10 2

10_1

10_3

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 2 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X5,14,6,15 X3,13,4,12 X13,3,14,2 X7,16,8,17 X9,18,10,19 X11,20,12,1 X15,6,16,7 X17,8,18,9 X19,10,20,11
Gauss code	-1, 4, -3, 1, -2, 8, -5, 9, -6, 10, -7, 3, -4, 2, -8, 5, -9, 6, -10, 7
Dowker-Thistlethwaite code	4 12 14 16 18 20 2 6 8 10
Conway Notation	[712]

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

[edit Notes on presentations of 10 2]

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 2"];

In[4]:=

PD[K]

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

Out[4]=

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

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 12 14 16 18 20 2 6 8 10

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[712]

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]=

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

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-15][3] Hyperbolic Volume 5.11484 A-Polynomial See Data:10 2/A-polynomial

[edit Notes for 10 2's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $3$ Topological 4 genus $3$ Concordance genus $4$ Rasmussen s-Invariant -6

[edit Notes for 10 2's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^4+3 t^3-3 t^2+3 t-3+3 t^{-1} -3 t^{-2} +3 t^{-3} - t^{-4}$
Conway polynomial	$-z^8-5 z^6-5 z^4+2 z^2+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 23, -6 }
Jones polynomial	$q^{-1} - q^{-2} +2 q^{-3} -2 q^{-4} +3 q^{-5} -3 q^{-6} +3 q^{-7} -3 q^{-8} +2 q^{-9} -2 q^{-10} + q^{-11}$
HOMFLY-PT polynomial (db, data sources)	$z^6 a^8+5 z^4 a^8+6 z^2 a^8+a^8-z^8 a^6-7 z^6 a^6-16 z^4 a^6-14 z^2 a^6-4 a^6+z^6 a^4+6 z^4 a^4+10 z^2 a^4+4 a^4$
Kauffman polynomial (db, data sources)	$z^2 a^{14}+2 z^3 a^{13}-z a^{13}+2 z^4 a^{12}-z^2 a^{12}+2 z^5 a^{11}-2 z^3 a^{11}-z a^{11}+2 z^6 a^{10}-4 z^4 a^{10}+2 z^7 a^9-6 z^5 a^9+2 z^3 a^9+z a^9+2 z^8 a^8-9 z^6 a^8+11 z^4 a^8-5 z^2 a^8+a^8+z^9 a^7-4 z^7 a^7+2 z^5 a^7+3 z^3 a^7-z a^7+3 z^8 a^6-18 z^6 a^6+33 z^4 a^6-21 z^2 a^6+4 a^6+z^9 a^5-6 z^7 a^5+10 z^5 a^5-3 z^3 a^5-2 z a^5+z^8 a^4-7 z^6 a^4+16 z^4 a^4-14 z^2 a^4+4 a^4$
The A2 invariant	$q^{32}-q^{26}-q^{24}-q^{22}-q^{20}+q^{18}+q^{14}+q^{10}+q^8+q^6+q^4$
The G2 invariant	$q^{182}-q^{180}+q^{178}-q^{176}-q^{174}-q^{170}+2 q^{168}-2 q^{166}+q^{164}+q^{158}-q^{156}+q^{154}-q^{152}+q^{150}-q^{146}+2 q^{144}+q^{140}+q^{134}+q^{128}-q^{120}+q^{118}-q^{116}-q^{114}-q^{110}-q^{106}-q^{104}-q^{102}-q^{98}-2 q^{92}+2 q^{90}-2 q^{88}+q^{86}-q^{84}-q^{82}+q^{80}-q^{78}+2 q^{76}-q^{74}-q^{70}-q^{64}+q^{56}+q^{54}-q^{52}+3 q^{50}-2 q^{48}+2 q^{46}+q^{44}-q^{42}+4 q^{40}-2 q^{38}+3 q^{36}+q^{34}+q^{32}+q^{30}-q^{28}+2 q^{26}+q^{22}$

Further Quantum Invariants

Further quantum knot invariants for 10_2.

A1 Invariants.

