9 25

9_24

9_26

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 25's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 25 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X3849 X5,12,6,13 X9,17,10,16 X13,18,14,1 X17,14,18,15 X15,11,16,10 X11,6,12,7 X7283
Gauss code	-1, 9, -2, 1, -3, 8, -9, 2, -4, 7, -8, 3, -5, 6, -7, 4, -6, 5
Dowker-Thistlethwaite code	4 8 12 2 16 6 18 10 14
Conway Notation	[22,21,2]

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

[edit Notes on presentations of 9 25]

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 25"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3849 X5,12,6,13 X9,17,10,16 X13,18,14,1 X17,14,18,15 X15,11,16,10 X11,6,12,7 X7283

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 12 2 16 6 18 10 14

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[22,21,2]

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]=

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

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 3 Super bridge index $\{4,7\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-10][-1] Hyperbolic Volume 11.3903 A-Polynomial See Data:9 25/A-polynomial

[edit Notes for 9 25's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 25's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3t^{2}+12t-17+12t^{-1}-3t^{-2}$
Conway polynomial	$1-3z^{4}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 47, -2 }
Jones polynomial	$q-2+5q^{-1}-7q^{-2}+8q^{-3}-8q^{-4}+7q^{-5}-5q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{8}+3z^{2}a^{6}+3a^{6}-2z^{4}a^{4}-4z^{2}a^{4}-3a^{4}-z^{4}a^{2}+a^{2}+z^{2}+1$
Kauffman polynomial (db, data sources)	$z^{5}a^{9}-2z^{3}a^{9}+za^{9}+3z^{6}a^{8}-7z^{4}a^{8}+4z^{2}a^{8}-a^{8}+3z^{7}a^{7}-4z^{5}a^{7}-2z^{3}a^{7}+za^{7}+z^{8}a^{6}+6z^{6}a^{6}-18z^{4}a^{6}+13z^{2}a^{6}-3a^{6}+6z^{7}a^{5}-10z^{5}a^{5}+5z^{3}a^{5}-za^{5}+z^{8}a^{4}+6z^{6}a^{4}-15z^{4}a^{4}+13z^{2}a^{4}-3a^{4}+3z^{7}a^{3}-3z^{5}a^{3}+3z^{3}a^{3}-za^{3}+3z^{6}a^{2}-3z^{4}a^{2}+2z^{2}a^{2}-a^{2}+2z^{5}a-2z^{3}a+z^{4}-2z^{2}+1$
The A2 invariant	$-q^{26}-q^{24}+2q^{22}+q^{18}+2q^{16}-2q^{14}-2q^{10}+q^{6}-q^{4}+3q^{2}+q^{-4}$
The G2 invariant	$q^{128}-2q^{126}+5q^{124}-8q^{122}+7q^{120}-4q^{118}-6q^{116}+19q^{114}-29q^{112}+34q^{110}-28q^{108}+4q^{106}+23q^{104}-50q^{102}+63q^{100}-55q^{98}+26q^{96}+12q^{94}-46q^{92}+60q^{90}-48q^{88}+22q^{86}+15q^{84}-38q^{82}+41q^{80}-18q^{78}-14q^{76}+48q^{74}-60q^{72}+50q^{70}-13q^{68}-30q^{66}+68q^{64}-87q^{62}+79q^{60}-45q^{58}-6q^{56}+48q^{54}-78q^{52}+76q^{50}-50q^{48}+8q^{46}+26q^{44}-46q^{42}+38q^{40}-14q^{38}-20q^{36}+43q^{34}-43q^{32}+21q^{30}+13q^{28}-44q^{26}+63q^{24}-54q^{22}+31q^{20}-q^{18}-27q^{16}+43q^{14}-43q^{12}+34q^{10}-14q^{8}+11q^{4}-16q^{2}+15-11q^{-2}+8q^{-4}-2q^{-6}-q^{-8}+3q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}$

Further Quantum Invariants

Further quantum knot invariants for 9_25.

A1 Invariants.

