9 7

9_6

9_8

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 7's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 7 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X3,12,4,13 X5,16,6,17 X7,18,8,1 X17,6,18,7 X9,14,10,15 X13,10,14,11 X15,8,16,9 X11,2,12,3
Gauss code	-1, 9, -2, 1, -3, 5, -4, 8, -6, 7, -9, 2, -7, 6, -8, 3, -5, 4
Dowker-Thistlethwaite code	4 12 16 18 14 2 10 8 6
Conway Notation	[342]

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

[edit Notes on presentations of 9 7]

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 7"];

In[4]:=

PD[K]

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

Out[4]=

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

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 12 16 18 14 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]=

[342]

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]=

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

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

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 8.01486 A-Polynomial See Data:9 7/A-polynomial

[edit Notes for 9 7'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 7's four dimensional invariants]

Polynomial invariants

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

Further Quantum Invariants

Further quantum knot invariants for 9_7.

A1 Invariants.

Weight	Invariant
1	$-q^{23}+q^{21}-q^{19}+q^{17}-q^{15}+q^{11}-q^{9}+2q^{7}+q^{3}$
2	$q^{64}-q^{62}-q^{60}+3q^{58}-q^{56}-4q^{54}+3q^{52}+2q^{50}-4q^{48}+2q^{46}+3q^{44}-4q^{42}+2q^{38}-q^{36}-2q^{34}+3q^{30}-4q^{28}-2q^{26}+5q^{24}-2q^{22}-2q^{20}+4q^{18}+2q^{12}+q^{6}$
3	$-q^{123}+q^{121}+q^{119}-q^{117}-2q^{115}+q^{113}+5q^{111}-7q^{107}-3q^{105}+6q^{103}+7q^{101}-4q^{99}-10q^{97}+q^{95}+10q^{93}+4q^{91}-10q^{89}-7q^{87}+9q^{85}+9q^{83}-6q^{81}-9q^{79}+5q^{77}+8q^{75}-3q^{73}-8q^{71}+6q^{67}+2q^{65}-3q^{63}-6q^{61}+3q^{59}+9q^{57}-12q^{53}-4q^{51}+10q^{49}+6q^{47}-11q^{45}-7q^{43}+5q^{41}+7q^{39}-2q^{37}-5q^{35}+q^{33}+2q^{31}+q^{29}+q^{25}+q^{21}+q^{19}+2q^{17}+q^{9}$
4	$q^{200}-q^{198}-q^{196}+q^{194}+2q^{190}-3q^{188}-3q^{186}+2q^{184}+2q^{182}+9q^{180}-3q^{178}-11q^{176}-5q^{174}-q^{172}+18q^{170}+9q^{168}-5q^{166}-13q^{164}-20q^{162}+9q^{160}+18q^{158}+16q^{156}-q^{154}-31q^{152}-16q^{150}+5q^{148}+32q^{146}+26q^{144}-21q^{142}-32q^{140}-18q^{138}+26q^{136}+40q^{134}-3q^{132}-30q^{130}-30q^{128}+13q^{126}+36q^{124}+6q^{122}-19q^{120}-25q^{118}+4q^{116}+24q^{114}+10q^{112}-10q^{110}-17q^{108}-3q^{106}+11q^{104}+16q^{102}+q^{100}-11q^{98}-18q^{96}-4q^{94}+27q^{92}+18q^{90}-4q^{88}-33q^{86}-27q^{84}+25q^{82}+35q^{80}+17q^{78}-31q^{76}-44q^{74}+6q^{72}+28q^{70}+31q^{68}-8q^{66}-35q^{64}-12q^{62}+5q^{60}+24q^{58}+9q^{56}-13q^{54}-9q^{52}-8q^{50}+8q^{48}+9q^{46}-q^{42}-7q^{40}+q^{38}+4q^{36}+2q^{34}+3q^{32}-3q^{30}+q^{26}+q^{24}+3q^{22}+q^{12}$
