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

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

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

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

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

.

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]=

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

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"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]=

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

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

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

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

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(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 $t^{r}q^{j}$ 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 $s=$-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)

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

The Coloured Jones Polynomials

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

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