Weight	Invariant
1	$-q^{17}+2q^{15}-2q^{13}+2q^{11}-q^{9}+q^{5}-2q^{3}+3q-q^{-1}+q^{-3}$
2	$q^{48}-2q^{46}-2q^{44}+7q^{42}-2q^{40}-9q^{38}+11q^{36}+3q^{34}-15q^{32}+7q^{30}+8q^{28}-12q^{26}+q^{24}+8q^{22}-3q^{20}-6q^{18}+3q^{16}+9q^{14}-10q^{12}-3q^{10}+16q^{8}-8q^{6}-8q^{4}+12q^{2}-2-5q^{-2}+4q^{-4}-q^{-8}+q^{-10}$
3	$-q^{93}+2q^{91}+2q^{89}-3q^{87}-7q^{85}+2q^{83}+16q^{81}+q^{79}-23q^{77}-12q^{75}+29q^{73}+29q^{71}-29q^{69}-46q^{67}+19q^{65}+59q^{63}-2q^{61}-68q^{59}-15q^{57}+66q^{55}+30q^{53}-55q^{51}-40q^{49}+44q^{47}+44q^{45}-29q^{43}-44q^{41}+11q^{39}+40q^{37}+7q^{35}-36q^{33}-28q^{31}+29q^{29}+46q^{27}-17q^{25}-61q^{23}+3q^{21}+70q^{19}+14q^{17}-69q^{15}-25q^{13}+55q^{11}+36q^{9}-39q^{7}-35q^{5}+24q^{3}+27q-8q^{-1}-18q^{-3}+3q^{-5}+9q^{-7}-q^{-9}-4q^{-11}+q^{-13}+q^{-15}-q^{-19}+q^{-21}$
4	$q^{152}-2q^{150}-2q^{148}+3q^{146}+3q^{144}+7q^{142}-9q^{140}-16q^{138}-q^{136}+11q^{134}+41q^{132}-q^{130}-47q^{128}-44q^{126}-10q^{124}+101q^{122}+72q^{120}-33q^{118}-123q^{116}-129q^{114}+98q^{112}+190q^{110}+105q^{108}-121q^{106}-295q^{104}-53q^{102}+211q^{100}+299q^{98}+33q^{96}-349q^{94}-252q^{92}+79q^{90}+374q^{88}+218q^{86}-243q^{84}-338q^{82}-85q^{80}+301q^{78}+291q^{76}-93q^{74}-289q^{72}-165q^{70}+170q^{68}+258q^{66}+32q^{64}-191q^{62}-191q^{60}+36q^{58}+196q^{56}+152q^{54}-78q^{52}-215q^{50}-129q^{48}+113q^{46}+288q^{44}+76q^{42}-208q^{40}-305q^{38}-32q^{36}+357q^{34}+254q^{32}-95q^{30}-388q^{28}-210q^{26}+269q^{24}+329q^{22}+82q^{20}-286q^{18}-287q^{16}+78q^{14}+232q^{12}+171q^{10}-100q^{8}-202q^{6}-36q^{4}+74q^{2}+119+7q^{-2}-73q^{-4}-33q^{-6}-2q^{-8}+40q^{-10}+13q^{-12}-14q^{-14}-4q^{-16}-7q^{-18}+7q^{-20}+2q^{-22}-3q^{-24}+2q^{-26}-2q^{-28}+q^{-30}-q^{-34}+q^{-36}$
5	$-q^{225}+2q^{223}+2q^{221}-3q^{219}-3q^{217}-3q^{215}+9q^{211}+16q^{209}+q^{207}-20q^{205}-29q^{203}-19q^{201}+19q^{199}+61q^{197}+65q^{195}-8q^{193}-97q^{191}-128q^{189}-59q^{187}+101q^{185}+232q^{183}+192q^{181}-50q^{179}-314q^{177}-380q^{175}-132q^{173}+317q^{171}+608q^{169}+433q^{167}-179q^{165}-764q^{163}-819q^{161}-168q^{159}+769q^{157}+1205q^{155}+673q^{153}-549q^{151}-1463q^{149}-1236q^{147}+92q^{145}+1502q^{143}+1753q^{141}+499q^{139}-1299q^{137}-2079q^{135}-1103q^{133}+880q^{131}+2172q^{129}+1600q^{127}-375q^{125}-2039q^{123}-1894q^{121}-105q^{119}+1723q^{117}+1983q^{115}+488q^{113}-1347q^{111}-1887q^{109}-723q^{107}+973q^{105}+1668q^{103}+846q^{101}-645q^{99}-1424q^{97}-882q^{95}+374q^{93}+1185q^{91}+903q^{89}-124q^{87}-979q^{85}-964q^{83}-132q^{81}+798q^{79}+1067q^{77}+447q^{75}-597q^{