9 19

9_18

9_20

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 19's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 19 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X5,10,6,11 X3948 X9,3,10,2 X13,16,14,17 X7,15,8,14 X15,7,16,6 X11,18,12,1 X17,12,18,13
Gauss code	-1, 4, -3, 1, -2, 7, -6, 3, -4, 2, -8, 9, -5, 6, -7, 5, -9, 8
Dowker-Thistlethwaite code	4 8 10 14 2 18 16 6 12
Conway Notation	[23112]

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

[edit Notes on presentations of 9 19]

Computer Talk[show]

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 19"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X5,10,6,11 X3948 X9,3,10,2 X13,16,14,17 X7,15,8,14 X15,7,16,6 X11,18,12,1 X17,12,18,13

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 10 14 2 18 16 6 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[23112]

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]=

${\textrm {BR}}(5,\{1,-2,1,-2,-2,-3,2,4,-3,4\})$

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]=

{ 5, 10, 5 }

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-5] Hyperbolic Volume 10.0325 A-Polynomial See Data:9 19/A-polynomial

[edit Notes for 9 19's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for 9 19's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{2}-10t+17-10t^{-1}+2t^{-2}$
Conway polynomial	$2z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 41, 0 }
Jones polynomial	$q^{4}-2q^{3}+4q^{2}-6q+7-7q^{-1}+6q^{-2}-4q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{4}+z^{4}a^{2}+z^{2}a^{2}+a^{2}+z^{4}-2z^{2}a^{-2}-a^{-2}+a^{-4}$
Kauffman polynomial (db, data sources)	$a^{2}z^{8}+z^{8}+3a^{3}z^{7}+5az^{7}+2z^{7}a^{-1}+3a^{4}z^{6}+3a^{2}z^{6}+2z^{6}a^{-2}+2z^{6}+a^{5}z^{5}-7a^{3}z^{5}-11az^{5}-z^{5}a^{-1}+2z^{5}a^{-3}-8a^{4}z^{4}-11a^{2}z^{4}+z^{4}a^{-4}-4z^{4}-2a^{5}z^{3}+4a^{3}z^{3}+10az^{3}+z^{3}a^{-1}-3z^{3}a^{-3}+4a^{4}z^{2}+8a^{2}z^{2}-3z^{2}a^{-2}-2z^{2}a^{-4}+3z^{2}-a^{3}z-3az-za^{-1}+za^{-3}-a^{2}+a^{-2}+a^{-4}$
The A2 invariant	$-q^{16}+q^{14}+q^{12}-q^{10}+2q^{8}+q^{2}-1+q^{-2}-2q^{-4}+q^{-8}-q^{-10}+q^{-12}+q^{-14}$
The G2 invariant	$q^{80}-2q^{78}+4q^{76}-7q^{74}+5q^{72}-3q^{70}-5q^{68}+16q^{66}-21q^{64}+24q^{62}-18q^{60}+2q^{58}+18q^{56}-34q^{54}+40q^{52}-31q^{50}+13q^{48}+11q^{46}-28q^{44}+33q^{42}-24q^{40}+8q^{38}+10q^{36}-23q^{34}+20q^{32}-6q^{30}-13q^{28}+30q^{26}-34q^{24}+27q^{22}-6q^{20}-20q^{18}+41q^{16}-51q^{14}+48q^{12}-25q^{10}-3q^{8}+30q^{6}-44q^{4}+44q^{2}-27+4q^{-2}+14q^{-4}-24q^{-6}+19q^{-8}-4q^{-10}-13q^{-12}+23q^{-14}-22q^{-16}+7q^{-18}+9q^{-20}-26q^{-22}+33q^{-24}-29q^{-26}+17q^{-28}-2q^{-30}-14q^{-32}+22q^{-34}-24q^{-36}+21q^{-38}-12q^{-40}+4q^{-42}+4q^{-44}-9q^{-46}+11q^{-48}-9q^{-50}+8q^{-52}-3q^{-54}+2q^{-58}-3q^{-60}+3q^{-62}-q^{-64}+q^{-66}$