Weight	Invariant
1	$q^{23}-q^{21}-q^{17}+q^9+q^5+q$
2	$q^{62}-q^{60}-q^{58}+q^{56}-q^{54}+q^{50}+q^{48}-q^{36}-q^{30}-q^{28}+q^{18}+2 q^{12}+q^{10}-q^8+q^6+q^4-q^2+ q^{-2}$
3	$q^{117}-q^{115}-q^{113}+q^{109}-q^{105}+2 q^{103}+q^{101}-q^{99}-3 q^{97}+2 q^{95}+2 q^{93}-q^{91}-3 q^{89}+3 q^{85}-3 q^{81}-q^{79}+2 q^{77}+q^{75}+q^{73}-q^{71}+q^{67}+3 q^{65}-2 q^{61}-q^{59}+q^{57}-q^{55}-2 q^{53}+q^{49}-q^{47}-q^{45}+q^{41}-q^{37}-q^{35}+q^{27}+q^{21}+2 q^{19}+q^{17}-q^{15}-q^{13}+2 q^{11}+2 q^9-2 q^5+2 q+ q^{-1} - q^{-3} - q^{-5} + q^{-9}$
4	$q^{188}-q^{186}-q^{184}+2 q^{178}-q^{176}+q^{174}-q^{170}-2 q^{166}+2 q^{164}+2 q^{162}-3 q^{158}-3 q^{156}+3 q^{154}+4 q^{152}-4 q^{148}-4 q^{146}+4 q^{144}+4 q^{142}-3 q^{138}-3 q^{136}+3 q^{134}+3 q^{132}+q^{130}-3 q^{128}-4 q^{126}+2 q^{122}+2 q^{120}-2 q^{116}-3 q^{114}-q^{112}+3 q^{110}+3 q^{108}+q^{106}-2 q^{104}-2 q^{102}+3 q^{98}+2 q^{96}-q^{94}-q^{92}+3 q^{88}+q^{86}-3 q^{84}-2 q^{82}+4 q^{78}+2 q^{76}-4 q^{74}-4 q^{72}-q^{70}+4 q^{68}+3 q^{66}-3 q^{64}-4 q^{62}-2 q^{60}+2 q^{58}+3 q^{56}-q^{54}-2 q^{52}-2 q^{50}+2 q^{46}+q^{44}-q^{40}-q^{38}+q^{36}+q^{30}+q^{28}+2 q^{26}-q^{22}+4 q^{16}+2 q^{14}-q^{12}-2 q^{10}-3 q^8+2 q^6+3 q^4+2 q^2-4 q^{-2} - q^{-4} + q^{-6} +2 q^{-8} +2 q^{-10} - q^{-12} - q^{-14} - q^{-16} + q^{-20}$
5	$q^{275}-q^{273}-q^{271}+q^{265}+q^{263}+q^{261}-q^{259}-q^{257}-2 q^{255}+q^{251}+3 q^{249}+2 q^{247}-2 q^{245}-4 q^{243}-3 q^{241}+2 q^{239}+5 q^{237}+4 q^{235}-2 q^{233}-5 q^{231}-5 q^{229}+6 q^{225}+5 q^{223}-2 q^{221}-2 q^{219}-2 q^{217}-q^{215}+q^{213}-2 q^{209}+q^{207}+4 q^{205}+3 q^{203}-q^{201}-5 q^{199}-5 q^{197}-q^{195}+6 q^{193}+8 q^{191}+3 q^{189}-5 q^{187}-7 q^{185}-5 q^{183}+q^{181}+6 q^{179}+6 q^{177}+2 q^{175}-q^{173}-6 q^{171}-6 q^{169}-3 q^{167}+q^{165}+6 q^{163}+7 q^{161}-5 q^{157}-6 q^{155}-3 q^{153}+3 q^{151}+6 q^{149}+2 q^{147}-3 q^{145}-3 q^{143}+2 q^{139}+2 q^{137}-2 q^{135}-4 q^{133}-q^{131}+2 q^{129}+5 q^{127}+4 q^{125}+q^{123}-3 q^{121}-4 q^{119}-q^{117}+3 q^{115}+5 q^{113}+4 q^{111}-q^{109}-5 q^{107}-7 q^{105}-q^{103}+5 q^{101}+7 q^{99}+3 q^{97}-4 q^{95}-10 q^{93}-6 q^{91}+3 q^{89}+9 q^{87}+7 q^{85}-q^{83}-9 q^{81}-10 q^{79}-3 q^{77}+7 q^{75}+9 q^{73}+4 q^{71}-3 q^{69}-8 q^{67}-6 q^{65}+5 q^{61}+5 q^{59}+q^{57}-2 q^{55}-4 q^{53}-3 q^{51}+2 q^{47}+3 q^{45}+q^{43}+q^{35}+q^{33}-q^{29}+2 q^{25}+2 q^{23}+3 q^{21}+q^{19}-2 q^{17}-4 q^{15}-2 q^{13}+q^{11}+4 q^9+5 q^7+2 q^5-2 q^3-5 q-4 q^{-1} +3 q^{-5} +5 q^{-7} +3 q^{-9} - q^{-11} -4 q^{-13} -3 q^{-15} - q^{-17} + q^{-19} +3 q^{-21} +2 q^{-23} - q^{-27} - q^{-29} - q^{-31} + q^{-35}$