73}-1215q^{71}-837q^{69}+336q^{67}+1349q^{65}+1278q^{63}+33q^{61}-1400q^{59}-1722q^{57}-511q^{55}+1301q^{53}+2085q^{51}+1038q^{49}-1011q^{47}-2255q^{45}-1542q^{43}+537q^{41}+2191q^{39}+1913q^{37}-10q^{35}-1840q^{33}-2044q^{31}-521q^{29}+1324q^{27}+1924q^{25}+878q^{23}-749q^{21}-1559q^{19}-1024q^{17}+235q^{15}+1098q^{13}+964q^{11}+102q^{9}-649q^{7}-735q^{5}-264q^{3}+292q+489q^{-1}+271q^{-3}-86q^{-5}-266q^{-7}-199q^{-9}-12q^{-11}+122q^{-13}+116q^{-15}+34q^{-17}-43q^{-19}-59q^{-21}-23q^{-23}+13q^{-25}+20q^{-27}+12q^{-29}+3q^{-31}-10q^{-33}-6q^{-35}+3q^{-37}+3q^{-43}-q^{-45}-2q^{-47}+q^{-49}-q^{-53}+q^{-55}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{26}-q^{24}+2q^{22}+q^{18}+2q^{16}-2q^{14}-2q^{10}+q^{6}-q^{4}+3q^{2}+q^{-4}$
1,1	$q^{68}-4q^{66}+12q^{64}-28q^{62}+52q^{60}-86q^{58}+130q^{56}-176q^{54}+212q^{52}-234q^{50}+232q^{48}-194q^{46}+126q^{44}-30q^{42}-80q^{40}+196q^{38}-303q^{36}+380q^{34}-432q^{32}+438q^{30}-410q^{28}+342q^{26}-244q^{24}+138q^{22}-19q^{20}-76q^{18}+154q^{16}-198q^{14}+214q^{12}-206q^{10}+174q^{8}-142q^{6}+107q^{4}-74q^{2}+50-30q^{-2}+21q^{-4}-10q^{-6}+6q^{-8}-2q^{-10}+q^{-12}$
2,0	$q^{66}+q^{64}-q^{62}-4q^{60}-2q^{58}+4q^{56}+2q^{54}-3q^{52}+8q^{48}+5q^{46}-8q^{44}-5q^{42}+4q^{40}-q^{38}-8q^{36}-3q^{34}+6q^{32}+q^{30}-q^{28}+4q^{26}+q^{24}-2q^{22}+5q^{20}+4q^{18}-7q^{16}-2q^{14}+7q^{12}+q^{10}-9q^{8}-2q^{6}+9q^{4}-5+q^{-2}+4q^{-4}+q^{-6}-q^{-8}+q^{-12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{54}-2q^{52}+q^{50}+3q^{48}-7q^{46}+3q^{44}+4q^{42}-11q^{40}+6q^{38}+7q^{36}-10q^{34}+4q^{32}+7q^{30}-5q^{28}-q^{26}+3q^{24}+2q^{22}-4q^{20}-4q^{18}+8q^{16}-5q^{14}-8q^{12}+12q^{10}-3q^{8}-7q^{6}+10q^{4}-3+4q^{-2}+q^{-4}-q^{-6}+q^{-8}$
1,0,0	$-q^{35}-q^{33}-q^{31}+2q^{29}+3q^{25}+q^{23}+2q^{21}-2q^{19}-q^{17}-2q^{15}-2q^{13}+2q^{7}-q^{5}+3q^{3}+q^{-1}+q^{-5}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{54}+2q^{52}-5q^{50}+7q^{48}-9q^{46}+11q^{44}-12q^{42}+11q^{40}-8q^{38}+5q^{36}+2q^{34}-6q^{32}+13q^{30}-17q^{28}+21q^{26}-23q^{24}+20q^{22}-18q^{20}+12q^{18}-8q^{16}+q^{14}+4q^{12}-8q^{10}+11q^{8}-11q^{6}+12q^{4}-8q^{2}+7-4q^{-2}+3q^{-4}-q^{-6}+q^{-8}$
1,0	$q^{88}-2q^{84}-2q^{82}+3q^{80}+5q^{78}-2q^{76}-8q^{74}-3q^{72}+9q^{70}+8q^{68}-7q^{66}-12q^{64}+q^{62}+13q^{60}+6q^{58}-10q^{56}-8q^{54}+5q^{52}+9q^{50}-2q^{48}-8q^{46}+8q^{42}+2q^{40}-8q^{38}-3q^{36}+7q^{34}+6q^{32}-6q^{30}-8q^{28}+4q^{26}+10q^{24}-2q^{22}-12q^{20}-3q^{18}+11q^{16}+8q^{14}-7q^{12}-11q^{10}+q^{8}+11q^{6}+4q^{4}-4q^{2}-5+q^{-2}+4q^{-4}+2q^{-6}-q^{-8}-q^{-10}+q^{-14}$

