10 126

10_125

10_127

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 126's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 126 at Knotilus!
	10_126 is also known as the pretzel knot P(-5,3,2).

Knot presentations

Planar diagram presentation	X4251 X8493 X5,14,6,15 X15,20,16,1 X9,16,10,17 X11,18,12,19 X17,10,18,11 X19,12,20,13 X13,6,14,7 X2837
Gauss code	1, -10, 2, -1, -3, 9, 10, -2, -5, 7, -6, 8, -9, 3, -4, 5, -7, 6, -8, 4
Dowker-Thistlethwaite code	4 8 -14 2 -16 -18 -6 -20 -10 -12
Conway Notation	[41,3,2-]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 10, width is 3, Braid index is 3		[{3, 10}, {2, 4}, {1, 3}, {13, 11}, {10, 12}, {11, 5}, {4, 6}, {5, 7}, {6, 8}, {7, 9}, {8, 13}, {12, 2}, {9, 1}]

[edit Notes on presentations of 10 126]

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["10 126"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X8493 X5,14,6,15 X15,20,16,1 X9,16,10,17 X11,18,12,19 X17,10,18,11 X19,12,20,13 X13,6,14,7 X2837

In[5]:=

GaussCode[K]

Out[5]=

1, -10, 2, -1, -3, 9, 10, -2, -5, 7, -6, 8, -9, 3, -4, 5, -7, 6, -8, 4

In[6]:=

DTCode[K]

Out[6]=

4 8 -14 2 -16 -18 -6 -20 -10 -12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[41,3,2-]

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]=

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

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]=

{ 3, 10, 3 }

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[{3, 10}, {2, 4}, {1, 3}, {13, 11}, {10, 12}, {11, 5}, {4, 6}, {5, 7}, {6, 8}, {7, 9}, {8, 13}, {12, 2}, {9, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-10][0] Hyperbolic Volume 6.90426 A-Polynomial See Data:10 126/A-polynomial

[edit Notes for 10 126's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for 10 126's four dimensional invariants]

Polynomial invariants

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

Further Quantum Invariants

Further quantum knot invariants for 10_126.

A1 Invariants.

