-388q^{135}-422q^{133}-140q^{131}+213q^{129}+390q^{127}+262q^{125}-44q^{123}-304q^{121}-338q^{119}-100q^{117}+215q^{115}+354q^{113}+205q^{111}-132q^{109}-361q^{107}-261q^{105}+87q^{103}+349q^{101}+288q^{99}-49q^{97}-356q^{95}-311q^{93}+44q^{91}+362q^{89}+335q^{87}-17q^{85}-376q^{83}-381q^{81}-16q^{79}+378q^{77}+426q^{75}+83q^{73}-346q^{71}-472q^{69}-180q^{67}+280q^{65}+491q^{63}+277q^{61}-165q^{59}-467q^{57}-379q^{55}+21q^{53}+390q^{51}+433q^{49}+129q^{47}-259q^{45}-429q^{43}-260q^{41}+102q^{39}+361q^{37}+336q^{35}+55q^{33}-241q^{31}-332q^{29}-176q^{27}+97q^{25}+275q^{23}+234q^{21}+28q^{19}-167q^{17}-220q^{15}-117q^{13}+61q^{11}+167q^{9}+140q^{7}+22q^{5}-87q^{3}-119q-65q^{-1}+23q^{-3}+75q^{-5}+67q^{-7}+15q^{-9}-32q^{-11}-45q^{-13}-29q^{-15}+4q^{-17}+25q^{-19}+22q^{-21}+5q^{-23}-6q^{-25}-11q^{-27}-9q^{-29}+q^{-31}+5q^{-33}+3q^{-35}+q^{-37}-2q^{-41}-q^{-43}+q^{-45}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{28}+q^{24}-q^{22}+q^{20}-q^{18}+q^{14}-q^{12}+2q^{10}-q^{8}+q^{6}+q^{4}+1$
1,1	$q^{76}-4q^{74}+8q^{72}-12q^{70}+22q^{68}-36q^{66}+46q^{64}-56q^{62}+72q^{60}-84q^{58}+84q^{56}-82q^{54}+77q^{52}-62q^{50}+32q^{48}+4q^{46}-43q^{44}+90q^{42}-132q^{40}+168q^{38}-192q^{36}+198q^{34}-190q^{32}+158q^{30}-126q^{28}+82q^{26}-30q^{24}-18q^{22}+57q^{20}-82q^{18}+106q^{16}-110q^{14}+104q^{12}-84q^{10}+70q^{8}-48q^{6}+29q^{4}-16q^{2}+8-2q^{-2}+q^{-4}$
2,0	$q^{70}-q^{66}+q^{62}-3q^{58}+q^{56}+2q^{54}-3q^{52}-q^{50}+4q^{48}+3q^{46}-4q^{44}-q^{42}+3q^{40}-2q^{38}-4q^{36}+2q^{34}+q^{32}-4q^{30}+q^{28}+2q^{26}-2q^{24}-2q^{22}+4q^{20}+3q^{18}-2q^{16}+q^{14}+5q^{12}-2q^{8}+q^{6}+2q^{4}-1+q^{-4}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{62}-2q^{60}-q^{58}+5q^{56}-3q^{54}-3q^{52}+8q^{50}-3q^{48}-6q^{46}+6q^{44}-2q^{42}-6q^{40}+4q^{38}+q^{36}-2q^{34}-q^{32}+3q^{30}+3q^{28}-6q^{26}+3q^{24}+5q^{22}-7q^{20}+5q^{16}-5q^{14}+2q^{12}+5q^{10}-2q^{8}+2q^{6}+2q^{4}-q^{2}+1$
1,0,0	$-q^{37}-q^{33}+q^{31}-q^{29}+2q^{27}-q^{25}+q^{23}-q^{15}+q^{13}-q^{11}+2q^{9}+2q^{5}+q$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{80}-2q^{76}-q^{74}+2q^{72}+2q^{70}-2q^{68}+5q^{64}-6q^{60}+2q^{56}-6q^{54}-3q^{52}+3q^{50}+q^{48}-4q^{46}+3q^{44}+5q^{42}-4q^{40}-q^{38}+7q^{36}-7q^{32}+4q^{30}+3q^{28}-4q^{26}-3q^{24}+3q^{22}+q^{20}-2q^{18}+2q^{16}+4q^{14}+q^{12}+q^{10}+3q^{8}+q^{6}+q^{2}$
1,0,0,0	$-q^{46}-q^{42}-q^{40}+q^{38}-q^{36}+2q^{34}+q^{30}+q^{28}-q^{22}-2q^{18}+q^{16}-q^{14}+2q^{12}+q^{10}+q^{8}+2q^{6}+q^{2}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{62}+2q^{60}-3q^{58}+5q^{56}-7q^{54}+7q^{52}-8q^{50}+7q^{48}-6q^{46}+4q^{44}-4q^{40}+8q^{38}-11q^{36}+14q^{34}-15q^{32}+15q^{30}-13q^{28}+10q^{26}-7q^{24}+3q^{22}+q^{20}-4q^{18}+7q^{16}-7q^{14}+8q^{12}-7q^{10}+6q^{8}-4q^{6}+4q^{4}-q^{2}+1$
1,0	$q^{100}-2q^{96}-2q^{94}+q^{92}+5q^{90}+2q^{88}-5q^{86}-5q^{84}+2q^{82}+8q^{80}+2q^{78}-7q^{76}-6q^{74}+3q^{72}+7q^{70}-q^{68}-7q^{66}-2q^{64}+6q^{62}+4q^{60}-5q^{58}-5q^{56}+3q^{54}+5q^{52}-2q^{50}-5q^{48}+q^{46}+5q^{44}-6q^{40}-q^{38}+7q^{36}+5q^{34}-5q^{32}-8q^{30}+q^{28}+9q^{26}+3q^{24}-6q^{22}-6q^{20}+4q^{18}+7q^{16}+q^{14}-4q^{12}-2q^{10}+3q^{8}+3q^{6}-q^{4}-q^{2}+q^{-2}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{86}-2q^{84}+q^{82}-2q^{80}+5q^{78}-5q^{76}+4q^{74}-5q^{72}+8q^{70}-5q^{68}+4q^{66}-6q^{64}+3q^{62}-2q^{60}-2q^{58}-6q^{54}+7q^{52}-8q^{50}+11q^{48}-11q^{46}+13q^{44}-9q^{42}+12q^{40}-9q^{38}+8q^{36}-5q^{34}+3q^{32}-3q^{30}-2q^{28}+2q^{26}-5q^{24}+4q^{22}-6q^{20}+8q^{18}-4q^{16}+7q^{14}-3q^{12}+6q^{10}-2q^{8}+3q^{6}-q^{4}+q^{2}$

