10 2

10_1

10_3

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10 2 Quick Notes

10 2 Further Notes and Views

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]

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]=

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

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

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 2]]
Out[2]=	10
In[3]:=	PD[Knot[10, 2]]
Out[3]=	PD[X[1, 4, 2, 5], X[5, 14, 6, 15], X[3, 13, 4, 12], X[13, 3, 14, 2], X[7, 16, 8, 17], X[9, 18, 10, 19], X[11, 20, 12, 1], X[15, 6, 16, 7], X[17, 8, 18, 9], X[19, 10, 20, 11]]
In[4]:=	GaussCode[Knot[10, 2]]
Out[4]=	GaussCode[-1, 4, -3, 1, -2, 8, -5, 9, -6, 10, -7, 3, -4, 2, -8, 5, -9, 6, -10, 7]
In[5]:=	BR[Knot[10, 2]]
Out[5]=	BR[3, {-1, -1, -1, -1, -1, -1, -1, 2, -1, 2}]
In[6]:=	alex = Alexander[Knot[10, 2]][t]
Out[6]=	-4 3 3 3 2 3 4 -3 - t + -- - -- + - + 3 t - 3 t + 3 t - t 3 2 t t t
In[7]:=	Conway[Knot[10, 2]][z]
Out[7]=	2 4 6 8 1 + 2 z - 5 z - 5 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 2]}
In[9]:=	{KnotDet[Knot[10, 2]], KnotSignature[Knot[10, 2]]}
Out[9]=	{23, -6}
In[10]:=	J=Jones[Knot[10, 2]][q]
Out[10]=	-11 2 2 3 3 3 3 2 2 -2 1 q - --- + -- - -- + -- - -- + -- - -- + -- - q + - 10 9 8 7 6 5 4 3 q q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 2]}
In[12]:=	A2Invariant[Knot[10, 2]][q]
Out[12]=	-32 -26 -24 -22 -20 -18 -14 -10 -8 -6 -4 q - q - q - q - q + q + q + q + q + q + q
In[13]:=	Kauffman[Knot[10, 2]][a, z]
Out[13]=	4 6 8 5 7 9 11 13 4 2 4 a + 4 a + a - 2 a z - a z + a z - a z - a z - 14 a z - 6 2 8 2 12 2 14 2 5 3 7 3 9 3 21 a z - 5 a z - a z + a z - 3 a z + 3 a z + 2 a z - 11 3 13 3 4 4 6 4 8 4 10 4 2 a z + 2 a z + 16 a z + 33 a z + 11 a z - 4 a z + 12 4 5 5 7 5 9 5 11 5 4 6 2 a z + 10 a z + 2 a z - 6 a z + 2 a z - 7 a z - 6 6 8 6 10 6 5 7 7 7 9 7 4 8 18 a z - 9 a z + 2 a z - 6 a z - 4 a z + 2 a z + a z + 6 8 8 8 5 9 7 9 3 a z + 2 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[10, 2]], Vassiliev[3][Knot[10, 2]]}
Out[14]=	{0, -2}
In[15]:=	Kh[Knot[10, 2]][q, t]
Out[15]=	-7 2 1 1 1 1 1 2 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + 5 23 8 21 7 19 7 19 6 17 6 17 5 q q t q t q t q t q t q t 1 1 2 2 1 1 2 1 ------ + ------ + ------ + ------ + ------ + ------ + ----- + ---- + 15 5 15 4 13 4 13 3 11 3 11 2 9 2 9 q t q t q t q t q t q t q t q t 2 1 t t ---- + -- + -- 7 5 q q t q