Further Quantum Invariants[show]

Further quantum knot invariants for 9_19.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+2q^{9}-q^{7}+2q^{5}-q^{3}+q^{-1}-2q^{-3}+2q^{-5}-q^{-7}+q^{-9}$
2	$q^{32}-2q^{30}-2q^{28}+6q^{26}-q^{24}-7q^{22}+7q^{20}+3q^{18}-10q^{16}+5q^{14}+6q^{12}-8q^{10}+q^{8}+5q^{6}-2q^{4}-4q^{2}+2+7q^{-2}-7q^{-4}-3q^{-6}+11q^{-8}-5q^{-10}-5q^{-12}+8q^{-14}-2q^{-16}-3q^{-18}+3q^{-20}-q^{-22}-q^{-24}+q^{-26}$
3	$-q^{63}+2q^{61}+2q^{59}-3q^{57}-6q^{55}+q^{53}+13q^{51}+2q^{49}-16q^{47}-10q^{45}+17q^{43}+20q^{41}-13q^{39}-30q^{37}+6q^{35}+33q^{33}+5q^{31}-35q^{29}-13q^{27}+34q^{25}+20q^{23}-27q^{21}-24q^{19}+21q^{17}+23q^{15}-13q^{13}-24q^{11}+5q^{9}+21q^{7}+6q^{5}-17q^{3}-17q+13q^{-1}+29q^{-3}-5q^{-5}-34q^{-7}-4q^{-9}+37q^{-11}+13q^{-13}-35q^{-15}-17q^{-17}+25q^{-19}+19q^{-21}-17q^{-23}-16q^{-25}+10q^{-27}+11q^{-29}-5q^{-31}-6q^{-33}+3q^{-35}+3q^{-37}-2q^{-39}-q^{-41}+2q^{-43}-q^{-47}-q^{-49}+q^{-51}$
4	$q^{104}-2q^{102}-2q^{100}+3q^{98}+3q^{96}+6q^{94}-8q^{92}-13q^{90}-q^{88}+7q^{86}+31q^{84}+q^{82}-30q^{80}-28q^{78}-13q^{76}+57q^{74}+44q^{72}-7q^{70}-56q^{68}-81q^{66}+31q^{64}+85q^{62}+70q^{60}-25q^{58}-138q^{56}-53q^{54}+60q^{52}+142q^{50}+61q^{48}-130q^{46}-126q^{44}-14q^{42}+147q^{40}+128q^{38}-73q^{36}-141q^{34}-73q^{32}+105q^{30}+139q^{28}-17q^{26}-110q^{24}-87q^{22}+54q^{20}+112q^{18}+27q^{16}-69q^{14}-85q^{12}+q^{10}+74q^{8}+76q^{6}-15q^{4}-84q^{2}-75+20q^{-2}+131q^{-4}+61q^{-6}-60q^{-8}-145q^{-10}-61q^{-12}+138q^{-14}+132q^{-16}+10q^{-18}-155q^{-20}-132q^{-22}+79q^{-24}+134q^{-26}+76q^{-28}-88q^{-30}-131q^{-32}+6q^{-34}+69q^{-36}+82q^{-38}-16q^{-40}-73q^{-42}-17q^{-44}+9q^{-46}+43q^{-48}+9q^{-50}-23q^{-52}-6q^{-54}-9q^{-56}+12q^{-58}+5q^{-60}-4q^{-62}+4q^{-64}-6q^{-66}+q^{-68}-q^{-72}+4q^{-74}-q^{-76}-q^{-80}-q^{-82}+q^{-84}$