6	$q^{378}-q^{376}-q^{374}+q^{368}+3 q^{364}-q^{362}-2 q^{360}-2 q^{358}-q^{356}+q^{354}+q^{352}+6 q^{350}-2 q^{346}-5 q^{344}-4 q^{342}+3 q^{338}+9 q^{336}+4 q^{334}-3 q^{332}-10 q^{330}-8 q^{328}-q^{326}+4 q^{324}+13 q^{322}+10 q^{320}-11 q^{316}-14 q^{314}-6 q^{312}-q^{310}+12 q^{308}+17 q^{306}+10 q^{304}-3 q^{302}-14 q^{300}-16 q^{298}-13 q^{296}+4 q^{294}+18 q^{292}+20 q^{290}+11 q^{288}-7 q^{286}-20 q^{284}-24 q^{282}-9 q^{280}+11 q^{278}+23 q^{276}+21 q^{274}+5 q^{272}-13 q^{270}-27 q^{268}-20 q^{266}-3 q^{264}+16 q^{262}+25 q^{260}+19 q^{258}+4 q^{256}-16 q^{254}-21 q^{252}-15 q^{250}-2 q^{248}+11 q^{246}+18 q^{244}+16 q^{242}+3 q^{240}-7 q^{238}-13 q^{236}-14 q^{234}-10 q^{232}+q^{230}+11 q^{228}+15 q^{226}+12 q^{224}+2 q^{222}-8 q^{220}-15 q^{218}-11 q^{216}-q^{214}+9 q^{212}+11 q^{210}+4 q^{208}-2 q^{206}-7 q^{204}-5 q^{202}+6 q^{198}+4 q^{196}-5 q^{194}-9 q^{192}-7 q^{190}+q^{188}+8 q^{186}+14 q^{184}+6 q^{182}-8 q^{180}-14 q^{178}-11 q^{176}+10 q^{172}+17 q^{170}+10 q^{168}-6 q^{166}-13 q^{164}-12 q^{162}-3 q^{160}+7 q^{158}+14 q^{156}+10 q^{154}-2 q^{152}-9 q^{150}-9 q^{148}-6 q^{146}+9 q^{142}+11 q^{140}+6 q^{138}-4 q^{134}-8 q^{132}-9 q^{130}-2 q^{128}+4 q^{126}+9 q^{124}+10 q^{122}+7 q^{120}-3 q^{118}-14 q^{116}-14 q^{114}-10 q^{112}+2 q^{110}+13 q^{108}+18 q^{106}+10 q^{104}-5 q^{102}-15 q^{100}-20 q^{98}-11 q^{96}+4 q^{94}+18 q^{92}+19 q^{90}+9 q^{88}-4 q^{86}-18 q^{84}-19 q^{82}-10 q^{80}+5 q^{78}+14 q^{76}+15 q^{74}+9 q^{72}-4 q^{70}-11 q^{68}-12 q^{66}-6 q^{64}+q^{62}+7 q^{60}+9 q^{58}+4 q^{56}+q^{54}-4 q^{52}-5 q^{50}-3 q^{48}+3 q^{44}+2 q^{42}+3 q^{40}+q^{38}-q^{36}+q^{32}+3 q^{30}+q^{28}+2 q^{26}-q^{24}-4 q^{22}-3 q^{20}-q^{18}+3 q^{16}+3 q^{14}+7 q^{12}+4 q^{10}-2 q^8-5 q^6-7 q^4-4 q^2-2+6 q^{-2} +8 q^{-4} +6 q^{-6} +2 q^{-8} -3 q^{-10} -6 q^{-12} -9 q^{-14} -2 q^{-16} +2 q^{-18} +5 q^{-20} +6 q^{-22} +4 q^{-24} + q^{-26} -5 q^{-28} -4 q^{-30} -3 q^{-32} - q^{-34} + q^{-36} +3 q^{-38} +3 q^{-40} - q^{-46} - q^{-48} - q^{-50} + q^{-54}$

