10 127: Difference between revisions

Revision as of 20:03, 29 August 2005

10_126

10_128

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Visit 10 127's page at the original Knot Atlas!

10 127 Quick Notes

10 127 Further Notes and Views

Knot presentations

Planar diagram presentation	X₁₄₂₅ X₃₈₄₉ X_14,6,15,5 X_15,20,16,1 X_9,16,10,17 X_11,18,12,19 X_17,10,18,11 X_19,12,20,13 X_6,14,7,13 X₇₂₈₃
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	[41,21,2-]

Minimum Braid Representative:

Length is 10, width is 3.

Braid index is 3.

A Morse Link Presentation:

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	[-13][3]
Hyperbolic Volume	8.89682
A-Polynomial	See Data:10 127/A-polynomial

[edit Notes for 10 127'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 10 127's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+4t^{2}-6t+7-6t^{-1}+4t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-2z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 29, -4 }
Jones polynomial	$2q^{-2}-2q^{-3}+4q^{-4}-5q^{-5}+5q^{-6}-5q^{-7}+3q^{-8}-2q^{-9}+q^{-10}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{8}+3z^{2}a^{8}+2a^{8}-z^{6}a^{6}-5z^{4}a^{6}-9z^{2}a^{6}-6a^{6}+2z^{4}a^{4}+7z^{2}a^{4}+5a^{4}$
Kauffman polynomial (db, data sources)	$z^{4}a^{12}-2z^{2}a^{12}+2z^{5}a^{11}-4z^{3}a^{11}+za^{11}+2z^{6}a^{10}-3z^{4}a^{10}+z^{2}a^{10}+2z^{7}a^{9}-5z^{5}a^{9}+7z^{3}a^{9}-2za^{9}+z^{8}a^{8}-2z^{6}a^{8}+4z^{4}a^{8}-2z^{2}a^{8}+2a^{8}+3z^{7}a^{7}-10z^{5}a^{7}+16z^{3}a^{7}-8za^{7}+z^{8}a^{6}-4z^{6}a^{6}+11z^{4}a^{6}-14z^{2}a^{6}+6a^{6}+z^{7}a^{5}-3z^{5}a^{5}+5z^{3}a^{5}-5za^{5}+3z^{4}a^{4}-9z^{2}a^{4}+5a^{4}$
The A2 invariant	$q^{30}+q^{26}-2q^{22}-q^{20}-3q^{18}+q^{12}+3q^{10}+q^{8}+2q^{6}$
The G2 invariant	$q^{162}-q^{160}+2q^{158}-3q^{156}+q^{154}-3q^{150}+5q^{148}-6q^{146}+6q^{144}-5q^{142}+q^{140}+3q^{138}-8q^{136}+12q^{134}-11q^{132}+9q^{130}-3q^{128}-5q^{126}+13q^{124}-13q^{122}+12q^{120}-3q^{118}-5q^{116}+11q^{114}-8q^{112}+2q^{110}+9q^{108}-14q^{106}+16q^{104}-8q^{102}-5q^{100}+15q^{98}-22q^{96}+21q^{94}-16q^{92}+2q^{90}+7q^{88}-17q^{86}+17q^{84}-17q^{82}+5q^{80}-11q^{76}+9q^{74}-9q^{72}+7q^{68}-13q^{66}+10q^{64}-4q^{62}-7q^{60}+17q^{58}-17q^{56}+14q^{54}-3q^{52}-4q^{50}+14q^{48}-12q^{46}+13q^{44}-4q^{42}+q^{40}+6q^{38}-5q^{36}+5q^{34}+2q^{30}+q^{28}$

Further Quantum Invariants

Further quantum knot invariants for 10_127.

A1 Invariants.