5	$-q^{155}+2q^{153}+2q^{151}-3q^{149}-3q^{147}-3q^{145}+q^{143}+8q^{141}+13q^{139}+q^{137}-17q^{135}-22q^{133}-13q^{131}+13q^{129}+40q^{127}+44q^{125}-3q^{123}-57q^{121}-72q^{119}-39q^{117}+43q^{115}+114q^{113}+106q^{111}-123q^{107}-173q^{105}-104q^{103}+78q^{101}+233q^{99}+226q^{97}+30q^{95}-227q^{93}-348q^{91}-200q^{89}+147q^{87}+423q^{85}+394q^{83}+14q^{81}-429q^{79}-553q^{77}-225q^{75}+338q^{73}+664q^{71}+439q^{69}-191q^{67}-687q^{65}-607q^{63}+4q^{61}+633q^{59}+715q^{57}+168q^{55}-531q^{53}-746q^{51}-292q^{49}+397q^{47}+709q^{45}+376q^{43}-277q^{41}-635q^{39}-394q^{37}+172q^{35}+533q^{33}+392q^{31}-91q^{29}-442q^{27}-362q^{25}+22q^{23}+351q^{21}+348q^{19}+48q^{17}-267q^{15}-349q^{13}-139q^{11}+184q^{9}+362q^{7}+257q^{5}-75q^{3}-379q-403q^{-1}-65q^{-3}+381q^{-5}+543q^{-7}+243q^{-9}-324q^{-11}-667q^{-13}-442q^{-15}+217q^{-17}+733q^{-19}+619q^{-21}-50q^{-23}-705q^{-25}-752q^{-27}-143q^{-29}+598q^{-31}+799q^{-33}+301q^{-35}-412q^{-37}-740q^{-39}-423q^{-41}+218q^{-43}+610q^{-45}+453q^{-47}-48q^{-49}-430q^{-51}-413q^{-53}-72q^{-55}+261q^{-57}+325q^{-59}+124q^{-61}-130q^{-63}-221q^{-65}-125q^{-67}+41q^{-69}+133q^{-71}+103q^{-73}+2q^{-75}-71q^{-77}-68q^{-79}-17q^{-81}+30q^{-83}+40q^{-85}+20q^{-87}-9q^{-89}-23q^{-91}-14q^{-93}+q^{-95}+7q^{-97}+9q^{-99}+6q^{-101}-4q^{-103}-6q^{-105}-q^{-107}-2q^{-109}+q^{-111}+4q^{-113}+q^{-115}-q^{-117}-q^{-121}-q^{-123}+q^{-125}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}+q^{14}+q^{12}-q^{10}+2q^{8}+q^{2}-1+q^{-2}-2q^{-4}+q^{-8}-q^{-10}+q^{-12}+q^{-14}$
1,1	$q^{44}-4q^{42}+10q^{40}-22q^{38}+40q^{36}-62q^{34}+86q^{32}-112q^{30}+129q^{28}-130q^{26}+120q^{24}-90q^{22}+48q^{20}+8q^{18}-70q^{16}+128q^{14}-177q^{12}+216q^{10}-234q^{8}+234q^{6}-214q^{4}+172q^{2}-122+64q^{-2}-9q^{-4}-38q^{-6}+78q^{-8}-94q^{-10}+104q^{-12}-100q^{-14}+92q^{-16}-78q^{-18}+62q^{-20}-50q^{-22}+36q^{-24}-26q^{-26}+17q^{-28}-10q^{-30}+6q^{-32}-2q^{-34}+q^{-36}$
2,0	$q^{42}-q^{40}-2q^{38}+3q^{34}+2q^{32}-5q^{30}-q^{28}+5q^{26}+3q^{24}-5q^{22}-3q^{20}+6q^{18}+3q^{16}-5q^{14}-q^{12}+5q^{10}-q^{8}-2q^{6}+q^{4}-q^{2}-1+2q^{-2}+2q^{-4}-5q^{-6}-2q^{-8}+8q^{-10}+2q^{-12}-6q^{-14}+q^{-16}+7q^{-18}+q^{-20}-5q^{-22}-2q^{-24}+2q^{-26}-2q^{-30}+q^{-34}+q^{-36}$