G2 Invariants.

Weight

Invariant

1,0

$q^{128}-2q^{126}+5q^{124}-8q^{122}+7q^{120}-4q^{118}-6q^{116}+19q^{114}-29q^{112}+34q^{110}-28q^{108}+4q^{106}+23q^{104}-50q^{102}+63q^{100}-55q^{98}+26q^{96}+12q^{94}-46q^{92}+60q^{90}-48q^{88}+22q^{86}+15q^{84}-38q^{82}+41q^{80}-18q^{78}-14q^{76}+48q^{74}-60q^{72}+50q^{70}-13q^{68}-30q^{66}+68q^{64}-87q^{62}+79q^{60}-45q^{58}-6q^{56}+48q^{54}-78q^{52}+76q^{50}-50q^{48}+8q^{46}+26q^{44}-46q^{42}+38q^{40}-14q^{38}-20q^{36}+43q^{34}-43q^{32}+21q^{30}+13q^{28}-44q^{26}+63q^{24}-54q^{22}+31q^{20}-q^{18}-27q^{16}+43q^{14}-43q^{12}+34q^{10}-14q^{8}+11q^{4}-16q^{2}+15-11q^{-2}+8q^{-4}-2q^{-6}-q^{-8}+3q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}$

.

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 25"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-3t^{2}+12t-17+12t^{-1}-3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$1-3z^{4}$

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]=

{ 47, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{8}+3z^{2}a^{6}+3a^{6}-2z^{4}a^{4}-4z^{2}a^{4}-3a^{4}-z^{4}a^{2}+a^{2}+z^{2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{5}a^{9}-2z^{3}a^{9}+za^{9}+3z^{6}a^{8}-7z^{4}a^{8}+4z^{2}a^{8}-a^{8}+3z^{7}a^{7}-4z^{5}a^{7}-2z^{3}a^{7}+za^{7}+z^{8}a^{6}+6z^{6}a^{6}-18z^{4}a^{6}+13z^{2}a^{6}-3a^{6}+6z^{7}a^{5}-10z^{5}a^{5}+5z^{3}a^{5}-za^{5}+z^{8}a^{4}+6z^{6}a^{4}-15z^{4}a^{4}+13z^{2}a^{4}-3a^{4}+3z^{7}a^{3}-3z^{5}a^{3}+3z^{3}a^{3}-za^{3}+3z^{6}a^{2}-3z^{4}a^{2}+2z^{2}a^{2}-a^{2}+2z^{5}a-2z^{3}a+z^{4}-2z^{2}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n134,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n25,}