G2 Invariants.

Weight

Invariant

1,0

$q^{148}-2q^{146}+3q^{144}-4q^{142}+2q^{140}-q^{138}-2q^{136}+9q^{134}-12q^{132}+15q^{130}-13q^{128}+5q^{126}+2q^{124}-13q^{122}+23q^{120}-27q^{118}+23q^{116}-13q^{114}-q^{112}+15q^{110}-23q^{108}+25q^{106}-21q^{104}+5q^{102}+7q^{100}-17q^{98}+16q^{96}-5q^{94}-9q^{92}+23q^{90}-24q^{88}+14q^{86}+3q^{84}-26q^{82}+44q^{80}-44q^{78}+30q^{76}-4q^{74}-21q^{72}+43q^{70}-45q^{68}+33q^{66}-17q^{64}-6q^{62}+22q^{60}-28q^{58}+22q^{56}-6q^{54}-9q^{52}+19q^{50}-19q^{48}+7q^{46}+8q^{44}-23q^{42}+31q^{40}-28q^{38}+12q^{36}+10q^{34}-26q^{32}+36q^{30}-30q^{28}+17q^{26}-q^{24}-13q^{22}+21q^{20}-19q^{18}+14q^{16}-3q^{14}-2q^{12}+5q^{10}-5q^{8}+4q^{6}-q^{4}+q^{2}$

.