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^{34}-2 q^{32}+2 q^{28}-5 q^{26}+4 q^{24}+4 q^{22}-6 q^{20}+5 q^{18}+5 q^{16}-8 q^{14}+q^{12}+4 q^{10}-5 q^8-q^6+3 q^4+3 q^2-1- q^{-2} +7 q^{-4} -4 q^{-6} -6 q^{-8} +8 q^{-10} -3 q^{-12} -6 q^{-14} +6 q^{-16} -3 q^{-20} +3 q^{-22} + q^{-24} - q^{-26} + q^{-28} }
1,0,0	$-q^{21}+q^{19}+q^{15}-q^{13}+2q^{11}+q^{7}+q^{3}-q^{-1}+q^{-3}-2q^{-5}-q^{-9}+q^{-11}-q^{-13}+q^{-15}+q^{-17}+q^{-19}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+2q^{32}-4q^{30}+6q^{28}-7q^{26}+8q^{24}-8q^{22}+8q^{20}-5q^{18}+3q^{16}+2q^{14}-5q^{12}+10q^{10}-13q^{8}+15q^{6}-15q^{4}+15q^{2}-13+9q^{-2}-5q^{-4}+2q^{-8}-6q^{-10}+7q^{-12}-8q^{-14}+8q^{-16}-6q^{-18}+5q^{-20}-3q^{-22}+3q^{-24}-q^{-26}+q^{-28}$
1,0	$q^{56}-2q^{52}-2q^{50}+2q^{48}+4q^{46}-2q^{44}-6q^{42}-q^{40}+8q^{38}+6q^{36}-5q^{34}-8q^{32}+2q^{30}+9q^{28}+4q^{26}-7q^{24}-6q^{22}+3q^{20}+6q^{18}-2q^{16}-6q^{14}+5q^{10}+q^{8}-5q^{6}-2q^{4}+6q^{2}+5-4q^{-2}-5q^{-4}+4q^{-6}+7q^{-8}-2q^{-10}-8q^{-12}-2q^{-14}+8q^{-16}+5q^{-18}-5q^{-20}-8q^{-22}+7q^{-26}+3q^{-28}-3q^{-30}-4q^{-32}+3q^{-36}+2q^{-38}-q^{-40}-q^{-42}+q^{-46}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-2q^{78}+4q^{76}-7q^{74}+5q^{72}-3q^{70}-5q^{68}+16q^{66}-21q^{64}+24q^{62}-18q^{60}+2q^{58}+18q^{56}-34q^{54}+40q^{52}-31q^{50}+13q^{48}+11q^{46}-28q^{44}+33q^{42}-24q^{40}+8q^{38}+10q^{36}-23q^{34}+20q^{32}-6q^{30}-13q^{28}+30q^{26}-34q^{24}+27q^{22}-6q^{20}-20q^{18}+41q^{16}-51q^{14}+48q^{12}-25q^{10}-3q^{8}+30q^{6}-44q^{4}+44q^{2}-27+4q^{-2}+14q^{-4}-24q^{-6}+19q^{-8}-4q^{-10}-13q^{-12}+23q^{-14}-22q^{-16}+7q^{-18}+9q^{-20}-26q^{-22}+33q^{-24}-29q^{-26}+17q^{-28}-2q^{-30}-14q^{-32}+22q^{-34}-24q^{-36}+21q^{-38}-12q^{-40}+4q^{-42}+4q^{-44}-9q^{-46}+11q^{-48}-9q^{-50}+8q^{-52}-3q^{-54}+2q^{-58}-3q^{-60}+3q^{-62}-q^{-64}+q^{-66}$