Weight	Invariant
1	$-q^{17}-q^{13}+q^{11}+q^{7}+2q^{5}+q-q^{-1}$
2	$q^{48}+q^{42}-q^{40}-2q^{38}+q^{36}-3q^{32}+q^{28}-2q^{26}+2q^{22}+q^{16}+3q^{14}-q^{12}+3q^{8}-q^{6}-q^{4}+2q^{2}-q^{-2}$
3	$-q^{93}+q^{83}+q^{81}-q^{77}+q^{75}+3q^{73}+q^{71}-3q^{69}-3q^{67}+2q^{65}+4q^{63}-6q^{59}-3q^{57}+3q^{55}+4q^{53}-4q^{51}-5q^{49}+q^{47}+4q^{45}-2q^{43}-3q^{41}+3q^{37}-q^{31}+q^{29}+4q^{27}-q^{25}-4q^{23}+7q^{19}+2q^{17}-4q^{15}-3q^{13}+5q^{11}+4q^{9}-q^{7}-3q^{5}+2q+2q^{-1}-q^{-3}-2q^{-5}-q^{-7}+q^{-11}$
5	$-q^{225}+q^{217}+q^{215}-q^{207}+2q^{203}+q^{201}-2q^{195}-3q^{193}-q^{191}+2q^{189}+3q^{187}+3q^{185}-5q^{181}-8q^{179}-5q^{177}+q^{175}+9q^{173}+11q^{171}+6q^{169}-5q^{167}-15q^{165}-16q^{163}-4q^{161}+12q^{159}+23q^{157}+20q^{155}+3q^{153}-20q^{151}-32q^{149}-21q^{147}+7q^{145}+33q^{143}+40q^{141}+15q^{139}-25q^{137}-47q^{135}-34q^{133}+6q^{131}+45q^{129}+50q^{127}+11q^{125}-35q^{123}-53q^{121}-27q^{119}+22q^{117}+49q^{115}+32q^{113}-8q^{111}-38q^{109}-31q^{107}+2q^{105}+26q^{103}+22q^{101}+4q^{99}-16q^{97}-16q^{95}-2q^{93}+8q^{91}+6q^{89}+q^{87}-4q^{85}-6q^{83}-q^{81}+3q^{79}+q^{75}-3q^{73}-8q^{71}-8q^{69}+q^{67}+14q^{65}+14q^{63}+3q^{61}-16q^{59}-28q^{57}-12q^{55}+22q^{53}+40q^{51}+26q^{49}-14q^{47}-48q^{45}-41q^{43}+7q^{41}+49q^{39}+52q^{37}+12q^{35}-39q^{33}-55q^{31}-26q^{29}+20q^{27}+49q^{25}+40q^{23}+2q^{21}-31q^{19}-36q^{17}-19q^{15}+11q^{13}+29q^{11}+24q^{9}+6q^{7}-12q^{5}-20q^{3}-15q-q^{-1}+11q^{-3}+12q^{-5}+7q^{-7}-q^{-9}-7q^{-11}-8q^{-13}-3q^{-15}+2q^{-17}+3q^{-19}+3q^{-21}+q^{-23}-q^{-25}-q^{-27}$
6	$q^{312}-q^{304}-q^{302}-q^{300}+q^{298}+q^{292}-q^{288}-2q^{286}+q^{284}+q^{282}+2q^{278}+q^{276}-3q^{272}-q^{270}+4q^{264}+5q^{262}+4q^{260}-3q^{258}-4q^{256}-6q^{254}-8q^{252}-2q^{250}+6q^{248}+14q^{246}+10q^{244}+7q^{242}-4q^{240}-18q^{238}-24q^{236}-18q^{234}+14q^{230}+34q^{228}+34q^{226}+15q^{224}-14q^{222}-39q^{220}-48q^{218}-42q^{216}+q^{214}+43q^{212}+70q^{210}+64q^{208}+25q^{206}-36q^{204}-93q^{202}-93q^{200}-49q^{198}+30q^{196}+100q^{194}+127q^{192}+79q^{190}-25q^{188}-113q^{186}-146q^{184}-96q^{182}+11q^{180}+127q^{178}+166q^{176}+100q^{174}-21q^{172}-133q^{170}-165q^{168}-98q^{166}+39q^{164}+142q^{162}+148q^{160}+65q^{158}-53q^{156}-130q^{154}-120q^{152}-26q^{150}+69q^{148}+105q^{146}+72q^{144}-60q^{140}-73q^{138}-29q^{136}+20q^{134}+43q^{132}+34q^{130}+8q^{128}-15q^{126}-25q^{124}-11q^{122}+3q^{120}+8q^{118}+5q^{116}-q^{114}-5q^{112}-6q^{110}-2q^