A2 Invariants.

Weight	Invariant
1,0	$q^{32}-q^{26}-q^{24}-q^{22}-q^{20}+q^{18}+q^{14}+q^{10}+q^8+q^6+q^4$
1,1	$q^{92}-2 q^{90}+2 q^{88}-2 q^{86}+3 q^{84}-4 q^{82}+2 q^{80}-2 q^{78}+2 q^{76}-2 q^{74}+4 q^{72}+q^{68}-2 q^{66}-2 q^{64}-2 q^{60}+2 q^{58}+2 q^{54}+2 q^{52}+2 q^{50}-2 q^{48}+2 q^{46}-6 q^{44}+6 q^{42}-10 q^{40}+8 q^{38}-11 q^{36}+8 q^{34}-10 q^{32}+4 q^{30}-4 q^{28}+2 q^{26}+4 q^{24}-4 q^{22}+9 q^{20}-6 q^{18}+10 q^{16}-6 q^{14}+6 q^{12}-2 q^{10}+4 q^8+q^4$
2,0	$q^{80}-q^{74}-q^{72}-q^{70}-q^{68}+q^{66}+q^{64}+q^{62}+2 q^{58}+q^{56}+q^{54}+q^{50}-2 q^{46}-2 q^{44}-2 q^{42}-2 q^{40}-3 q^{38}-q^{36}-q^{34}+q^{30}+2 q^{28}+q^{26}+q^{24}+2 q^{22}+2 q^{20}+q^{14}+q^{12}+q^8+q^6+q^4$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{76}-q^{74}-q^{72}+q^{70}-q^{68}-q^{66}+q^{64}+q^{62}+q^{58}+q^{50}+q^{48}-q^{40}-2 q^{38}-q^{36}-2 q^{34}-2 q^{32}-2 q^{30}-2 q^{28}+q^{24}+q^{22}+3 q^{20}+3 q^{18}+2 q^{16}+2 q^{14}+2 q^{12}+q^8$
1,0,0	$q^{41}+q^{37}-q^{35}-2 q^{31}-q^{29}-q^{27}-q^{25}+q^{19}+2 q^{15}+q^{13}+2 q^{11}+q^9+q^7$
1,0,1	$q^{122}-2 q^{120}+q^{118}+q^{116}-2 q^{114}+3 q^{112}-2 q^{110}-q^{108}+q^{106}-2 q^{104}+q^{102}-q^{100}+4 q^{96}+q^{92}-2 q^{88}-2 q^{84}+q^{82}-3 q^{80}+2 q^{78}-q^{76}+3 q^{72}-4 q^{70}+5 q^{68}-5 q^{66}+4 q^{64}-q^{62}+5 q^{60}+3 q^{58}+6 q^{54}-6 q^{52}+6 q^{50}-10 q^{48}-q^{46}-8 q^{44}-7 q^{42}-3 q^{40}-7 q^{38}-3 q^{34}+3 q^{32}+q^{30}+6 q^{28}+2 q^{26}+6 q^{24}+4 q^{22}+5 q^{20}+6 q^{18}+2 q^{16}+4 q^{14}+q^{10}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{94}-q^{90}-2 q^{84}-2 q^{82}+q^{74}+3 q^{72}+2 q^{70}+2 q^{68}+2 q^{66}+2 q^{64}+q^{60}-q^{56}-q^{54}-q^{52}-q^{50}-3 q^{48}-4 q^{46}-5 q^{44}-4 q^{42}-6 q^{40}-4 q^{38}-q^{36}+q^{34}+2 q^{32}+4 q^{30}+5 q^{28}+5 q^{26}+4 q^{24}+4 q^{22}+3 q^{20}+2 q^{18}+q^{16}+q^{14}$
1,0,0,0	$q^{50}+q^{46}-q^{40}-q^{38}-q^{36}-2 q^{34}-q^{32}-2 q^{30}-q^{26}+q^{24}+q^{22}+2 q^{20}+2 q^{18}+2 q^{16}+2 q^{14}+q^{12}+q^{10}$

B2 Invariants.

Weight	Invariant
0,1	$q^{76}-q^{74}+q^{72}-q^{70}+q^{68}-q^{66}+q^{64}-q^{62}-q^{58}+q^{50}-q^{48}+2 q^{46}-2 q^{44}+2 q^{42}-3 q^{40}+2 q^{38}-3 q^{36}+2 q^{34}-2 q^{32}+q^{24}-q^{22}+3 q^{20}-q^{18}+2 q^{16}+2 q^{12}+q^8$
1,0	$q^{122}-q^{118}-q^{116}+q^{112}-q^{108}-q^{106}+q^{102}+q^{100}+q^{92}+q^{82}+q^{80}-q^{76}+q^{72}-q^{68}-q^{66}-q^{62}-q^{60}-q^{58}-2 q^{52}-2 q^{50}+q^{46}-q^{42}-q^{40}+q^{38}+2 q^{36}+q^{34}-q^{32}+q^{30}+2 q^{28}+2 q^{26}+q^{20}+2 q^{18}+q^{10}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{106}-q^{104}-q^{100}+q^{98}-q^{96}-q^{92}+q^{90}+q^{86}+q^{82}+q^{68}+2 q^{64}-q^{62}+2 q^{60}-2 q^{58}+2 q^{56}-2 q^{54}+q^{52}-4 q^{50}-4 q^{46}-2 q^{44}-4 q^{42}-3 q^{40}-2 q^{38}-q^{36}+q^{34}+q^{32}+5 q^{30}+2 q^{28}+5 q^{26}+2 q^{24}+4 q^{22}+q^{20}+2 q^{18}+q^{14}$

G2 Invariants.