.

Computer Talk[show]

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 19"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{2}-10t+17-10t^{-1}+2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{2}a^{4}+z^{4}a^{2}+z^{2}a^{2}+a^{2}+z^{4}-2z^{2}a^{-2}-a^{-2}+a^{-4}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{8}+z^{8}+3a^{3}z^{7}+5az^{7}+2z^{7}a^{-1}+3a^{4}z^{6}+3a^{2}z^{6}+2z^{6}a^{-2}+2z^{6}+a^{5}z^{5}-7a^{3}z^{5}-11az^{5}-z^{5}a^{-1}+2z^{5}a^{-3}-8a^{4}z^{4}-11a^{2}z^{4}+z^{4}a^{-4}-4z^{4}-2a^{5}z^{3}+4a^{3}z^{3}+10az^{3}+z^{3}a^{-1}-3z^{3}a^{-3}+4a^{4}z^{2}+8a^{2}z^{2}-3z^{2}a^{-2}-2z^{2}a^{-4}+3z^{2}-a^{3}z-3az-za^{-1}+za^{-3}-a^{2}+a^{-2}+a^{-4}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Computer Talk[show]

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 19"];

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]=

{ $2t^{2}-10t+17-10t^{-1}+2t^{-2}$, $q^{4}-2q^{3}+4q^{2}-6q+7-7q^{-1}+6q^{-2}-4q^{-3}+3q^{-4}-q^{-5}$ }

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:

(-2, -1)

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 $-8$ $-8$ $32$ ${\frac {68}{3}}$ ${\frac {4}{3}}$ $64$ ${\frac {304}{3}}$ ${\frac {160}{3}}$ $-8$ $-{\frac {256}{3}}$ $32$ $-{\frac {544}{3}}$ $-{\frac {32}{3}}$ $-{\frac {991}{15}}$ $-{\frac {1676}{15}}$ ${\frac {5276}{45}}$ $-{\frac {113}{9}}$ ${\frac {449}{15}}$

