10 6

10_5

10_7

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 6's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 6 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X11,14,12,15 X3,13,4,12 X13,3,14,2 X5,16,6,17 X7,18,8,19 X9,20,10,1 X17,6,18,7 X19,8,20,9 X15,10,16,11
Gauss code	-1, 4, -3, 1, -5, 8, -6, 9, -7, 10, -2, 3, -4, 2, -10, 5, -8, 6, -9, 7
Dowker-Thistlethwaite code	4 12 16 18 20 14 2 10 6 8
Conway Notation	[532]

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

[edit Notes on presentations of 10 6]

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 6"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X11,14,12,15 X3,13,4,12 X13,3,14,2 X5,16,6,17 X7,18,8,19 X9,20,10,1 X17,6,18,7 X19,8,20,9 X15,10,16,11

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 12 16 18 20 14 2 10 6 8

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[532]

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]=

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

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-13][1] Hyperbolic Volume 8.39094 A-Polynomial See Data:10 6/A-polynomial

[edit Notes for 10 6'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 10 6's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+6t^{2}-7t+7-7t^{-1}+6t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-6z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 37, -4 }
Jones polynomial	$1-q^{-1}+3q^{-2}-4q^{-3}+5q^{-4}-6q^{-5}+6q^{-6}-5q^{-7}+3q^{-8}-2q^{-9}+q^{-10}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{8}+3z^{2}a^{8}+a^{8}-z^{6}a^{6}-4z^{4}a^{6}-4z^{2}a^{6}-a^{6}-z^{6}a^{4}-4z^{4}a^{4}-4z^{2}a^{4}-2a^{4}+z^{4}a^{2}+4z^{2}a^{2}+3a^{2}$
Kauffman polynomial (db, data sources)	$z^{4}a^{12}-2z^{2}a^{12}+2z^{5}a^{11}-4z^{3}a^{11}+za^{11}+2z^{6}a^{10}-3z^{4}a^{10}+z^{2}a^{10}+2z^{7}a^{9}-4z^{5}a^{9}+4z^{3}a^{9}+2z^{8}a^{8}-7z^{6}a^{8}+12z^{4}a^{8}-5z^{2}a^{8}+a^{8}+z^{9}a^{7}-3z^{7}a^{7}+5z^{5}a^{7}-2z^{3}a^{7}+3z^{8}a^{6}-12z^{6}a^{6}+18z^{4}a^{6}-10z^{2}a^{6}+a^{6}+z^{9}a^{5}-4z^{7}a^{5}+8z^{5}a^{5}-10z^{3}a^{5}+3za^{5}+z^{8}a^{4}-2z^{6}a^{4}-3z^{4}a^{4}+5z^{2}a^{4}-2a^{4}+z^{7}a^{3}-3z^{5}a^{3}+2za^{3}+z^{6}a^{2}-5z^{4}a^{2}+7z^{2}a^{2}-3a^{2}$
The A2 invariant	$q^{30}-q^{22}+q^{20}-q^{18}-q^{14}-2q^{12}+q^{10}+2q^{6}+q^{4}+q^{2}+1$
The G2 invariant	$q^{162}-q^{160}+2q^{158}-3q^{156}+q^{154}-3q^{150}+5q^{148}-6q^{146}+6q^{144}-4q^{142}+4q^{138}-7q^{136}+9q^{134}-8q^{132}+6q^{130}-4q^{128}+7q^{124}-9q^{122}+13q^{120}-10q^{118}+6q^{116}-7q^{112}+9q^{110}-9q^{108}+5q^{106}+5q^{104}-9q^{102}+7q^{100}-q^{98}-7q^{96}+14q^{94}-17q^{92}+11q^{90}-3q^{88}-7q^{86}+18q^{84}-20q^{82}+18q^{80}-11q^{78}-2q^{76}+9q^{74}-15q^{72}+15q^{70}-14q^{68}+5q^{66}+5q^{64}-11q^{62}+10q^{60}-7q^{58}-4q^{56}+10q^{54}-13q^{52}+5q^{50}-9q^{46}+18q^{44}-17q^{42}+10q^{40}-q^{38}-8q^{36}+14q^{34}-13q^{32}+11q^{30}-4q^{28}+2q^{26}+4q^{24}-5q^{22}+7q^{20}-4q^{18}+4q^{16}+2q^{10}-q^{8}+2q^{6}+q^{2}$

Further Quantum Invariants

Further quantum knot invariants for 10_6.