Weight

Invariant

1,0

$q^{182}-q^{180}+q^{178}-q^{176}-q^{174}-q^{170}+2 q^{168}-2 q^{166}+q^{164}+q^{158}-q^{156}+q^{154}-q^{152}+q^{150}-q^{146}+2 q^{144}+q^{140}+q^{134}+q^{128}-q^{120}+q^{118}-q^{116}-q^{114}-q^{110}-q^{106}-q^{104}-q^{102}-q^{98}-2 q^{92}+2 q^{90}-2 q^{88}+q^{86}-q^{84}-q^{82}+q^{80}-q^{78}+2 q^{76}-q^{74}-q^{70}-q^{64}+q^{56}+q^{54}-q^{52}+3 q^{50}-2 q^{48}+2 q^{46}+q^{44}-q^{42}+4 q^{40}-2 q^{38}+3 q^{36}+q^{34}+q^{32}+q^{30}-q^{28}+2 q^{26}+q^{22}$

. </div></div>

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 2"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

$-z^8-5 z^6-5 z^4+2 z^2+1$

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]=

{ 23, -6 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^6 a^8+5 z^4 a^8+6 z^2 a^8+a^8-z^8 a^6-7 z^6 a^6-16 z^4 a^6-14 z^2 a^6-4 a^6+z^6 a^4+6 z^4 a^4+10 z^2 a^4+4 a^4$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^2 a^{14}+2 z^3 a^{13}-z a^{13}+2 z^4 a^{12}-z^2 a^{12}+2 z^5 a^{11}-2 z^3 a^{11}-z a^{11}+2 z^6 a^{10}-4 z^4 a^{10}+2 z^7 a^9-6 z^5 a^9+2 z^3 a^9+z a^9+2 z^8 a^8-9 z^6 a^8+11 z^4 a^8-5 z^2 a^8+a^8+z^9 a^7-4 z^7 a^7+2 z^5 a^7+3 z^3 a^7-z a^7+3 z^8 a^6-18 z^6 a^6+33 z^4 a^6-21 z^2 a^6+4 a^6+z^9 a^5-6 z^7 a^5+10 z^5 a^5-3 z^3 a^5-2 z a^5+z^8 a^4-7 z^6 a^4+16 z^4 a^4-14 z^2 a^4+4 a^4$

"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["10 2"];

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]=

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

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:

(2, -2)

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 $8$ $-16$ $32$ $\frac{268}{3}$ $\frac{164}{3}$ $-128$ $-\frac{1600}{3}$ $-\frac{544}{3}$ $-272$ $\frac{256}{3}$ $128$ $\frac{2144}{3}$ $\frac{1312}{3}$ $\frac{38551}{15}$ $-\frac{13084}{15}$ $\frac{137164}{45}$ $\frac{1817}{9}$ $\frac{6151}{15}$

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 $t^rq^j$ 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 $s=$-6 is the signature of 10 2. 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 -3                       0 -5                 2 1   1 -7               1 1     0 -9             2 1       1 -11           1 1         0 -13         2 2           0 -15       1 1             0 -17     1 2               -1 -19   1 1                 0 -21   1                   -1 -23 1                     1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-7$ $i=-5$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