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=$0 is the signature of 9 19. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                   1 1 7                 1   -1 5               3 1   2 3             3 1     -2 1           4 3       1 -1         4 4         0 -3       2 3           -1 -5     2 4             2 -7   1 2               -1 -9   2                 2 -11 1                   -1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{12}-2q^{11}+5q^{9}-8q^{8}+q^{7}+15q^{6}-21q^{5}+q^{4}+31q^{3}-35q^{2}-3q+45-40q^{-1}-9q^{-2}+47q^{-3}-33q^{-4}-13q^{-5}+38q^{-6}-19q^{-7}-14q^{-8}+23q^{-9}-6q^{-10}-10q^{-11}+9q^{-12}-3q^{-14}+q^{-15}$
3	$q^{24}-2q^{23}+q^{21}+3q^{20}-5q^{19}-q^{18}+6q^{17}+3q^{16}-14q^{15}+22q^{13}+2q^{12}-40q^{11}-q^{10}+58q^{9}+8q^{8}-82q^{7}-19q^{6}+106q^{5}+32q^{4}-123q^{3}-49q^{2}+135q+66-139q^{-1}-79q^{-2}+135q^{-3}+89q^{-4}-124q^{-5}-95q^{-6}+106q^{-7}+100q^{-8}-88q^{-9}-97q^{-10}+61q^{-11}+97q^{-12}-41q^{-13}-83q^{-14}+14q^{-15}+75q^{-16}-q^{-17}-55q^{-18}-13q^{-19}+39q^{-20}+16q^{-21}-22q^{-22}-16q^{-23}+12q^{-24}+10q^{-25}-4q^{-26}-5q^{-27}+3q^{-29}-q^{-30}$
4	$q^{40}-2q^{39}+q^{37}-q^{36}+6q^{35}-7q^{34}+q^{33}+2q^{32}-8q^{31}+16q^{30}-15q^{29}+10q^{28}+9q^{27}-29q^{26}+19q^{25}-32q^{24}+42q^{23}+43q^{22}-63q^{21}-7q^{20}-88q^{19}+99q^{18}+141q^{17}-76q^{16}-70q^{15}-225q^{14}+142q^{13}+305q^{12}-18q^{11}-125q^{10}-436q^{9}+119q^{8}+470q^{7}+104q^{6}-119q^{5}-635q^{4}+35q^{3}+555q^{2}+225q-49-746q^{-1}-60q^{-2}+546q^{-3}+294q^{-4}+42q^{-5}-748q^{-6}-133q^{-7}+460q^{-8}+310q^{-9}+138q^{-10}-663q^{-11}-191q^{-12}+319q^{-13}+287q^{-14}+231q^{-15}-507q^{-16}-225q^{-17}+141q^{-18}+219q^{-19}+299q^{-20}-306q^{-21}-206q^{-22}-20q^{-23}+107q^{-24}+295q^{-25}-115q^{-26}-125q^{-27}-102q^{-28}-6q^{-29}+210q^{-30}-2q^{-31}-30q^{-32}-87q^{-33}-60q^{-34}+98q^{-35}+23q^{-36}+19q^{-37}-36q^{-38}-47q^{-39}+28q^{-40}+8q^{-41}+17q^{-42}-5q^{-43}-17q^{-44}+4q^{-45}+5q^{-47}-3q^{-49}+q^{-50}$
5	$q^{60}-2q^{59}+q^{57}-q^{56}+2q^{55}+4q^{54}-5q^{53}-3q^{52}+2q^{51}-6q^{50}+4q^{49}+14q^{48}-2q^{47}-5q^{46}-4q^{45}-21q^{44}-5q^{43}+28q^{42}+27q^{41}+15q^{40}-14q^{39}-68q^{38}-56q^{37}+25q^{36}+100q^{35}+116q^{34}+16q^{33}-160q^{32}-222q^{31}-71q^{30}+191q^{29}+370q^{28}+217q^{27}-224q^{26}-555q^{25}-412q^{24}+174q^{23}+752q^{22}+718q^{21}-67q^{20}-947q^{19}-1053q^{18}-143q^{17}+1080q^{16}+1431q^{15}+431q^{14}-1148q^{13}-1794q^{12}-752q^{11}+1127q^{10}+2086q^{9}+1100q^{8}-1034q^{7}-2310q^{6}-1411q^{5}+902q^{4}+2429q^{3}+1667q^{2}-734q-2472-1857q^{-1}+564q^{-2}+2453q^{-3}+1971q^{-4}-402q^{-5}-2367q^{-6}-2035q^{-7}+241q^{-8}+2243q^{-9}+2053q^{-10}-87q^{-11}-2067q^{-12}-2032q^{-13}-88q^{-14}+1859q^{-15}+1973q^{-16}+264q^{-17}-1584q