A1 Invariants.

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

A2 Invariants.

Weight	Invariant
1,0	$q^{30}-q^{22}+q^{20}-q^{18}-q^{14}-2q^{12}+q^{10}+2q^{6}+q^{4}+q^{2}+1$
1,1	$q^{84}-2q^{82}+4q^{80}-8q^{78}+11q^{76}-14q^{74}+18q^{72}-20q^{70}+23q^{68}-22q^{66}+22q^{64}-28q^{62}+26q^{60}-26q^{58}+24q^{56}-20q^{54}+11q^{52}+6q^{50}-20q^{48}+38q^{46}-54q^{44}+68q^{42}-76q^{40}+82q^{38}-79q^{36}+70q^{34}-58q^{32}+40q^{30}-19q^{28}-4q^{26}+24q^{24}-36q^{22}+43q^{20}-50q^{18}+42q^{16}-38q^{14}+28q^{12}-22q^{10}+18q^{8}-8q^{6}+10q^{4}-2q^{2}+4+q^{-4}$
2,0	$q^{76}-q^{70}+q^{66}-2q^{60}-q^{58}-3q^{52}+4q^{48}+2q^{46}+q^{42}+4q^{40}-q^{38}-3q^{36}+q^{34}-q^{30}-3q^{24}-q^{22}+2q^{20}-q^{18}-3q^{16}+3q^{12}-q^{10}-q^{8}+2q^{6}+3q^{4}+q^{2}+1+q^{-2}+q^{-4}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{68}-q^{66}+q^{62}-3q^{60}+3q^{56}-3q^{54}-q^{52}+6q^{50}-2q^{48}-2q^{46}+4q^{44}-q^{42}-q^{40}+q^{38}+2q^{36}-2q^{32}+3q^{30}-5q^{26}-7q^{20}-q^{18}+q^{16}-2q^{14}+4q^{12}+3q^{10}+2q^{8}+3q^{6}+2q^{4}+1$
1,0,0	$q^{39}+q^{35}-q^{33}+q^{31}-q^{29}+q^{27}-q^{25}-q^{21}-2q^{19}-q^{17}-2q^{15}+q^{13}+3q^{9}+q^{7}+2q^{5}+q^{3}+q$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{86}-q^{82}+q^{80}+q^{78}-3q^{76}-2q^{74}+2q^{72}-3q^{68}+4q^{64}-2q^{60}+3q^{58}+q^{56}-2q^{54}+2q^{52}+3q^{50}-q^{48}+q^{46}+5q^{44}+q^{42}-q^{40}+q^{38}+3q^{36}-4q^{34}-6q^{32}-4q^{30}-6q^{28}-7q^{26}-5q^{24}-2q^{22}+4q^{18}+4q^{16}+5q^{14}+5q^{12}+5q^{10}+3q^{8}+2q^{6}+q^{4}+q^{2}$
1,0,0,0	$q^{48}+q^{44}+q^{38}-q^{36}+q^{34}-q^{32}-q^{28}-q^{26}-2q^{24}-2q^{22}-q^{20}-2q^{18}+q^{16}+3q^{12}+2q^{10}+2q^{8}+2q^{6}+q^{4}+q^{2}$

B2 Invariants.