{108}+3q^{106}+7q^{104}+4q^{102}-q^{100}-9q^{98}-16q^{96}-20q^{94}-7q^{92}+24q^{90}+34q^{88}+31q^{86}+2q^{84}-39q^{82}-70q^{80}-52q^{78}+19q^{76}+79q^{74}+104q^{72}+57q^{70}-38q^{68}-128q^{66}-135q^{64}-42q^{62}+77q^{60}+161q^{58}+142q^{56}+32q^{54}-110q^{52}-179q^{50}-131q^{48}-12q^{46}+117q^{44}+171q^{42}+125q^{40}+2q^{38}-108q^{36}-143q^{34}-105q^{32}-10q^{30}+83q^{28}+123q^{26}+88q^{24}+18q^{22}-47q^{20}-87q^{18}-77q^{16}-27q^{14}+27q^{12}+54q^{10}+55q^{8}+35q^{6}-3q^{4}-33q^{2}-39-27q^{-2}-9q^{-4}+9q^{-6}+24q^{-8}+23q^{-10}+11q^{-12}-q^{-14}-10q^{-16}-13q^{-18}-12q^{-20}-3q^{-22}+3q^{-24}+5q^{-26}+5q^{-28}+4q^{-30}+2q^{-32}-2q^{-34}-q^{-36}-q^{-38}-q^{-40}-q^{-42}+q^{-46}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{24}-q^{22}-2q^{20}-q^{18}+q^{16}+q^{14}+3q^{12}+2q^{10}+2q^{8}+q^{6}-q^{4}-1$
1,1	$q^{68}+2q^{64}-2q^{62}+4q^{60}-4q^{58}+4q^{56}-8q^{54}+7q^{52}-8q^{50}+8q^{48}-6q^{46}+q^{44}+2q^{42}-10q^{40}+10q^{38}-18q^{36}+14q^{34}-18q^{32}+12q^{30}-13q^{28}+10q^{26}-2q^{24}+6q^{22}+10q^{20}-2q^{18}+16q^{16}-6q^{14}+10q^{12}-8q^{10}+2q^{8}-4q^{6}-q^{4}-2q^{2}+q^{-4}$
2,0	$q^{62}+q^{60}+2q^{58}+2q^{56}+2q^{54}-q^{50}-3q^{48}-4q^{46}-7q^{44}-6q^{42}-3q^{40}-2q^{38}+q^{34}+4q^{32}+3q^{30}+4q^{28}+4q^{26}+5q^{24}+2q^{22}+3q^{20}+q^{18}-q^{16}-q^{14}-q^{8}+q^{4}-1$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{54}+q^{50}+q^{48}-3q^{40}-q^{38}-2q^{36}-6q^{34}-4q^{32}-4q^{30}-3q^{28}+5q^{24}+7q^{22}+9q^{20}+7q^{18}+7q^{16}+q^{14}-2q^{12}-q^{10}-4q^{8}-4q^{6}+q^{-2}$
1,0,0	$-q^{31}-q^{29}-3q^{27}-2q^{25}-2q^{23}+q^{21}+2q^{19}+4q^{17}+4q^{15}+3q^{13}+2q^{11}-2q^{5}-q$
1,0,1	$q^{88}+2q^{84}+q^{80}+q^{78}-2q^{76}+q^{74}-2q^{72}-3q^{70}+q^{68}-4q^{66}+5q^{64}+q^{62}+3q^{60}+9q^{58}-4q^{56}+11q^{54}-9q^{52}-3q^{50}-12q^{48}-18q^{46}-14q^{44}-20q^{42}-7q^{40}-11q^{38}+12q^{36}+3q^{34}+24q^{32}+17q^{30}+19q^{28}+24q^{26}+3q^{24}+15q^{22}-4q^{20}-2q^{18}-5q^{16}-8q^{14}-5q^{12}-3q^{10}-3q^{8}-q^{6}-q^{2}+2+q^{-4}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{68}+q^{66}+2q^{64}+3q^{62}+3q^{60}+2q^{58}+3q^{56}-4q^{52}-6q^{50}-8q^{48}-12q^{46}-15q^{44}-12q^{42}-9q^{40}-6q^{38}+q^{36}+11q^{34}+14q^{32}+18q^{30}+21q^{28}+17q^{26}+10q^{24}+5q^{22}-q^{20}-7q^{18}-10q^{16}-7q^{14}-5q^{12}-4q^{10}-q^{8}+2q^{6}+q^{4}+q^{2}+1$
1,0,0,0	$-q^{38}-q^{36}-3q^{34}-3q^{32}-3q^{30}-2q^{28}+q^{26}+2q^{24}+5q^{22}+5q^{20}+5q^{18}+3q^{16}+2q^{14}-q^{10}-q^{8}-2q^{6}-q^{2}$