^{-18}-1891q^{-19}-449q^{-20}+1293q^{-21}+1732q^{-22}+622q^{-23}-931q^{-24}-1558q^{-25}-761q^{-26}+604q^{-27}+1278q^{-28}+837q^{-29}-232q^{-30}-1011q^{-31}-843q^{-32}-25q^{-33}+667q^{-34}+757q^{-35}+264q^{-36}-381q^{-37}-618q^{-38}-351q^{-39}+104q^{-40}+429q^{-41}+388q^{-42}+62q^{-43}-238q^{-44}-322q^{-45}-172q^{-46}+82q^{-47}+240q^{-48}+183q^{-49}+19q^{-50}-126q^{-51}-165q^{-52}-73q^{-53}+58q^{-54}+114q^{-55}+69q^{-56}-3q^{-57}-59q^{-58}-65q^{-59}-13q^{-60}+32q^{-61}+36q^{-62}+12q^{-63}-5q^{-64}-18q^{-65}-17q^{-66}+5q^{-67}+10q^{-68}+3q^{-69}-5q^{-72}+3q^{-74}-q^{-75}$
6	$q^{84}-2q^{83}+q^{81}-q^{80}+2q^{79}+6q^{77}-9q^{76}-3q^{75}+4q^{74}-7q^{73}+5q^{72}+5q^{71}+24q^{70}-21q^{69}-12q^{68}+5q^{67}-27q^{66}+q^{65}+20q^{64}+74q^{63}-27q^{62}-25q^{61}-7q^{60}-89q^{59}-31q^{58}+49q^{57}+196q^{56}+17q^{55}-23q^{54}-51q^{53}-269q^{52}-168q^{51}+67q^{50}+454q^{49}+246q^{48}+118q^{47}-103q^{46}-697q^{45}-637q^{44}-120q^{43}+844q^{42}+874q^{41}+752q^{40}+116q^{39}-1384q^{38}-1763q^{37}-1019q^{36}+999q^{35}+1922q^{34}+2314q^{33}+1243q^{32}-1859q^{31}-3543q^{30}-3125q^{29}+165q^{28}+2796q^{27}+4727q^{26}+3756q^{25}-1278q^{24}-5230q^{23}-6201q^{22}-2076q^{21}+2590q^{20}+7062q^{19}+7201q^{18}+693q^{17}-5842q^{16}-9126q^{15}-5123q^{14}+1061q^{13}+8292q^{12}+10321q^{11}+3355q^{10}-5164q^{9}-10853q^{8}-7759q^{7}-1079q^{6}+8234q^{5}+12163q^{4}+5616q^{3}-3831q^{2}-11252q-9251-2914q^{-1}+7437q^{-2}+12695q^{-3}+6961q^{-4}-2533q^{-5}-10799q^{-6}-9712q^{-7}-4130q^{-8}+6408q^{-9}+12358q^{-10}+7589q^{-11}-1401q^{-12}-9859q^{-13}-9555q^{-14}-4989q^{-15}+5152q^{-16}+11426q^{-17}+7868q^{-18}-143q^{-19}-8380q^{-20}-8972q^{-21}-5790q^{-22}+3411q^{-23}+9815q^{-24}+7871q^{-25}+1419q^{-26}-6164q^{-27}-7790q^{-28}-6448q^{-29}+1157q^{-30}+7355q^{-31}+7270q^{-32}+2976q^{-33}-3310q^{-34}-5756q^{-35}-6442q^{-36}-1100q^{-37}+4217q^{-38}+5691q^{-39}+3790q^{-40}-491q^{-41}-3019q^{-42}-5286q^{-43}-2470q^{-44}+1183q^{-45}+3279q^{-46}+3298q^{-47}+1297q^{-48}-405q^{-49}-3171q^{-50}-2394q^{-51}-712q^{-52}+933q^{-53}+1773q^{-54}+1543q^{-55}+1079q^{-56}-1078q^{-57}-1268q^{-58}-1073q^{-59}-366q^{-60}+262q^{-61}+767q^{-62}+1182q^{-63}+60q^{-64}-157q^{-65}-525q^{-66}-498q^{-67}-415q^{-68}+6q^{-69}+601q^{-70}+228q^{-71}+268q^{-72}-15q^{-73}-152q^{-74}-365q^{-75}-224q^{-76}+143q^{-77}+49q^{-78}+196q^{-79}+107q^{-80}+58q^{-81}-139q^{-82}-140q^{-83}+7q^{-84}-38q^{-85}+56q^{-86}+52q^{-87}+64q^{-88}-28q^{-89}-43q^{-90}-q^{-91}-26q^{-92}+6q^{-93}+9q^{-94}+26q^{-95}-5q^{-96}-10q^{-97}+4q^{-98}-7q^{-99}+5q^{-102}-3q^{-104}+q^{-105}$