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 20"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$1-2q^{-1}+4q^{-2}-5q^{-3}+7q^{-4}-7q^{-5}+6q^{-6}-5q^{-7}+3q^{-8}-q^{-9}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{3}a^{11}+3z^{4}a^{10}-z^{2}a^{10}+5z^{5}a^{9}-5z^{3}a^{9}+2za^{9}+5z^{6}a^{8}-6z^{4}a^{8}+3z^{2}a^{8}-a^{8}+3z^{7}a^{7}-7z^{3}a^{7}+2za^{7}+z^{8}a^{6}+5z^{6}a^{6}-16z^{4}a^{6}+10z^{2}a^{6}-2a^{6}+5z^{7}a^{5}-12z^{5}a^{5}+5z^{3}a^{5}+z^{8}a^{4}+z^{6}a^{4}-11z^{4}a^{4}+11z^{2}a^{4}-2a^{4}+2z^{7}a^{3}-7z^{5}a^{3}+6z^{3}a^{3}+z^{6}a^{2}-4z^{4}a^{2}+5z^{2}a^{2}-2a^{2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_149, K11n26,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n90,}

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 20"];

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]=

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

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]=

{10_149, 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]=

{K11n90,}

Vassiliev invariants

V2 and V3:

(2, -4)

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 $8$ $-32$ $32$ ${\frac {412}{3}}$ ${\frac {68}{3}}$ $-256$ $-{\frac {1856}{3}}$ $-{\frac {224}{3}}$ $-128$ ${\frac {256}{3}}$ $512$ ${\frac {3296}{3}}$ ${\frac {544}{3}}$ ${\frac {42751}{15}}$ $-{\frac {4124}{15}}$ ${\frac {64924}{45}}$ ${\frac {833}{9}}$ ${\frac {2911}{15}}$