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 25"];

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]=

{ $-3t^{2}+12t-17+12t^{-1}-3t^{-2}$, $q-2+5q^{-1}-7q^{-2}+8q^{-3}-8q^{-4}+7q^{-5}-5q^{-6}+3q^{-7}-q^{-8}$ }

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]=

{K11n134,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11n25,}

Vassiliev invariants

V2 and V3:

(0, -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 $0$ $-8$ $0$ $64$ $24$ $0$ $-{\frac {656}{3}}$ $-{\frac {128}{3}}$ $-72$ $0$ $32$ $0$ $0$ $528$ $-{\frac {248}{3}}$ $344$ $48$ $32$

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 $s=$-2 is the signature of 9 25. 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 3                   1 1 1                 1   -1 -1               4 1   3 -3             4 2     -2 -5           4 3       1 -7         4 4         0 -9       3 4           -1 -11     2 4             2 -13   1 3               -2 -15   2                 2 -17 1                   -1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{4}-2q^{3}+q^{2}+5q-11+4q^{-1}+19q^{-2}-31q^{-3}+4q^{-4}+43q^{-5}-50q^{-6}-3q^{-7}+62q^{-8}-56q^{-9}-12q^{-10}+65q^{-11}-45q^{-12}-19q^{-13}+52q^{-14}-25q^{-15}-20q^{-16}+30q^{-17}-7q^{-18}-12q^{-19}+10q^{-20}-3q^{-22}+q^{-23}$
3	$q^{9}-2q^{8}+q^{7}+q^{6}+q^{5}-7q^{4}+4q^{3}+11q^{2}-5q-28+14q^{-1}+46q^{-2}-8q^{-3}-87q^{-4}+10q^{-5}+121q^{-6}+11q^{-7}-167q^{-8}-34q^{-9}+204q^{-10}+67q^{-11}-234q^{-12}-98q^{-13}+248q^{-14}+130q^{-15}-251q^{-16}-155q^{-17}+240q^{-18}+173q^{-19}-218q^{-20}-184q^{-21}+185q^{-22}+188q^{-23}-145q^{-24}-184q^{-25}+101q^{-26}+173q^{-27}-60q^{-28}-148q^{-29}+20q^{-30}+120q^{-31}+6q^{-32}-87q^{-33}-20q^{-34}+55q^{-35}+23q^{-36}-29q^{-37}-20q^{-38}+14q^{-39}+12q^{-40}-5q^{-41}-5q^{-42}+3q^{-44}-q^{-45}$
4	$q^{16}-2q^{15}+q^{14}+q^{13}-3q^{12}+5q^{11}-7q^{10}+6q^{9}+6q^{8}-17q^{7}+8q^{6}-17q^{5}+33q^{4}+33q^{3}-59q^{2}-23q-57+113q^{-1}+145q^{-2}-104q^{-3}-133q^{-4}-223q^{-5}+215q^{-6}+416q^{-7}-43q^{-8}-287q^{-9}-588q^{-10}+216q^{-11}+784q^{-12}+204q^{-13}-347q^{-14}-1067q^{-15}+38q^{-16}+1077q^{-17}+553q^{-18}-244q^{-19}-1456q^{-20}-235q^{-21}+1174q^{-22}+837q^{-23}-32q^{-24}-1631q^{-25}-477q^{-26}+1088q^{-27}+974q^{-28}+198q^{-29}-1587q^{-30}-637q^{-31}+861q^{-32}+974q^{-33}+421q^{-34}-1361q^{-35}-725q^{-36}+526q^{-37}+850q^{-38}+617q^{-39}-977q^{-40}-715q^{-41}+140q^{-42}+597q^{-43}+712q^{-44}-516q^{-45}-559q^{-46}-155q^{-47}+266q^{-48}+615q^{-49}-134q^{-50}-293q^{-51}-243q^{-52}+2q^{-53}+373q^{-54}+40q^{-55}-67q^{-56}-158q^{-57}-90q^{-58}+146q^{-59}+46q^{-60}+23q^{-61}-53q^{-62}-61q^{-63}+35q^{-64}+12q^{-65}+20q^{-66}-7q^{-67}-19q^{-68}+5q^{-69}+5q^{-71}-3q^{-73}+q^{-74}$