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{74}+q^{70}+2q^{66}-q^{64}+q^{62}-2q^{60}+q^{58}-2q^{56}-2q^{52}-2q^{50}-3q^{48}-6q^{46}-4q^{44}-7q^{42}-3q^{40}-6q^{38}+3q^{36}+q^{34}+10q^{32}+7q^{30}+12q^{28}+8q^{26}+9q^{24}+5q^{22}+q^{20}-q^{18}-4q^{16}-2q^{14}-5q^{12}-2q^{10}-4q^{8}+q^{6}-q^{4}+q^{2}+q^{-2}$

G2 Invariants.

Weight

Invariant

1,0

$q^{128}+q^{124}-q^{122}+q^{120}-q^{116}+2q^{114}-2q^{112}+2q^{110}-2q^{108}-3q^{102}+4q^{100}-5q^{98}+q^{96}-q^{94}-4q^{92}+3q^{90}-5q^{88}-q^{86}+q^{84}-5q^{82}+2q^{80}-2q^{78}-4q^{76}+6q^{74}-5q^{72}+3q^{70}-3q^{66}+6q^{64}-4q^{62}+5q^{60}+2q^{56}+4q^{54}-2q^{52}+7q^{50}-q^{48}+3q^{46}+3q^{44}-2q^{42}+5q^{40}+2q^{38}-2q^{36}+6q^{34}-3q^{32}+q^{30}+2q^{28}-5q^{26}+5q^{24}-4q^{22}+q^{20}-3q^{16}+2q^{14}-2q^{12}-q^{8}-q^{4}-q^{2}+q^{-4}$

.