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 $t^{r}q^{j}$ 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 20. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                   1 1 -1                 1   -1 -3               3 1   2 -5             3 2     -1 -7           4 2       2 -9         3 3         0 -11       3 4           -1 -13     2 3             1 -15   1 3               -2 -17   2                 2 -19 1                   -1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{2}-2q-1+7q^{-1}-5q^{-2}-8q^{-3}+18q^{-4}-5q^{-5}-21q^{-6}+29q^{-7}-36q^{-9}+35q^{-10}+7q^{-11}-45q^{-12}+33q^{-13}+14q^{-14}-43q^{-15}+25q^{-16}+14q^{-17}-30q^{-18}+15q^{-19}+7q^{-20}-14q^{-21}+6q^{-22}+2q^{-23}-3q^{-24}+q^{-25}$
3	$q^{6}-2q^{5}-q^{4}+2q^{3}+6q^{2}-4q-11+q^{-1}+21q^{-2}+3q^{-3}-27q^{-4}-17q^{-5}+38q^{-6}+29q^{-7}-37q^{-8}-50q^{-9}+39q^{-10}+65q^{-11}-28q^{-12}-87q^{-13}+23q^{-14}+96q^{-15}-4q^{-16}-113q^{-17}-6q^{-18}+114q^{-19}+28q^{-20}-123q^{-21}-41q^{-22}+120q^{-23}+57q^{-24}-115q^{-25}-67q^{-26}+105q^{-27}+71q^{-28}-90q^{-29}-70q^{-30}+76q^{-31}+58q^{-32}-57q^{-33}-47q^{-34}+44q^{-35}+32q^{-36}-31q^{-37}-20q^{-38}+21q^{-39}+12q^{-40}-15q^{-41}-5q^{-42}+8q^{-43}+2q^{-44}-3q^{-45}-2q^{-46}+3q^{-47}-q^{-48}$
4	$q^{12}-2q^{11}-q^{10}+2q^{9}+q^{8}+7q^{7}-8q^{6}-9q^{5}+2q^{3}+33q^{2}-7q-25-22q^{-1}-19q^{-2}+77q^{-3}+24q^{-4}-16q^{-5}-59q^{-6}-94q^{-7}+101q^{-8}+74q^{-9}+51q^{-10}-62q^{-11}-204q^{-12}+65q^{-13}+87q^{-14}+160q^{-15}+6q^{-16}-292q^{-17}-13q^{-18}+27q^{-19}+252q^{-20}+124q^{-21}-314q^{-22}-86q^{-23}-90q^{-24}+296q^{-25}+252q^{-26}-280q^{-27}-134q^{-28}-226q^{-29}+298q^{-30}+366q^{-31}-212q^{-32}-166q^{-33}-354q^{-34}+273q^{-35}+454q^{-36}-122q^{-37}-178q^{-38}-455q^{-39}+214q^{-40}+490q^{-41}-21q^{-42}-150q^{-43}-494q^{-44}+130q^{-45}+439q^{-46}+47q^{-47}-74q^{-48}-431q^{-49}+49q^{-50}+312q^{-51}+52q^{-52}+3q^{-53}-294q^{-54}+13q^{-55}+175q^{-56}+10q^{-57}+36q^{-58}-155q^{-59}+13q^{-60}+81q^{-61}-21q^{-62}+29q^{-63}-65q^{-64}+16q^{-65}+32q^{-66}-22q^{-67}+14q^{-68}-22q^{-69}+9q^{-70}+11q^{-71}-10q^{-72}+4q^{-73}-5q^{-74}+3q^{-75}+2q^{-76}-3q^{-77}+q^{-78}$
5	$q^{20}-2q^{19}-q^{18}+2q^{17}+q^{16}+2q^{15}+3q^{14}-6q^{13}-11q^{12}+6q^{10}+13q^{9}+20q^{8}-3q^{7}-32q^{6}-33q^{5}-10q^{4}+26q^{3}+67q^{2}+49q-24-85q^{-1}-98q^{-2}-28q^{-3}+99q^{-4}+158q^{-5}+94q^{-6}-58q^{-7}-204q^{-8}-206q^{-9}-4q^{-10}+211q^{-11}+289q^{-12}+148q^{-13}-163q^{-14}-384q^{-15}-277q^{-16}+55q^{-17}+380q^{-18}+444q^{-19}+118q^{-20}-359q^{-21}-536q^{-22}-307q^{-23}+211q^{-24}+614q^{-25}+506q^{-26}-55q^{-27}-579q^{-28}-676q^{-29}-189q^{-30}+526q^{-31}+808q^{-32}+387q^{-33}-365q^{-34}-887q^{-35}-649q^{-36}+234q^{-37}+934q^{-38}+816q^{-39}-22q^{-40}-935q^{-41}-1043q^{-42}-131q^{-43}+939q^{-44}+1175q^{-45}+330q^{-46}-908q^{-47}-1361q^{-48}-486q^{-49}+894q^{-50}+1482q^{-51}+667q^{-52}-847q^{-53}-1623q^{-54}-834q^{-55}+794q^{-56}+