5	$q^{25}-2q^{24}+q^{23}+q^{22}-3q^{21}+q^{20}+5q^{19}-5q^{18}+q^{17}+4q^{16}-12q^{15}-3q^{14}+18q^{13}+4q^{12}+9q^{11}-3q^{10}-48q^{9}-39q^{8}+34q^{7}+81q^{6}+91q^{5}+3q^{4}-182q^{3}-226q^{2}-33q+261+448q^{-1}+221q^{-2}-379q^{-3}-782q^{-4}-504q^{-5}+347q^{-6}+1199q^{-7}+1083q^{-8}-245q^{-9}-1645q^{-10}-1763q^{-11}-188q^{-12}+2009q^{-13}+2710q^{-14}+801q^{-15}-2245q^{-16}-3608q^{-17}-1711q^{-18}+2213q^{-19}+4540q^{-20}+2724q^{-21}-1967q^{-22}-5262q^{-23}-3790q^{-24}+1500q^{-25}+5784q^{-26}+4773q^{-27}-920q^{-28}-6046q^{-29}-5602q^{-30}+289q^{-31}+6106q^{-32}+6206q^{-33}+325q^{-34}-5975q^{-35}-6615q^{-36}-884q^{-37}+5728q^{-38}+6824q^{-39}+1369q^{-40}-5355q^{-41}-6884q^{-42}-1814q^{-43}+4896q^{-44}+6809q^{-45}+2224q^{-46}-4328q^{-47}-6602q^{-48}-2625q^{-49}+3640q^{-50}+6267q^{-51}+3003q^{-52}-2837q^{-53}-5780q^{-54}-3320q^{-55}+1944q^{-56}+5103q^{-57}+3543q^{-58}-1002q^{-59}-4285q^{-60}-3580q^{-61}+116q^{-62}+3314q^{-63}+3398q^{-64}+662q^{-65}-2310q^{-66}-3008q^{-67}-1176q^{-68}+1331q^{-69}+2422q^{-70}+1442q^{-71}-512q^{-72}-1754q^{-73}-1427q^{-74}-79q^{-75}+1094q^{-76}+1215q^{-77}+402q^{-78}-532q^{-79}-895q^{-80}-515q^{-81}+157q^{-82}+564q^{-83}+457q^{-84}+53q^{-85}-283q^{-86}-340q^{-87}-134q^{-88}+115q^{-89}+209q^{-90}+119q^{-91}-19q^{-92}-98q^{-93}-94q^{-94}-16q^{-95}+49q^{-96}+50q^{-97}+12q^{-98}-9q^{-99}-21q^{-100}-20q^{-101}+7q^{-102}+12q^{-103}+2q^{-104}-5q^{-107}+3q^{-109}-q^{-110}$
6	$q^{36}-2q^{35}+q^{34}+q^{33}-3q^{32}+q^{31}+q^{30}+7q^{29}-10q^{28}-q^{27}+9q^{26}-13q^{25}+q^{24}+9q^{23}+28q^{22}-24q^{21}-19q^{20}+13q^{19}-46q^{18}-4q^{17}+51q^{16}+119q^{15}-15q^{14}-69q^{13}-52q^{12}-220q^{11}-82q^{10}+166q^{9}+468q^{8}+248q^{7}-36q^{6}-281q^{5}-904q^{4}-671q^{3}+129q^{2}+1296q+1406+843q^{-1}-286q^{-2}-2463q^{-3}-2814q^{-4}-1280q^{-5}+2022q^{-6}+3948q^{-7}+4080q^{-8}+1692q^{-9}-3997q^{-10}-7111q^{-11}-6048q^{-12}+341q^{-13}+6591q^{-14}+10268q^{-15}+7942q^{-16}-2671q^{-17}-11812q^{-18}-14534q^{-19}-6166q^{-20}+6068q^{-21}+17110q^{-22}+18345q^{-23}+3858q^{-24}-13313