Weight	Invariant
0,1	$q^{68}-q^{66}+2q^{64}-3q^{62}+3q^{60}-4q^{58}+5q^{56}-5q^{54}+5q^{52}-4q^{50}+2q^{48}-2q^{44}+5q^{42}-7q^{40}+9q^{38}-10q^{36}+10q^{34}-10q^{32}+7q^{30}-6q^{28}+3q^{26}-2q^{24}-2q^{22}+3q^{20}-5q^{18}+5q^{16}-4q^{14}+6q^{12}-3q^{10}+4q^{8}-q^{6}+2q^{4}+1$
1,0	$q^{110}-q^{106}-q^{104}+q^{102}+2q^{100}-q^{98}-3q^{96}-2q^{94}+2q^{92}+4q^{90}-4q^{86}-3q^{84}+3q^{82}+6q^{80}-5q^{76}-2q^{74}+4q^{72}+3q^{70}-3q^{68}-3q^{66}+2q^{64}+4q^{62}-3q^{58}+3q^{54}+q^{52}-3q^{50}-q^{48}+2q^{46}+2q^{44}-3q^{42}-5q^{40}+5q^{36}+q^{34}-6q^{32}-6q^{30}+2q^{28}+5q^{26}-4q^{22}-2q^{20}+4q^{18}+3q^{16}+q^{14}-q^{12}+q^{10}+2q^{8}+2q^{6}+q^{-2}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{94}-q^{92}+q^{90}-2q^{88}+2q^{86}-3q^{84}+2q^{82}-3q^{80}+4q^{78}-4q^{76}+3q^{74}-3q^{72}+5q^{70}-2q^{68}+q^{66}+4q^{60}-4q^{58}+5q^{56}-6q^{54}+8q^{52}-7q^{50}+8q^{48}-8q^{46}+7q^{44}-5q^{42}+5q^{40}-5q^{38}+q^{36}-3q^{34}-3q^{32}-2q^{30}-6q^{28}+q^{26}-6q^{24}+3q^{22}-3q^{20}+7q^{18}+6q^{14}+q^{12}+5q^{10}+q^{8}+2q^{6}+q^{2}$

G2 Invariants.

Weight

Invariant

1,0

$q^{162}-q^{160}+2q^{158}-3q^{156}+q^{154}-3q^{150}+5q^{148}-6q^{146}+6q^{144}-4q^{142}+4q^{138}-7q^{136}+9q^{134}-8q^{132}+6q^{130}-4q^{128}+7q^{124}-9q^{122}+13q^{120}-10q^{118}+6q^{116}-7q^{112}+9q^{110}-9q^{108}+5q^{106}+5q^{104}-9q^{102}+7q^{100}-q^{98}-7q^{96}+14q^{94}-17q^{92}+11q^{90}-3q^{88}-7q^{86}+18q^{84}-20q^{82}+18q^{80}-11q^{78}-2q^{76}+9q^{74}-15q^{72}+15q^{70}-14q^{68}+5q^{66}+5q^{64}-11q^{62}+10q^{60}-7q^{58}-4q^{56}+10q^{54}-13q^{52}+5q^{50}-9q^{46}+18q^{44}-17q^{42}+10q^{40}-q^{38}-8q^{36}+14q^{34}-13q^{32}+11q^{30}-4q^{28}+2q^{26}+4q^{24}-5q^{22}+7q^{20}-4q^{18}+4q^{16}+2q^{10}-q^{8}+2q^{6}+q^{2}$

.

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["10 6"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+6t^{2}-7t+7-7t^{-1}+6t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 37, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{4}a^{12}-2z^{2}a^{12}+2z^{5}a^{11}-4z^{3}a^{11}+za^{11}+2z^{6}a^{10}-3z^{4}a^{10}+z^{2}a^{10}+2z^{7}a^{9}-4z^{5}a^{9}+4z^{3}a^{9}+2z^{8}a^{8}-7z^{6}a^{8}+12z^{4}a^{8}-5z^{2}a^{8}+a^{8}+z^{9}a^{7}-3z^{7}a^{7}+5z^{5}a^{7}-2z^{3}a^{7}+3z^{8}a^{6}-12z^{6}a^{6}+18z^{4}a^{6}-10z^{2}a^{6}+a^{6}+z^{9}a^{5}-4z^{7}a^{5}+8z^{5}a^{5}-10z^{3}a^{5}+3za^{5}+z^{8}a^{4}-2z^{6}a^{4}-3z^{4}a^{4}+5z^{2}a^{4}-2a^{4}+z^{7}a^{3}-3z^{5}a^{3}+2za^{3}+z^{6}a^{2}-5z^{4}a^{2}+7z^{2}a^{2}-3a^{2}$

"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 6"];

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]=

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

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:

(-1, 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 $-4$ $32$ $8$ $-{\frac {110}{3}}$ ${\frac {158}{3}}$ $-128$ $-{\frac {1120}{3}}$ $-{\frac {544}{3}}$ $-256$ $-{\frac {32}{3}}$ $512$ ${\frac {440}{3}}$ $-{\frac {632}{3}}$ ${\frac {71009}{30}}$ $-{\frac {1298}{15}}$ ${\frac {68818}{45}}$ ${\frac {5983}{18}}$ ${\frac {929}{30}}$

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 10 6. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                     1 1 -1                       0 -3                 3 1   2 -5               2 1     -1 -7             3 2       1 -9           3 2         -1 -11         3 3           0 -13       2 3             1 -15     1 3               -2 -17   1 2                 1 -19   1                   -1 -21 1                     1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{2}-q+3q^{-1}-4q^{-2}-q^{-3}+9q^{-4}-8q^{-5}-5q^{-6}+17q^{-7}-9q^{-8}-13q^{-9}+23q^{-10}-7q^{-11}-20q^{-12}+26q^{-13}-4q^{-14}-23q^{-15}+25q^{-16}-q^{-17}-19q^{-18}+17q^{-19}-11q^{-21}+8q^{-22}-5q^{-24}+4q^{-25}-2q^{-27}+q^{-28}$
3	$q^{6}-q^{5}+2q^{2}-3q+2q^{-1}+4q^{-2}-8q^{-3}-2q^{-4}+7q^{-5}+12q^{-6}-15q^{-7}-12q^{-8}+10q^{-9}+24q^{-10}-12q^{-11}-26q^{-12}+2q^{-13}+34q^{-14}+q^{-15}-31q^{-16}-13q^{-17}+32q^{-18}+19q^{-19}-27q^{-20}-26q^{-21}+23q^{-22}+32q^{-23}-20q^{-24}-35q^{-25}+15q^{-26}+39q^{-27}-13q^{-28}-38q^{-29}+8q^{-30}+36q^{-31}-5q^{-32}-28q^{-33}-q^{-34}+23q^{-35}-q^{-36}-11q^{-37}-2q^{-38}+6q^{-39}-q^{-40}-q^{-41}+3q^{-42}-4q^{-44}-q^{-45}+5q^{-46}+q^{-47}-3q^{-48}-3q^{-49}+3q^{-50}+q^{-51}-2q^{-53}+q^{-54}$
4	$q^{12}-q^{11}-q^{8}+3q^{7}-3q^{6}+q^{5}+2q^{4}-4q^{3}+5q^{2}-8q+3+10q^{-1}-6q^{-2}+7q^{-3}-22q^{-4}-q^{-5}+23q^{-6}+q^{-7}+20q^{-8}-44q^{-9}-19q^{-10}+27q^{-11}+10q^{-12}+53q^{-13}-52q^{-14}-39q^{-15}+11q^{-16}-4q^{-17}+89q^{-18}-37q^{-19}-34q^{-20}-3q^{-21}-46q^{-22}+93q^{-23}-19q^{-24}+5q^{-25}+9q^{-26}-95q^{-27}+65q^{-28}-21q^{-29}+53q^{-30}+46q^{-31}-126q^{-32}+26q^{-33}-41q^{-34}+91q^{-35}+86q^{-36}-142q^{-37}-4q^{-38}-59q^{-39}+113q^{-40}+113q^{-41}-148q^{-42}-24q^{-43}-72q^{-44}+122q^{-45}+129q^{-46}-136q^{-47}-36q^{-48}-88q^{-49}+107q^{-50}+138q^{-51}-95q^{-52}-33q^{-53}-104q^{-54}+63q^{-55}+122q^{-56}-40q^{-57}-2q^{-58}-100q^{-59}+7q^{-60}+78q^{-61}-2q^{-62}+32q^{-63}-69q^{-64}-21q^{-65}+30q^{-66}+q^{-67}+45q^{-68}-31q^{-69}-19q^{-70}+3q^{-71}-8q^{-72}+34q^{-73}-9q^{-74}-7q^{-75}-3q^{-76}-11q^{-77}+17q^{-78}-q^{-79}-q^{-81}-7q^{-82}+5q^{-83}+q^{-85}-2q^{-87}+q^{-88}$

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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