The Coloured Jones Polynomials

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

<p>

$n$	$J_n$
2	$1- q^{-1} - q^{-2} +3 q^{-3} - q^{-4} -3 q^{-5} +5 q^{-6} -5 q^{-8} +5 q^{-9} + q^{-10} -6 q^{-11} +5 q^{-12} + q^{-13} -6 q^{-14} +4 q^{-15} + q^{-16} -5 q^{-17} +4 q^{-18} -4 q^{-20} +4 q^{-21} -4 q^{-23} +4 q^{-24} + q^{-25} -4 q^{-26} +3 q^{-27} -2 q^{-29} + q^{-30}$
3	$q^3-q^2-q+3 q^{-1} -3 q^{-3} -2 q^{-4} +5 q^{-5} +2 q^{-6} -3 q^{-7} -5 q^{-8} +5 q^{-9} +4 q^{-10} -2 q^{-11} -6 q^{-12} +4 q^{-13} +4 q^{-14} - q^{-15} -7 q^{-16} +4 q^{-17} +4 q^{-18} -2 q^{-19} -7 q^{-20} +5 q^{-21} +5 q^{-22} -3 q^{-23} -8 q^{-24} +5 q^{-25} +7 q^{-26} -4 q^{-27} -10 q^{-28} +6 q^{-29} +9 q^{-30} -6 q^{-31} -11 q^{-32} +8 q^{-33} +12 q^{-34} -8 q^{-35} -12 q^{-36} +7 q^{-37} +14 q^{-38} -8 q^{-39} -11 q^{-40} +4 q^{-41} +12 q^{-42} -5 q^{-43} -8 q^{-44} + q^{-45} +9 q^{-46} -3 q^{-47} -5 q^{-48} + q^{-49} +4 q^{-50} - q^{-51} -3 q^{-52} +2 q^{-53} + q^{-54} -2 q^{-56} + q^{-57}$
4	$q^8-q^7-q^6+4 q^3-q^2-2 q-2-3 q^{-1} +8 q^{-2} + q^{-3} - q^{-4} -3 q^{-5} -8 q^{-6} +9 q^{-7} +2 q^{-8} +2 q^{-9} - q^{-10} -12 q^{-11} +9 q^{-12} + q^{-13} +3 q^{-14} + q^{-15} -13 q^{-16} +9 q^{-17} +3 q^{-19} +2 q^{-20} -15 q^{-21} +9 q^{-22} + q^{-23} +4 q^{-24} +3 q^{-25} -17 q^{-26} +7 q^{-27} + q^{-28} +5 q^{-29} +7 q^{-30} -18 q^{-31} +3 q^{-32} - q^{-33} +6 q^{-34} +13 q^{-35} -17 q^{-36} -2 q^{-37} -4 q^{-38} +6 q^{-39} +19 q^{-40} -15 q^{-41} -6 q^{-42} -6 q^{-43} +5 q^{-44} +23 q^{-45} -13 q^{-46} -9 q^{-47} -7 q^{-48} +5 q^{-49} +26 q^{-50} -12 q^{-51} -12 q^{-52} -9 q^{-53} +5 q^{-54} +29 q^{-55} -10 q^{-56} -12 q^{-57} -13 q^{-58} +3 q^{-59} +30 q^{-60} -8 q^{-61} -10 q^{-62} -13 q^{-63} + q^{-64} +26 q^{-65} -7 q^{-66} -6 q^{-67} -11 q^{-68} + q^{-69} +20 q^{-70} -7 q^{-71} -3 q^{-72} -7 q^{-73} + q^{-74} +12 q^{-75} -7 q^{-76} + q^{-77} -3 q^{-78} +6 q^{-80} -7 q^{-81} +4 q^{-82} - q^{-83} +2 q^{-85} -5 q^{-86} +3 q^{-87} + q^{-89} -2 q^{-91} + q^{-92}$
5	$q^{15}-q^{14}-q^{13}+q^{10}+3 q^9-3 q^7-2 q^6-2 q^5+6 q^3+4 q^2-q-4-5 q^{-1} -4 q^{-2} +5 q^{-3} +7 q^{-4} +3 q^{-5} - q^{-6} -6 q^{-7} -7 q^{-8} +2 q^{-9} +5 q^{-10} +5 q^{-11} +2 q^{-12} -4 q^{-13} -8 q^{-14} +2 q^{-15} +3 q^{-16} +4 q^{-17} +3 q^{-18} -3 q^{-19} -8 q^{-20} + q^{-21} +3 q^{-22} +4 q^{-23} +4 q^{-24} - q^{-25} -9 q^{-26} - q^{-27} +3 q^{-29} +6 q^{-30} +2 q^{-31} -5 q^{-32} - q^{-33} -5 q^{-34} -3 q^{-35} +4 q^{-36} +7 q^{-37} +2 q^{-38} +4 q^{-39} -7 q^{-40} -13 q^{-41} -3 q^{-42} +8 q^{-43} +10 q^{-44} +12 q^{-45} -5 q^{-46} -19 q^{-47} -12 q^{-48} +4 q^{-49} +16 q^{-50} +19 q^{-51} - q^{-52} -21 q^{-53} -18 q^{-54} -2 q^{-55} +18 q^{-56} +23 q^{-57} +4 q^{-58} -20 q^{-59} -20 q^{-60} -6 q^{-61} +15 q^{-62} +24 q^{-63} +8 q^{-64} -17 q^{-65} -19 q^{-66} -9 q^{-67} +12 q^{-68} +21 q^{-69} +10 q^{-70} -13 q^{-71} -19 q^{-72} -11 q^{-73} +9 q^{-74} +21 q^{-75} +15 q^{-76} -9 q^{-77} -22 q^{-78} -17 q^{-79} +6 q^{-80} +22 q^{-81} +20 q^{-82} -2 q^{-83} -23 q^{-84} -22 q^{-85} +2 q^{-86} +19 q^{-87} +20 q^{-88} +3 q^{-89} -20 