5	$-q^{295}+q^{293}+q^{291}-q^{289}+q^{281}+2q^{279}-2q^{277}-5q^{275}-2q^{273}+q^{271}+6q^{269}+9q^{267}+5q^{265}-9q^{263}-17q^{261}-11q^{259}+q^{257}+19q^{255}+24q^{253}+13q^{251}-12q^{249}-30q^{247}-27q^{245}-9q^{243}+20q^{241}+40q^{239}+35q^{237}+2q^{235}-35q^{233}-54q^{231}-39q^{229}+11q^{227}+61q^{225}+75q^{223}+27q^{221}-51q^{219}-96q^{217}-73q^{215}+18q^{213}+105q^{211}+112q^{209}+21q^{207}-92q^{205}-132q^{203}-62q^{201}+67q^{199}+140q^{197}+92q^{195}-40q^{193}-132q^{191}-105q^{189}+11q^{187}+112q^{185}+107q^{183}+6q^{181}-91q^{179}-95q^{177}-15q^{175}+67q^{173}+81q^{171}+19q^{169}-50q^{167}-63q^{165}-18q^{163}+35q^{161}+48q^{159}+20q^{157}-20q^{155}-44q^{153}-26q^{151}+8q^{149}+35q^{147}+43q^{145}+12q^{143}-36q^{141}-62q^{139}-36q^{137}+31q^{135}+86q^{133}+72q^{131}-15q^{129}-102q^{127}-111q^{125}-14q^{123}+111q^{121}+142q^{119}+50q^{117}-93q^{115}-160q^{113}-92q^{111}+65q^{109}+157q^{107}+110q^{105}-18q^{103}-127q^{101}-124q^{99}-19q^{97}+87q^{95}+109q^{93}+40q^{91}-44q^{89}-84q^{87}-53q^{85}+11q^{83}+55q^{81}+45q^{79}+5q^{77}-27q^{75}-35q^{73}-14q^{71}+13q^{69}+22q^{67}+13q^{65}-3q^{63}-15q^{61}-11q^{59}+q^{57}+7q^{55}+8q^{53}+4q^{51}-6q^{49}-5q^{47}+2q^{43}+4q^{41}+5q^{39}-q^{37}-2q^{35}+q^{29}+3q^{27}+q^{25}+q^{15}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{34}-q^{28}+q^{26}-q^{18}+q^{16}+q^{12}+2q^{10}+q^{6}$
1,1	$q^{92}-2q^{90}+4q^{88}-8q^{86}+15q^{84}-20q^{82}+26q^{80}-34q^{78}+35q^{76}-34q^{74}+28q^{72}-16q^{70}+3q^{68}+14q^{66}-30q^{64}+42q^{62}-53q^{60}+58q^{58}-58q^{56}+56q^{54}-46q^{52}+36q^{50}-22q^{48}+6q^{46}+q^{44}-18q^{42}+18q^{40}-24q^{38}+21q^{36}-20q^{34}+20q^{32}-14q^{30}+16q^{28}-10q^{26}+12q^{24}-6q^{22}+8q^{20}-2q^{18}+4q^{16}+q^{12}$
2,0	$q^{86}+q^{78}-3q^{74}-q^{72}+q^{70}+q^{68}-q^{66}+3q^{62}+q^{60}-2q^{58}-q^{56}+q^{54}-q^{52}-q^{50}-q^{46}-3q^{44}-2q^{42}-3q^{38}-q^{36}+4q^{34}+2q^{32}-q^{30}+2q^{28}+4q^{26}+2q^{24}-q^{22}+2q^{20}+2q^{18}+q^{12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{74}-q^{72}+q^{68}-2q^{66}+2q^{64}+2q^{62}-3q^{60}+2q^{58}+q^{56}-5q^{54}-q^{52}+q^{50}-3q^{48}-q^{46}+q^{44}+q^{42}+4q^{36}-2q^{34}-3q^{32}+3q^{30}-2q^{28}-3q^{26}+4q^{24}+2q^{22}+q^{20}+3q^{18}+2q^{16}+q^{12}$
1,0,0	$-q^{45}-q^{41}-q^{37}+q^{35}+q^{31}-q^{25}-q^{23}+q^{21}+2q^{17}+q^{15}+2q^{13}+q^{9}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{96}-q^{92}+q^{90}+q^{88}-2q^{86}+4q^{82}+2q^{80}-2q^{78}+2q^{74}-3q^{72}-6q^{70}-q^{68}-2q^{66}-5q^{64}-q^{62}+q^{60}-q^{58}+4q^{54}+3q^{52}+q^{48}+3q^{46}-2q^{44}-5q^{42}-q^{40}-q^{36}+3q^{32}+5q^{30}+3q^{28}+3q^{26}+3q^{24}+2q^{22}+q^{18}$
1,0,0,0	$-q^{56}-q^{52}-q^{50}-q^{46}+q^{44}+q^{40}+q^{38}-q^{32}-q^{30}-q^{28}+q^{26}+2q^{22}+2q^{20}+q^{18}+2q^{16}+q^{12}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{74}+q^{72}-2q^{70}+3q^{68}-4q^{66}+4q^{64}-4q^{62}+3q^{60}-2q^{58}+q^{56}+q^{54}-3q^{52}+5q^{50}-7q^{48}+7q^{46}-7q^{44}+7q^{42}-6q^{40}+4q^{38}-2q^{36}+q^{32}-3q^{30}+4q^{28}-3q^{26}+4q^{24}-2q^{22}+3q^{20}-q^{18}+2q^{16}+q^{12}$
1,0	$q^{120}-q^{116}-q^{114}+q^{112}+2q^{110}-q^{108}-3q^{106}+4q^{102}+3q^{100}-3q^{98}-4q^{96}+q^{94}+4q^{92}+q^{90}-4q^{88}-3q^{86}+q^{84}+2q^{82}-q^{80}-3q^{78}+2q^{74}-3q^{70}-q^{68}+3q^{66}+2q^{64}-2q^{62}-2q^{60}+2q^{58}+3q^{56}-q^{54}-4q^{52}-q^{50}+4q^{48}+2q^{46}-3q^{44}-3q^{42}+4q^{38}+2q^{36}-q^{32}+q^{30}+2q^{28}+2q^{26}+q^{18}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{102}-q^{100}+q^{98}-2q^{96}+3q^{94}-3q^{92}+3q^{90}-3q^{88}+4q^{86}-2q^{84}+2q^{82}-q^{80}-q^{76}-4q^{74}+q^{72}-5q^{70}+3q^{68}-7q^{66}+5q^{64}-5q^{62}+7q^{60}-4q^{58}+5q^{56}-2q^{54}+4q^{52}-q^{46}-3q^{44}+2q^{42}-4q^{40}+q^{38}-3q^{36}+4q^{34}+4q^{30}+q^{28}+4q^{26}+q^{24}+2q^{22}+q^{18}$