q^{-25}-23788q^{-26}-16818q^{-27}+302q^{-28}+20740q^{-29}+29289q^{-30}+14325q^{-31}-9763q^{-32}-29726q^{-33}-27651q^{-34}-8996q^{-35}+19526q^{-36}+36660q^{-37}+24647q^{-38}-2982q^{-39}-30850q^{-40}-34893q^{-41}-17980q^{-42}+15180q^{-43}+39209q^{-44}+31596q^{-45}+3769q^{-46}-28659q^{-47}-37739q^{-48}-24145q^{-49}+10297q^{-50}+38309q^{-51}+34815q^{-52}+8783q^{-53}-25138q^{-54}-37536q^{-55}-27577q^{-56}+5901q^{-57}+35503q^{-58}+35603q^{-59}+12527q^{-60}-20802q^{-61}-35488q^{-62}-29521q^{-63}+1281q^{-64}+30969q^{-65}+34848q^{-66}+16129q^{-67}-14860q^{-68}-31473q^{-69}-30505q^{-70}-4448q^{-71}+23891q^{-72}+32060q^{-73}+19627q^{-74}-6779q^{-75}-24549q^{-76}-29514q^{-77}-10684q^{-78}+14058q^{-79}+25993q^{-80}+21279q^{-81}+2141q^{-82}-14620q^{-83}-24835q^{-84}-14902q^{-85}+3352q^{-86}+16539q^{-87}+18845q^{-88}+8669q^{-89}-3934q^{-90}-16341q^{-91}-14471q^{-92}-4504q^{-93}+6245q^{-94}+12251q^{-95}+10001q^{-96}+3585q^{-97}-6897q^{-98}-9567q^{-99}-6822q^{-100}-880q^{-101}+4656q^{-102}+6712q^{-103}+5644q^{-104}-498q^{-105}-3620q^{-106}-4667q^{-107}-2986q^{-108}-165q^{-109}+2389q^{-110}+3804q^{-111}+1480q^{-112}-58q^{-113}-1619q^{-114}-1905q^{-115}-1366q^{-116}-17q^{-117}+1418q^{-118}+950q^{-119}+732q^{-120}-54q^{-121}-510q^{-122}-820q^{-123}-452q^{-124}+252q^{-125}+207q^{-126}+391q^{-127}+194q^{-128}+39q^{-129}-248q^{-130}-225q^{-131}+3q^{-132}-31q^{-133}+92q^{-134}+80q^{-135}+76q^{-136}-45q^{-137}-57q^{-138}-q^{-139}-29q^{-140}+9q^{-141}+12q^{-142}+29q^{-143}-7q^{-144}-12q^{-145}+5q^{-146}-7q^{-147}+5q^{-150}-3q^{-152}+q^{-153}$
7	$q^{49}-2q^{48}+q^{47}+q^{46}-3q^{45}+q^{44}+q^{43}+3q^{42}+2q^{41}-12q^{40}+4q^{39}+8q^{38}-9q^{37}+2q^{36}+2q^{35}+15q^{34}+8q^{33}-47q^{32}-4q^{31}+20q^{30}-8q^{29}+23q^{28}+17q^{27}+58q^{26}+23q^{25}-143q^{24}-96q^{23}-36q^{22}-9q^{21}+152q^{20}+191q^{19}+277q^{18}+144q^{17}-377q^{16}-540q^{15}-567q^{14}-327q^{13}+414q^{12}+954q^{11}+1430q^{10}+1109q^{9}-402q^{8}-1739q^{7}-2768q^{6}-2573q^{5}-380q^{4}+2316q^{3}+5012q^{2}+5548q+2388-2370q^{-1}-7721q^{-2}-10214q^{-3}-6877q^{-4}+623q^{-5}+10484q^{-6}+16861q^{-7}+14375q^{-8}+4244q^{-9}-11540q^{-10}-24697q^{-11}-25884q^{-12}-13868q^{-13}+9592q^{-14}+32412q^{-15}+40381q^{-16}+28983q^{-17}-1848q^{-18}-37526q^{-19}-57242q^{-20}-49960q^{-21}-12323q^{-22}+38102q^{-23}+73166q^{-24}+74995q^{-25}+34149q^{-26}-31469q^{-27}-86279q^{-28}-102321q^{-