1711q^{-57}+1004q^{-58}-698q^{-59}-1762q^{-60}-1149q^{-61}+556q^{-62}+1745q^{-63}+1264q^{-64}-392q^{-65}-1634q^{-66}-1326q^{-67}+203q^{-68}+1463q^{-69}+1298q^{-70}-27q^{-71}-1211q^{-72}-1212q^{-73}-113q^{-74}+959q^{-75}+1031q^{-76}+193q^{-77}-682q^{-78}-832q^{-79}-226q^{-80}+470q^{-81}+613q^{-82}+197q^{-83}-290q^{-84}-414q^{-85}-156q^{-86}+169q^{-87}+262q^{-88}+103q^{-89}-101q^{-90}-144q^{-91}-53q^{-92}+47q^{-93}+75q^{-94}+29q^{-95}-31q^{-96}-35q^{-97}-4q^{-98}+16q^{-99}+10q^{-100}-2q^{-101}-3q^{-102}-9q^{-103}+3q^{-104}+11q^{-105}-5q^{-106}-5q^{-107}+4q^{-108}-2q^{-109}-q^{-110}+5q^{-111}-3q^{-112}-2q^{-113}+3q^{-114}-q^{-115}$
6	$q^{30}-2q^{29}-q^{28}+2q^{27}+q^{26}+2q^{25}-2q^{24}+5q^{23}-8q^{22}-11q^{21}+3q^{20}+5q^{19}+14q^{18}+3q^{17}+25q^{16}-17q^{15}-41q^{14}-24q^{13}-12q^{12}+27q^{11}+23q^{10}+109q^{9}+22q^{8}-57q^{7}-90q^{6}-109q^{5}-49q^{4}-22q^{3}+243q^{2}+184q+88-62q^{-1}-217q^{-2}-287q^{-3}-325q^{-4}+204q^{-5}+320q^{-6}+435q^{-7}+283q^{-8}-9q^{-9}-430q^{-10}-849q^{-11}-249q^{-12}+18q^{-13}+593q^{-14}+793q^{-15}+731q^{-16}+5q^{-17}-1053q^{-18}-838q^{-19}-865q^{-20}+33q^{-21}+803q^{-22}+1568q^{-23}+1062q^{-24}-397q^{-25}-804q^{-26}-1744q^{-27}-1183q^{-28}-186q^{-29}+1676q^{-30}+2041q^{-31}+949q^{-32}+289q^{-33}-1778q^{-34}-2289q^{-35}-1890q^{-36}+688q^{-37}+2133q^{-38}+2187q^{-39}+2082q^{-40}-690q^{-41}-2542q^{-42}-3509q^{-43}-996q^{-44}+1158q^{-45}+2661q^{-46}+3815q^{-47}+1083q^{-48}-1848q^{-49}-4466q^{-50}-2685q^{-51}-431q^{-52}+2345q^{-53}+5003q^{-54}+2900q^{-55}-651q^{-56}-4768q^{-57}-3983q^{-58}-2053q^{-59}+1658q^{-60}+5676q^{-61}+4418q^{-62}+550q^{-63}-4773q^{-64}-4945q^{-65}-3433q^{-66}+1016q^{-67}+6136q^{-68}+5677q^{-69}+1565q^{-70}-4761q^{-71}-5802q^{-72}-4645q^{-73}+475q^{-74}+6536q^{-75}+6838q^{-76}+2552q^{-77}-4613q^{-78}-6563q^{-79}-5846q^{-80}-262q^{-81}+6595q^{-82}+7795q^{-83}+3705q^{-84}-3898q^{-85}-6802q^{-86}-6858q^{-87}-1426q^{-88}+5798q^{-89}+8017q^{-90}+4789q^{-91}-2421q^{-92}-5975q^{-93}-7076q^{-94}-2684q^{-95}+4049q^{-96}+6995q^{-97}+5132q^{-98}-696q^{-99}-4123q^{-100}-6036q^{-101}-3274q^{-102}+2024q^{-103}+4932q^{-104}+4330q^{-105}+434q^{-106}-2061q^{-107}-4086q^{-108}-2838q^{-109}+604q^{-110}+2741q^{-111}+2797q^{-112}+655q^{-113}-632q^{-114}-2152q^{-115}-1811q^{-116}+37q^{-117}+1200q^{-118}+1372q^{-119}+365q^{-120}-12q^{-121}-878q^{-122}-878q^{-123}-27q^{-124}+421q^{-125}+509q^{-126}+74q^{-127}+118q^{-128}-275q^{-129}-337q^{-130}+20q^{-131}+123q^{-132}+139q^{-133}-41q^{-134}+86q^{-135}-63q^{-136}-108q^{-137}+32q^{-138}+32q^{-139}+25q^{-140}-45q^{-141}+42q^{-142}-9q^{-143}-30q^{-144}+19q^{-145}+5q^{-146}+4q^{-147}-22q^{-148}+16q^{-149}+q^{-150}-10q^{-151}+8q^{-152}-q^{-153}+q^{-154}-5q^{-155}+3q^{-156}+2q^{-157}-3q^{-158}+q^{-159}$