q^{-90} -18 q^{-91} +2 q^{-92} +14 q^{-93} +14 q^{-94} + q^{-95} -18 q^{-96} -10 q^{-97} +7 q^{-98} +12 q^{-99} +7 q^{-100} -3 q^{-101} -18 q^{-102} -6 q^{-103} +11 q^{-104} +13 q^{-105} +4 q^{-106} -6 q^{-107} -16 q^{-108} -5 q^{-109} +9 q^{-110} +12 q^{-111} +4 q^{-112} -6 q^{-113} -9 q^{-114} -4 q^{-115} +3 q^{-116} +7 q^{-117} +4 q^{-118} -3 q^{-119} -3 q^{-120} -3 q^{-121} +2 q^{-123} +3 q^{-124} - q^{-125} + q^{-126} -2 q^{-127} -2 q^{-128} + q^{-129} + q^{-130} + q^{-132} -2 q^{-134} + q^{-135}$
6	$q^{24}-q^{23}-q^{22}+q^{19}+4 q^{17}-q^{16}-3 q^{15}-2 q^{14}-2 q^{13}-q^{11}+10 q^{10}+2 q^9-q^8-3 q^7-5 q^6-4 q^5-8 q^4+13 q^3+5 q^2+4 q+1-3 q^{-1} -6 q^{-2} -16 q^{-3} +11 q^{-4} +2 q^{-5} +6 q^{-6} +4 q^{-7} +3 q^{-8} -3 q^{-9} -20 q^{-10} +11 q^{-11} -2 q^{-12} +4 q^{-13} +3 q^{-14} +6 q^{-15} -21 q^{-17} +13 q^{-18} -4 q^{-19} +3 q^{-20} +2 q^{-21} +8 q^{-22} +2 q^{-23} -22 q^{-24} +14 q^{-25} -7 q^{-26} +9 q^{-29} +7 q^{-30} -19 q^{-31} +19 q^{-32} -9 q^{-33} -6 q^{-34} -7 q^{-35} +3 q^{-36} +8 q^{-37} -12 q^{-38} +32 q^{-39} -3 q^{-40} -7 q^{-41} -16 q^{-42} -12 q^{-43} - q^{-44} -11 q^{-45} +46 q^{-46} +10 q^{-47} +3 q^{-48} -17 q^{-49} -26 q^{-50} -16 q^{-51} -20 q^{-52} +51 q^{-53} +20 q^{-54} +18 q^{-55} -9 q^{-56} -31 q^{-57} -27 q^{-58} -32 q^{-59} +47 q^{-60} +20 q^{-61} +29 q^{-62} + q^{-63} -28 q^{-64} -28 q^{-65} -37 q^{-66} +41 q^{-67} +13 q^{-68} +30 q^{-69} +5 q^{-70} -24 q^{-71} -22 q^{-72} -32 q^{-73} +39 q^{-74} +4 q^{-75} +24 q^{-76} +2 q^{-77} -24 q^{-78} -15 q^{-79} -20 q^{-80} +43 q^{-81} -3 q^{-82} +14 q^{-83} -7 q^{-84} -25 q^{-85} -8 q^{-86} -4 q^{-87} +50 q^{-88} -10 q^{-89} +4 q^{-90} -18 q^{-91} -28 q^{-92} -2 q^{-93} +10 q^{-94} +58 q^{-95} -16 q^{-96} -3 q^{-97} -26 q^{-98} -30 q^{-99} +2 q^{-100} +19 q^{-101} +60 q^{-102} -22 q^{-103} -8 q^{-104} -28 q^{-105} -25 q^{-106} +8 q^{-107} +26 q^{-108} +58 q^{-109} -32 q^{-110} -18 q^{-111} -32 q^{-112} -18 q^{-113} +18 q^{-114} +36 q^{-115} +61 q^{-116} -36 q^{-117} -28 q^{-118} -43 q^{-119} -22 q^{-120} +19 q^{-121} +42 q^{-122} +71 q^{-123} -23 q^{-124} -26 q^{-125} -50 q^{-126} -35 q^{-127} +6 q^{-128} +36 q^{-129} +76 q^{-130} -3 q^{-131} -11 q^{-132} -44 q^{-133} -44 q^{-134} -13 q^{-135} +19 q^{-136} +69 q^{-137} +11 q^{-138} +7 q^{-139} -28 q^{-140} -42 q^{-141} -25 q^{-142} - q^{-143} +54 q^{-144} +15 q^{-145} +20 q^{-146} -10 q^{-147} -33 q^{-148} -27 q^{-149} -15 q^{-150} +37 q^{-151} +12 q^{-152} +22 q^{-153} + q^{-154} -20 q^{-155} -20 q^{-156} -20 q^{-157} +24 q^{-158} +7 q^{-159} +14 q^{-160} +4 q^{-161} -9 q^{-162} -10 q^{-163} -17 q^{-164} +15 q^{-165} +2 q^{-166} +7 q^{-167} +2 q^{-168} -2 q^{-169} -3 q^{-170} -12 q^{-171} +9 q^{-172} - q^{-173} +3 q^{-174} + q^{-175} + q^{-176} - q^{-177} -6 q^{-178} +4 q^{-179} - q^{-180} + q^{-181} + q^{-183} -2 q^{-185} + q^{-186}$