7	$q^{112}-2q^{111}+q^{109}-q^{108}+2q^{107}+2q^{105}+2q^{104}-9q^{103}-q^{102}+3q^{101}-6q^{100}+7q^{99}+3q^{98}+11q^{97}+11q^{96}-28q^{95}-7q^{94}+2q^{93}-19q^{92}+12q^{91}+9q^{90}+38q^{89}+41q^{88}-56q^{87}-29q^{86}-16q^{85}-51q^{84}+25q^{83}+27q^{82}+96q^{81}+115q^{80}-91q^{79}-96q^{78}-111q^{77}-139q^{76}+62q^{75}+122q^{74}+276q^{73}+308q^{72}-112q^{71}-286q^{70}-463q^{69}-495q^{68}+34q^{67}+402q^{66}+869q^{65}+980q^{64}+139q^{63}-599q^{62}-1430q^{61}-1736q^{60}-661q^{59}+632q^{58}+2222q^{57}+2998q^{56}+1670q^{55}-351q^{54}-3062q^{53}-4786q^{52}-3517q^{51}-588q^{50}+3816q^{49}+7082q^{48}+6256q^{47}+2521q^{46}-3931q^{45}-9630q^{44}-10112q^{43}-5829q^{42}+3173q^{41}+12096q^{40}+14679q^{39}+10528q^{38}-900q^{37}-13795q^{36}-19766q^{35}-16639q^{34}-2931q^{33}+14414q^{32}+24606q^{31}+23543q^{30}+8369q^{29}-13428q^{28}-28609q^{27}-30788q^{26}-15023q^{25}+10894q^{24}+31380q^{23}+37534q^{22}+22153q^{21}-6984q^{20}-32519q^{19}-43182q^{18}-29277q^{17}+2194q^{16}+32266q^{15}+47435q^{14}+35554q^{13}+2811q^{12}-30755q^{11}-50085q^{10}-40694q^{9}-7636q^{8}+28531q^{7}+51410q^{6}+44492q^{5}+11758q^{4}-26022q^{3}-51586q^{2}-46997q-15071+23460q^{-1}+51043q^{-2}+48493q^{-3}+17596q^{-4}-21149q^{-5}-50051q^{-6}-49152q^{-7}-19460q^{-8}+18995q^{-9}+48723q^{-10}+49321q^{-11}+20961q^{-12}-16936q^{-13}-47183q^{-14}-49153q^{-15}-22238q^{-16}+14779q^{-17}+45278q^{-18}+48678q^{-19}+23585q^{-20}-12258q^{-21}-42961q^{-22}-47959q^{-23}-25002q^{-24}+9292q^{-25}+39963q^{-26}+46799q^{-27}+26559q^{-28}-5721q^{-29}-36200q^{-30}-45063q^{-31}-28062q^{-32}+1588q^{-33}+31519q^{-34}+42587q^{-35}+29294q^{-36}+2862q^{-37}-25974q^{-38}-39041q^{-39}-29920q^{-40}-7486q^{-41}+19703q^{-42}+34560q^{-43}+29637q^{-44}+11556q^{-45}-13068q^{-46}-28889q^{-47}-28087q^{-48}-14950q^{-49}+6456q^{-50}+22647q^{-51}+25292q^{-52}+16844q^{-53}-625q^{-54}-15869q^{-55}-21155q^{-56}-17348q^{-57}-4126q^{-58}+9515q^{-59}+16312q^{-60}+16060q^{-61}+7132q^{-62}-3882q^{-63}-11017q^{-64}-13576q^{-65}-8487q^{-66}-263q^{-67}+6155q^{-68}+10143q^{-69}+8141q^{-70}+2916q^{-71}-2143q^{-72}-6607q^{-73}-6670q^{-74}-3880q^{-75}-595q^{-76}+3349q^{-77}+4603q^{-78}+3714q^{-79}+2037q^{-80}-1038q^{-81}-2529q^{-82}-2688q^{-83}-2354q^{-84}-432q^{-85}+844q^{-86}+1549q^{-87}+1994q^{-88}+958q^{-89}+142q^{-90}-484q^{-91}-1255q^{-92}-945q^{-93}-644q^{-94}-187q^{-95}+659q^{-96}+643q^{-97}+618q^{-98}+447q^{-99}-136q^{-100}-264q^{-101}-485q^{-102}-517q^{-103}-64q^{-104}+77q^{-105}+242q^{-106}+349q^{-107}+137q^{-108}+105q^{-109}-88q^{-110}-260q^{-111}-122q^{-112}-86q^{-113}+21q^{-114}+110q^{-115}+55q^{-116}+91q^{-117}+41q^{-118}-65q^{-119}-47q^{-120}-50q^{-121}-12q^{-122}+28q^{-123}-3q^{-124}+26q^{-125}+25q^{-126}-6q^{-127}-9q^{-128}-17q^{-129}-4q^{-130}+10q^{-131}-4q^{-132}+7q^{-134}-5q^{-137}+3q^{-139}-q^{-140}$

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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