7	$q^{42}-2q^{41}-q^{40}+2q^{39}+q^{38}+2q^{37}-2q^{36}+3q^{34}-8q^{33}-8q^{32}+2q^{31}+5q^{30}+16q^{29}+5q^{28}+q^{27}+14q^{26}-23q^{25}-36q^{24}-27q^{23}-14q^{22}+40q^{21}+41q^{20}+42q^{19}+78q^{18}-69q^{16}-112q^{15}-153q^{14}-32q^{13}+31q^{12}+104q^{11}+267q^{10}+202q^{9}+88q^{8}-98q^{7}-371q^{6}-348q^{5}-296q^{4}-127q^{3}+347q^{2}+536q+633+471q^{-1}-158q^{-2}-528q^{-3}-909q^{-4}-1003q^{-5}-343q^{-6}+262q^{-7}+1027q^{-8}+1527q^{-9}+1042q^{-10}+407q^{-11}-740q^{-12}-1855q^{-13}-1795q^{-14}-1400q^{-15}-67q^{-16}+1661q^{-17}+2346q^{-18}+2576q^{-19}+1350q^{-20}-880q^{-21}-2302q^{-22}-3469q^{-23}-2945q^{-24}-678q^{-25}+1477q^{-26}+3901q^{-27}+4430q^{-28}+2558q^{-29}+225q^{-30}-3270q^{-31}-5354q^{-32}-4701q^{-33}-2643q^{-34}+1750q^{-35}+5395q^{-36}+6267q^{-37}+5374q^{-38}+877q^{-39}-4237q^{-40}-7151q^{-41}-8021q^{-42}-4017q^{-43}+1983q^{-44}+6786q^{-45}+10027q^{-46}+7477q^{-47}+1237q^{-48}-5317q^{-49}-11107q^{-50}-10603q^{-51}-4985q^{-52}+2639q^{-53}+11066q^{-54}+13216q^{-55}+8885q^{-56}+726q^{-57}-9928q^{-58}-14829q^{-59}-12558q^{-60}-4718q^{-61}+7866q^{-62}+15709q^{-63}+15733q^{-64}+8624q^{-65}-5173q^{-66}-15529q^{-67}-18237q^{-68}-12589q^{-69}+2095q^{-70}+14934q^{-71}+20144q^{-72}+15948q^{-73}+969q^{-74}-13680q^{-75}-21411q^{-76}-19067q^{-77}-3977q^{-78}+12473q^{-79}+22377q^{-80}+21542q^{-81}+6560q^{-82}-11132q^{-83}-23000q^{-84}-23780q^{-85}-8926q^{-86}+10107q^{-87}+23703q^{-88}+25666q^{-89}+10835q^{-90}-9234q^{-91}-24359q^{-92}-27500q^{-93}-12663q^{-94}+8611q^{-95}+25234q^{-96}+29289q^{-97}+14367q^{-98}-8002q^{-99}-26076q^{-100}-31165q^{-101}-16233q^{-102}+7241q^{-103}+26777q^{-104}+33006q^{-105}+18333q^{-106}-6020q^{-107}-27075q^{-108}-34654q^{-109}-20609q^{-110}+4190q^{-111}+26566q^{-112}+35774q^{-113}+23007q^{-114}-1698q^{-115}-25097q^{-116}-36023q^{-117}-25085q^{-118}-1340q^{-119}+22486q^{-120}+35116q^{-121}+26516q^{-122}+4549q^{-123}-18913q^{-124}-32827q^{-125}-26913q^{-126}-7552q^{-127}+14704q^{-128}+29362q^{-129}+26036q^{-130}+9756q^{-131}-10333q^{-132}-24876q^{-133}-23885q^{-134}-11045q^{-135}+6306q^{-136}+20052q^{-137}+20748q^{-138}+11104q^{-139}-3086q^{-140}-15200q^{-141}-16953q^{-142}-10261q^{-143}+730q^{-144}+10941q^{-145}+13142q^{-146}+8685q^{-147}+584q^{-148}-7446q^{-149}-9521q^{-150}-6816q^{-151}-1223q^{-152}+4803q^{-153}+6555q^{-154}+5001q^{-155}+1293q^{-156}-3000q^{-157}-4253q^{-158}-3373q^{-159}-1084q^{-160}+1769q^{-161}+2570q^{-162}+2169q^{-163}+843q^{-164}-1064q^{-165}-1513q^{-166}-1265q^{-167}-526q^{-168}+610q^{-169}+775q^{-170}+701q^{-171}+358q^{-172}-354q^{-173}-407q^{-174}-363q^{-175}-191q^{-176}+225q^{-177}+170q^{-178}+147q^{-179}+109q^{-180}-110q^{-181}-56q^{-182}-82q^{-183}-68q^{-184}+92q^{-185}+22q^{-186}+2q^{-187}+24q^{-188}-29q^{-189}+17q^{-190}-17q^{-191}-29q^{-192}+38q^{-193}-q^{-194}-12q^{-195}+3q^{-196}-6q^{-197}+15q^{-198}-5q^{-199}-12q^{-200}+14q^{-201}-2q^{-202}-5q^{-203}+q^{-204}-q^{-205}+5q^{-206}-3q^{-207}-2q^{-208}+3q^{-209}-q^{-210}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

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