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["10 126"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-2t^{2}+4t-5+4t^{-1}-2t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{6}+4z^{4}+5z^{2}+1$

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]=

{ 19, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{4}a^{6}-4z^{2}a^{6}-4a^{6}+z^{6}a^{4}+6z^{4}a^{4}+12z^{2}a^{4}+7a^{4}-z^{4}a^{2}-3z^{2}a^{2}-2a^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"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["10 126"];

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]=

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

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]=

{}

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, -9)

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 $20$ $-72$ $200$ ${\frac {1222}{3}}$ ${\frac {194}{3}}$ $-1440$ $-2416$ $-384$ $-392$ ${\frac {4000}{3}}$ $2592$ ${\frac {24440}{3}}$ ${\frac {3880}{3}}$ ${\frac {87967}{6}}$ $-{\frac {310}{3}}$ ${\frac {56342}{9}}$ ${\frac {3109}{18}}$ ${\frac {5119}{6}}$

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 $s=$-2 is the signature of 10 126. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$-1+q^{-1}+2q^{-2}-4q^{-3}+q^{-4}+6q^{-5}-7q^{-6}+10q^{-8}-9q^{-9}-q^{-10}+10q^{-11}-7q^{-12}-3q^{-13}+8q^{-14}-4q^{-15}-4q^{-16}+5q^{-17}-q^{-18}-3q^{-19}+2q^{-20}-q^{-22}+q^{-23}$
3	$q^{4}-q^{3}-q^{2}-q+2+2q^{-1}-q^{-2}-3q^{-3}-q^{-4}+4q^{-5}+4q^{-6}-2q^{-7}-9q^{-8}+3q^{-9}+10q^{-10}+3q^{-11}-16q^{-12}-q^{-13}+13q^{-14}+8q^{-15}-19q^{-16}-3q^{-17}+14q^{-18}+8q^{-19}-16q^{-20}-6q^{-21}+11q^{-22}+9q^{-23}-10q^{-24}-9q^{-25}+5q^{-26}+10q^{-27}-2q^{-28}-10q^{-29}-q^{-30}+7q^{-31}+4q^{-32}-6q^{-33}-3q^{-34}+2q^{-35}+4q^{-36}-2q^{-37}-q^{-38}+2q^{-40}-q^{-41}+q^{-44}-q^{-45}$
4	$-q^{8}+q^{7}+2q^{6}-q^{4}-5q^{3}-q^{2}+5q+4+4q^{-1}-8q^{-2}-10q^{-3}+3q^{-4}+4q^{-5}+15q^{-6}+q^{-7}-15q^{-8}-7q^{-9}-10q^{-10}+22q^{-11}+19q^{-12}-7q^{-13}-13q^{-14}-33q^{-15}+18q^{-16}+34q^{-17}+6q^{-18}-10q^{-19}-51q^{-20}+11q^{-21}+39q^{-22}+13q^{-23}-3q^{-24}-60q^{-25}+8q^{-26}+40q^{-27}+14q^{-28}-q^{-29}-59q^{-30}+6q^{-31}+36q^{-32}+14q^{-33}+5q^{-34}-54q^{-35}+26q^{-37}+14q^{-38}+16q^{-39}-41q^{-40}-8q^{-41}+8q^{-42}+9q^{-43}+27q^{-44}-21q^{-45}-9q^{-46}-7q^{-47}-3q^{-48}+26q^{-49}-4q^{-50}-10q^{-52}-11q^{-53}+14q^{-54}+7q^{-56}-3q^{-57}-9q^{-58}+5q^{-59}-3q^{-60}+5q^{-61}+q^{-62}-4q^{-63}+3q^{-64}-3q^{-65}+q^{-66}+q^{-67}-2q^{-68}+2q^{-69}-q^{-70}-q^{-73}+q^{-74}$
5	$-q^{11}+2q^{9}+2q^{8}-q^{6}-6q^{5}-5q^{4}+3q^{3}+8q^{2}+8q+4-7q^{-1}-17q^{-2}-11q^{-3}+3q^{-4}+16q^{-5}+22q^{-6}+11q^{-7}-12q^{-8}-29q^{-9}-27q^{-10}-q^{-11}+27q^{-12}+44q^{-13}+26q^{-14}-20q^{-15}-56q^{-16}-47q^{-17}-2q^{-18}+60q^{-19}+77q^{-20}+20q^{-21}-59q^{-22}-89q^{-23}-50q^{-24}+53q^{-25}+111q^{-26}+60q^{-27}-45q^{-28}-107q^{-29}-84q^{-30}+37q^{-31}+123q^{-32}+79q^{-33}-34q^{-34}-107q^{-35}-97q^{-36}+28q^{-37}+123q^{-38}+84q^{-39}-30q^{-40}-108q^{-41}-94q^{-42}+24q^{-43}+118q^{-44}+86q^{-45}-25q^{-46}-103q^{-47}-92q^{-48}+14q^{-49}+104q^{-50}+86q^{-51}-5q^{-52}-85q^{-53}-88q^{-54}-10q^{-55}+71q^{-56}+79q^{-57}+25q^{-58}-45q^{-59}-71q^{-60}-37q^{-61}+22q^{-62}+53q^{-63}+43q^{-64}+q^{-65}-32q^{-66}-42q^{-67}-17q^{-68}+13q^{-69}+30q^{-70}+23q^{-71}+8q^{-72}-17q^{-73}-24q^{-74}-13q^{-75}+2q^{-76}+12q^{-77}+20q^{-78}+6q^{-79}-7q^{-80}-10q^{-81}-9q^{-82}-4q^{-83}+8q^{-84}+7q^{-85}+3q^{-86}+q^{-87}-4q^{-88}-6q^{-89}+q^{-91}+4q^{-93}+q^{-94}-3q^{-95}-3q^{-98}+2q^{-99}+2q^{-100}-q^{-101}+q^{-103}-2q^{-104}+q^{-106}+q^{-109}-q^{-110}$
6	$q^{20}-q^{19}-q^{18}-q^{14}+5q^{13}+q^{12}-2q^{9}-6q^{8}-10q^{7}+4q^{6}+4q^{5}+9q^{4}+12q^{3}+10q^{2}-5q-25-14q^{-1}-14q^{-2}-3q^{-3}+18q^{-4}+40q^{-5}+33q^{-6}-5q^{-7}-15q^{-8}-41q^{-9}-57q^{-10}-32q^{-11}+30q^{-12}+73q^{-13}+60q^{-14}+55q^{-15}-6q^{-16}-97q^{-17}-125q^{-18}-65q^{-19}+35q^{-20}+95q^{-21}+165q^{-22}+117q^{-23}-51q^{-24}-179q^{-25}-194q^{-26}-84q^{-27}+47q^{-28}+234q^{-29}+259q^{-30}+59q^{-31}-160q^{-32}-278q^{-33}-203q^{-34}-46q^{-35}+241q^{-36}+349q^{-37}+154q^{-38}-113q^{-39}-303q^{-40}-263q^{-41}-117q^{-42}+223q^{-43}+380q^{-44}+195q^{-45}-84q^{-46}-300q^{-47}-273q^{-48}-148q^{-49}+210q^{-50}+384q^{-51}+202q^{-52}-76q^{-53}-295q^{-54}-270q^{-55}-152q^{-56}+205q^{-57}+380q^{-58}+203q^{-59}-72q^{-60}-289q^{-61}-267q^{-62}-157q^{-63}+191q^{-64}+366q^{-65}+213q^{-66}-49q^{-67}-263q^{-68}-258q^{-69}-180q^{-70}+142q^{-71}+322q^{-72}+226q^{-73}+11q^{-74}-191q^{-75}-225q^{-76}-216q^{-77}+47q^{-78}+228q^{-79}+216q^{-80}+88q^{-81}-73q^{-82}-142q^{-83}-222q^{-84}-56q^{-85}+91q^{-86}+149q^{-87}+120q^{-88}+39q^{-89}-21q^{-90}-156q^{-91}-95q^{-92}-25q^{-93}+42q^{-94}+70q^{-95}+72q^{-96}+67q^{-97}-52q^{-98}-47q^{-99}-52q^{-100}-28q^{-101}-9q^{-102}+28q^{-103}+67q^{-104}+5q^{-105}+14q^{-106}-13q^{-107}-22q^{-108}-36q^{-109}-14q^{-110}+24q^{-111}-q^{-112}+23q^{-113}+12q^{-114}+7q^{-115}-17q^{-116}-14q^{-117}+4q^{-118}-15q^{-119}+5q^{-120}+6q^{-121}+13q^{-122}-3q^{-123}-3q^{-124}+7q^{-125}-11q^{-126}-3q^{-127}-2q^{-128}+7q^{-129}-q^{-130}-q^{-131}+8q^{-132}-4q^{-133}-2q^{-134}-3q^{-135}+3q^{-136}-q^{-137}-2q^{-138}+6q^{-139}-q^{-140}-q^{-141}-2q^{-142}+q^{-143}-2q^{-145}+3q^{-146}-q^{-149}-q^{-152}+q^{-153}$

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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