G2 Invariants.

Weight

Invariant

1,0

$q^{176}-q^{174}+2q^{172}-3q^{170}+2q^{168}-q^{166}-2q^{164}+7q^{162}-8q^{160}+9q^{158}-7q^{156}+6q^{152}-12q^{150}+13q^{148}-11q^{146}+4q^{144}+4q^{142}-10q^{140}+10q^{138}-6q^{136}+5q^{132}-9q^{130}+5q^{128}-7q^{124}+11q^{122}-12q^{120}+8q^{118}-q^{116}-8q^{114}+13q^{112}-16q^{110}+15q^{108}-7q^{106}-q^{104}+9q^{102}-13q^{100}+14q^{98}-7q^{96}+q^{94}+5q^{92}-7q^{90}+5q^{88}+q^{86}-6q^{84}+8q^{82}-7q^{80}+4q^{76}-9q^{74}+10q^{72}-8q^{70}+4q^{68}-q^{66}-4q^{64}+6q^{62}-6q^{60}+7q^{58}-3q^{56}+3q^{54}+q^{52}-q^{50}+4q^{48}-3q^{46}+4q^{44}-q^{42}+q^{40}+q^{38}-q^{36}+2q^{34}+q^{30}$

.

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 7"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{2}-7t+9-7t^{-1}+3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$3z^{4}+5z^{2}+1$

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]=

{ 29, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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 7"];

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]=

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

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]=

{}

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:

(5, -12)

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 $20$ $-96$ $200$ ${\frac {1654}{3}}$ ${\frac {218}{3}}$ $-1920$ $-3552$ $-576$ $-448$ ${\frac {4000}{3}}$ $4608$ 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{33080}{3}} ${\frac {4360}{3}}$ ${\frac {140719}{6}}$ ${\frac {1954}{3}}$ ${\frac {77486}{9}}$ ${\frac {4789}{18}}$ ${\frac {6415}{6}}$