29}-61840q^{-30}+17521q^{-31}+93437q^{-32}+128109q^{-33}+93677q^{-34}+3851q^{-35}-93581q^{-36}-150173q^{-37}-126202q^{-38}-30131q^{-39}+86429q^{-40}+165930q^{-41}+156317q^{-42}+58993q^{-43}-73228q^{-44}-174769q^{-45}-181702q^{-46}-87249q^{-47}+56185q^{-48}+176989q^{-49}+200831q^{-50}+112483q^{-51}-37527q^{-52}-173938q^{-53}-213557q^{-54}-133252q^{-55}+19431q^{-56}+167410q^{-57}+220610q^{-58}+149010q^{-59}-3390q^{-60}-159021q^{-61}-223169q^{-62}-160099q^{-63}-10179q^{-64}+150009q^{-65}+222740q^{-66}+167515q^{-67}+21258q^{-68}-141101q^{-69}-220251q^{-70}-172291q^{-71}-30665q^{-72}+132140q^{-73}+216507q^{-74}+175603q^{-75}+39315q^{-76}-122795q^{-77}-211642q^{-78}-178042q^{-79}-48156q^{-80}+112108q^{-81}+205276q^{-82}+180019q^{-83}+58062q^{-84}-99274q^{-85}-196848q^{-86}-181260q^{-87}-69252q^{-88}+83491q^{-89}+185321q^{-90}+181081q^{-91}+81626q^{-92}-64344q^{-93}-169860q^{-94}-178470q^{-95}-94293q^{-96}+42096q^{-97}+149911q^{-98}+172010q^{-99}+105726q^{-100}-17597q^{-101}-125246q^{-102}-160638q^{-103}-114329q^{-104}-7281q^{-105}+96865q^{-106}+143639q^{-107}+117833q^{-108}+30299q^{-109}-66111q^{-110}-121194q^{-111}-115123q^{-112}-49018q^{-113}+35809q^{-114}+94658q^{-115}+105401q^{-116}+61050q^{-117}-8534q^{-118}-66080q^{-119}-89559q^{-120}-65447q^{-121}-12985q^{-122}+38731q^{-123}+69413q^{-124}+62059q^{-125}+26964q^{-126}-15169q^{-127}-47756q^{-128}-52635q^{-129}-33026q^{-130}-2300q^{-131}+27660q^{-132}+39557q^{-133}+32200q^{-134}+12759q^{-135}-11470q^{-136}-25676q^{-137}-26572q^{-138}-16839q^{-139}+373q^{-140}+13644q^{-141}+18824q^{-142}+16002q^{-143}+5406q^{-144}-4743q^{-145}-11038q^{-146}-12491q^{-147}-7209q^{-148}-564q^{-149}+5046q^{-150}+8191q^{-151}+6327q^{-152}+2770q^{-153}-1113q^{-154}-4371q^{-155}-4446q^{-156}-3090q^{-157}-820q^{-158}+1866q^{-159}+2551q^{-160}+2323q^{-161}+1311q^{-162}-374q^{-163}-1079q^{-164}-1465q^{-165}-1226q^{-166}-177q^{-167}+368q^{-168}+717q^{-169}+757q^{-170}+279q^{-171}+71q^{-172}-258q^{-173}-484q^{-174}-237q^{-175}-99q^{-176}+92q^{-177}+198q^{-178}+94q^{-179}+115q^{-180}+37q^{-181}-100q^{-182}-75q^{-183}-62q^{-184}-4q^{-185}+42q^{-186}-3q^{-187}+29q^{-188}+29q^{-189}-9q^{-190}-12q^{-191}-20q^{-192}-2q^{-193}+12q^{-194}-5q^{-195}+7q^{-197}-5q^{-200}+3q^{-202}-q^{-203}$

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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