7	$q^{35}-q^{34}-q^{33}+q^{30}+q^{28}+3 q^{27}-q^{26}-3 q^{25}-2 q^{24}-3 q^{23}+q^{22}+q^{20}+9 q^{19}+3 q^{18}-q^{17}-3 q^{16}-8 q^{15}-3 q^{14}-4 q^{13}-4 q^{12}+11 q^{11}+8 q^{10}+6 q^9+5 q^8-9 q^7-4 q^6-8 q^5-13 q^4+6 q^3+5 q^2+8 q+13-3 q^{-1} -5 q^{-3} -17 q^{-4} +3 q^{-5} -2 q^{-6} +2 q^{-7} +15 q^{-8} - q^{-9} +4 q^{-10} - q^{-11} -16 q^{-12} +5 q^{-13} -3 q^{-14} -4 q^{-15} +13 q^{-16} - q^{-17} +6 q^{-18} +2 q^{-19} -17 q^{-20} +8 q^{-21} -3 q^{-22} -8 q^{-23} +9 q^{-24} -3 q^{-25} +7 q^{-26} +7 q^{-27} -13 q^{-28} +12 q^{-29} +2 q^{-30} -11 q^{-31} + q^{-32} -13 q^{-33} - q^{-34} +7 q^{-35} -9 q^{-36} +24 q^{-37} +17 q^{-38} -2 q^{-39} -23 q^{-41} -20 q^{-42} -12 q^{-43} -16 q^{-44} +29 q^{-45} +36 q^{-46} +20 q^{-47} +17 q^{-48} -17 q^{-49} -33 q^{-50} -36 q^{-51} -39 q^{-52} +15 q^{-53} +39 q^{-54} +38 q^{-55} +40 q^{-56} +4 q^{-57} -25 q^{-58} -46 q^{-59} -58 q^{-60} -8 q^{-61} +26 q^{-62} +38 q^{-63} +50 q^{-64} +21 q^{-65} -8 q^{-66} -38 q^{-67} -59 q^{-68} -19 q^{-69} +13 q^{-70} +30 q^{-71} +44 q^{-72} +21 q^{-73} -3 q^{-74} -31 q^{-75} -50 q^{-76} -13 q^{-77} +17 q^{-78} +30 q^{-79} +38 q^{-80} +10 q^{-81} -14 q^{-82} -41 q^{-83} -47 q^{-84} -2 q^{-85} +33 q^{-86} +47 q^{-87} +44 q^{-88} +4 q^{-89} -34 q^{-90} -66 q^{-91} -61 q^{-92} + q^{-93} +52 q^{-94} +74 q^{-95} +66 q^{-96} +8 q^{-97} -50 q^{-98} -97 q^{-99} -86 q^{-100} -8 q^{-101} +64 q^{-102} +103 q^{-103} +93 q^{-104} +21 q^{-105} -58 q^{-106} -122 q^{-107} -114 q^{-108} -24 q^{-109} +69 q^{-110} +126 q^{-111} +119 q^{-112} +35 q^{-113} -59 q^{-114} -141 q^{-115} -138 q^{-116} -40 q^{-117} +68 q^{-118} +143 q^{-119} +142 q^{-120} +50 q^{-121} -58 q^{-122} -152 q^{-123} -157 q^{-124} -56 q^{-125} +61 q^{-126} +153 q^{-127} +160 q^{-128} +63 q^{-129} -55 q^{-130} -155 q^{-131} -168 q^{-132} -67 q^{-133} +56 q^{-134} +158 q^{-135} +171 q^{-136} +67 q^{-137} -59 q^{-138} -160 q^{-139} -177 q^{-140} -67 q^{-141} +64 q^{-142} +171 q^{-143} +183 q^{-144} +66 q^{-145} -71 q^{-146} -177 q^{-147} -194 q^{-148} -70 q^{-149} +72 q^{-150} +189 q^{-151} +205 q^{-152} +77 q^{-153} -74 q^{-154} -189 q^{-155} -212 q^{-156} -88 q^{-157} +61 q^{-158} +190 q^{-159} +221 q^{-160} +94 q^{-161} -56 q^{-162} -176 q^{-163} -210 q^{-164} -106 q^{-165} +34 q^{-166} +166 q^{-167} +211 q^{-168} +104 q^{-169} -33 q^{-170} -145 q^{-171} -187 q^{-172} -107 q^{-173} +12 q^{-174} +134 q^{-175} +184 q^{-176} +96 q^{-177} -18 q^{-178} -115 q^{-179} -154 q^{-180} -95 q^{-181} +4 q^{-182} +105 q^{-183} +148 q^{-184} +81 q^{-185} -10 q^{-186} -87 q^{-187} -121 q^{-188} -76 q^{-189} +77 q^{-191} +110 q^{-192} +63 q^{-193} -4 q^{-194} -60 q^{-195} -85 q^{-196} -56 q^{-197} -4 q^{-198} +48 q^{-199} +74 q^{-200} +42 q^{-201} -34 q^{-203} -52 q^{-204} -34 q^{-205} -5 q^{-206} +26 q^{-207} +41 q^{-208} +24 q^{-209} +2 q^{-210} -17 q^{-211} -31 q^{-212} -14 q^{-213} - q^{-214} +10 q^{-215} +20 q^{-216} +11 q^{-217} +4 q^{-218} -9 q^{-219} -19 q^{-220} -3 q^{-221} +2 q^{-222} +3 q^{-223} +7 q^{-224} +5 q^{-225} +4 q^{-226} -4 q^{-227} -11 q^{-228} +3 q^{-230} - q^{-231} +3 q^{-232} + q^{-233} +3 q^{-234} - q^{-235} -5 q^{-236} +2 q^{-238} - q^{-239} + q^{-240} + q^{-242} -2 q^{-244} + q^{-245}$

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.

10_1 10_3