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 7. 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                 1 1 0 -7               2     2 -9             2 1     -1 -11           3 2       1 -13         2 2         0 -15       2 3           -1 -17     1 2             1 -19   1 2               -1 -21   1                 1 -23 1                   -1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{-4}-q^{-5}+3q^{-7}-3q^{-8}+7q^{-10}-9q^{-11}+14q^{-13}-16q^{-14}-2q^{-15}+21q^{-16}-19q^{-17}-4q^{-18}+22q^{-19}-16q^{-20}-6q^{-21}+18q^{-22}-9q^{-23}-7q^{-24}+12q^{-25}-3q^{-26}-6q^{-27}+5q^{-28}-2q^{-30}+q^{-31}$
3	$q^{-6}-q^{-7}+3q^{-10}-2q^{-11}-q^{-13}+4q^{-14}-3q^{-15}+q^{-16}+3q^{-18}-9q^{-19}+4q^{-20}+9q^{-21}+q^{-22}-21q^{-23}+26q^{-25}+5q^{-26}-35q^{-27}-8q^{-28}+38q^{-29}+14q^{-30}-41q^{-31}-17q^{-32}+41q^{-33}+19q^{-34}-37q^{-35}-23q^{-36}+33q^{-37}+24q^{-38}-26q^{-39}-26q^{-40}+19q^{-41}+27q^{-42}-11q^{-43}-26q^{-44}+3q^{-45}+24q^{-46}+3q^{-47}-20q^{-48}-6q^{-49}+13q^{-50}+9q^{-51}-9q^{-52}-7q^{-53}+4q^{-54}+5q^{-55}-2q^{-56}-2q^{-57}+2q^{-59}-q^{-60}$
4	$q^{-8}-q^{-9}+4q^{-13}-3q^{-14}-q^{-16}-3q^{-17}+10q^{-18}-4q^{-19}+2q^{-20}-4q^{-21}-11q^{-22}+16q^{-23}-3q^{-24}+11q^{-25}-5q^{-26}-27q^{-27}+15q^{-28}-7q^{-29}+33q^{-30}+10q^{-31}-46q^{-32}-2q^{-33}-30q^{-34}+60q^{-35}+49q^{-36}-49q^{-37}-24q^{-38}-80q^{-39}+73q^{-40}+97q^{-41}-31q^{-42}-34q^{-43}-132q^{-44}+67q^{-45}+126q^{-46}-9q^{-47}-25q^{-48}-163q^{-49}+53q^{-50}+133q^{-51}+3q^{-52}-10q^{-53}-168q^{-54}+39q^{-55}+119q^{-56}+10q^{-57}+10q^{-58}-154q^{-59}+19q^{-60}+90q^{-61}+16q^{-62}+35q^{-63}-124q^{-64}-4q^{-65}+47q^{-66}+16q^{-67}+62q^{-68}-81q^{-69}-18q^{-70}+3q^{-71}+2q^{-72}+73q^{-73}-34q^{-74}-12q^{-75}-24q^{-76}-19q^{-77}+58q^{-78}-4q^{-79}+5q^{-80}-22q^{-81}-28q^{-82}+29q^{-83}+3q^{-84}+13q^{-85}-8q^{-86}-19q^{-87}+10q^{-88}-q^{-89}+7q^{-90}-7q^{-92}+3q^{-93}-q^{-94}+2q^{-95}-2q^{-97}+q^{-98}$
5	$q^{-10}-q^{-11}+q^{-15}+3q^{-16}-3q^{-17}-q^{-18}-2q^{-20}+2q^{-21}+9q^{-22}-4q^{-23}-3q^{-24}-2q^{-25}-7q^{-26}+q^{-27}+19q^{-28}-4q^{-30}-8q^{-31}-19q^{-32}-3q^{-33}+31q^{-34}+16q^{-35}+5q^{-36}-17q^{-37}-46q^{-38}-24q^{-39}+39q^{-40}+48q^{-41}+45q^{-42}-7q^{-43}-90q^{-44}-88q^{-45}+8q^{-46}+88q^{-47}+129q^{-48}+62q^{-49}-112q^{-50}-194q^{-51}-97q^{-52}+85q^{-53}+238q^{-54}+190q^{-55}-65q^{-56}-286q^{-57}-254q^{-58}+17q^{-59}+305q^{-60}+333q^{-61}+27q^{-62}-317q^{-63}-379q^{-64}-80q^{-65}+314q^{-66}+420q^{-67}+114q^{-68}-303q^{-69}-434q^{-70}-147q^{-71}+288q^{-72}+446q^{-73}+162q^{-74}-272q^{-75}-442q^{-76}-174q^{-77}+254q^{-78}+428q^{-79}+186q^{-80}-232q^{-81}-414q^{-82}-187q^{-83}+201q^{-84}+383q^{-85}+199q^{-86}-163q^{-87}-352q^{-88}-201q^{-89}+119q^{-90}+303q^{-91}+203q^{-92}-66q^{-93}-251q^{-94}-196q^{-95}+18q^{-96}+187q^{-97}+176q^{-98}+26q^{-99}-119q^{-100}-148q^{-101}-55q^{-102}+58q^{-103}+106q^{-104}+66q^{-105}-6q^{-106}-57q^{-107}-62q^{-108}-29q^{-109}+15q^{-110}+43q^{-111}+39q^{-112}+21q^{-113}-14q^{-114}-43q^{-115}-35q^{-116}-7q^{-117}+24q^{-118}+40q^{-119}+23q^{-120}-10q^{-121}-30q^{-122}-27q^{-123}-5q^{-124}+22q^{-125}+20q^{-126}+8q^{-127}-5q^{-128}-16q^{-129}-10q^{-130}+4q^{-131}+8q^{-132}+2q^{-133}+3q^{-134}-2q^{-135}-6q^{-136}+q^{-137}+3q^{-138}-q^{-139}+q^{-141}-2q^{-142}+2q^{-144}-q^{-145}$

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_6 9_8