Weight	Invariant
1	$q^{21}-q^{19}+q^{17}-2q^{15}-q^{9}+2q^{7}+2q^{3}$
2	$q^{58}-q^{56}-q^{54}+2q^{52}-2q^{50}-2q^{48}+5q^{46}-q^{44}-5q^{42}+6q^{40}+q^{38}-4q^{36}+2q^{34}+2q^{32}-q^{30}-4q^{28}+2q^{26}+2q^{24}-6q^{22}+q^{20}+4q^{18}-5q^{16}+5q^{12}-q^{10}-q^{8}+3q^{6}+q^{4}$
3	$q^{111}-q^{109}-q^{107}+q^{103}-q^{99}+2q^{97}+2q^{95}-3q^{93}-5q^{91}+4q^{89}+9q^{87}-2q^{85}-15q^{83}-q^{81}+17q^{79}+7q^{77}-19q^{75}-10q^{73}+14q^{71}+12q^{69}-8q^{67}-12q^{65}+4q^{63}+9q^{61}+3q^{59}-7q^{57}-7q^{55}+5q^{53}+12q^{51}-4q^{49}-13q^{47}+q^{45}+17q^{43}-q^{41}-17q^{39}-5q^{37}+17q^{35}+7q^{33}-14q^{31}-12q^{29}+7q^{27}+13q^{25}-3q^{23}-10q^{21}-2q^{19}+8q^{17}+5q^{15}-2q^{13}-4q^{11}+2q^{9}+2q^{7}+2q^{5}$
5	$q^{265}-q^{263}-q^{261}-q^{257}+q^{255}+3q^{253}+2q^{251}-q^{249}-q^{247}-4q^{245}-5q^{243}+q^{241}+6q^{239}+6q^{237}+4q^{235}-2q^{233}-12q^{231}-14q^{229}-2q^{227}+14q^{225}+24q^{223}+18q^{221}-6q^{219}-38q^{217}-44q^{215}-10q^{213}+40q^{211}+75q^{209}+54q^{207}-26q^{205}-107q^{203}-112q^{201}-19q^{199}+119q^{197}+179q^{195}+89q^{193}-96q^{191}-231q^{189}-182q^{187}+38q^{185}+250q^{183}+257q^{181}+46q^{179}-214q^{177}-308q^{175}-128q^{173}+156q^{171}+297q^{169}+187q^{167}-71q^{165}-248q^{163}-207q^{161}+178q^{157}+187q^{155}+48q^{153}-99q^{151}-148q^{149}-78q^{147}+42q^{145}+101q^{143}+76q^{141}+4q^{139}-67q^{137}-79q^{135}-29q^{133}+49q^{131}+83q^{129}+46q^{127}-44q^{125}-104q^{123}-61q^{121}+51q^{119}+132q^{117}+93q^{115}-58q^{113}-171q^{111}-122q^{109}+50q^{107}+201q^{105}+177q^{103}-29q^{101}-224q^{99}-217q^{97}-17q^{95}+208q^{93}+262q^{91}+79q^{89}-174q^{87}-270q^{85}-143q^{83}+98q^{81}+250q^{79}+196q^{77}-14q^{75}-194q^{73}-214q^{71}-69q^{69}+109q^{67}+190q^{65}+129q^{63}-21q^{61}-134q^{59}-139q^{57}-55q^{55}+59q^{53}+114q^{51}+88q^{49}+9q^{47}-65q^{45}-88q^{43}-47q^{41}+15q^{39}+52q^{37}+58q^{35}+21q^{33}-23q^{31}-38q^{29}-26q^{27}-5q^{25}+17q^{23}+25q^{21}+11q^{19}-2q^{17}-6q^{15}-10q^{13}-4q^{11}+2q^{9}+4q^{7}+2q^{5}+2q^{3}$
6	$q^{366}-q^{364}-q^{362}-q^{358}+q^{356}+q^{354}+5q^{352}-3q^{348}-q^{346}-5q^{344}-4q^{342}-2q^{340}+10q^{338}+6q^{336}+2q^{334}+4q^{332}-6q^{330}-13q^{328}-15q^{326}+7q^{324}+11q^{322}+13q^{320}+22q^{318}+4q^{316}-24q^{314}-45q^{312}-17q^{310}+2q^{308}+30q^{306}+71q^{304}+56q^{302}-7q^{300}-90q^{298}-104q^{296}-90q^{294}-3q^{292}+146q^{290}+216q^{288}+154q^{286}-40q^{284}-218q^{282}-351q^{280}-265q^{278}+60q^{276}+396q^{274}+542q^{272}+336q^{270}-83q^{268}-596q^{266}-783q^{264}-445q^{262}+219q^{260}+834q^{258}+957q^{256}+530q^{254}-374q^{252}-1085q^{250}-1118q^{248}-441q^{246}+556q^{244}+1206q^{242}+1176q^{240}+301q^{238}-737q^{236}-1274q^{234}-1003q^{232}-114q^{230}+778q^{228}+1199q^{226}+772q^{224}-70q^{222}-778q^{220}-934q^{218}-517q^{216}+144q^{214}+678q^{212}+677q^{210}+301q^{208}-193q^{206}-479q^{204}-449q^{202}-176q^{200}+178q^{198}+338q^{196}+295q^{194}+79q^{192}-135q^{190}-255q^{188}-205q^{186}-29q^{184}+159q^{182}+245q^{180}+135q^{178}-60q^{176}-248q^{174}-246q^{172}-71q^{170}+219q^{168}+383q^{166}+233q^{164}-103q^{162}-440q^{160}-466q^{158}-173q^{156}+355q^{154}+683q^{152}+500q^{150}-49q^{148}-645q^{146}-812q^{144}-461q^{142}+322q^{140}+926q^{138}+888q^{136}+259q^{134}-593q^{132}-1055q^{130}-888q^{128}-48q^{126}+821q^{124}+1135q^{122}+742q^{120}-130q^{118}-884q^{116}-1141q^{114}-614q^{112}+250q^{110}+909q^{108}+1006q^{106}+507q^{104}-240q^{102}-863q^{100}-877q^{98}-423q^{96}+222q^{94}+688q^{92}+745q^{90}+406q^{88}-170q^{86}-528q^{84}-598q^{82}-356q^{80}+34q^{78}+376q^{76}+487q^{74}+305q^{72}+48q^{70}-220q^{68}-345q^{66}-303q^{64}-96q^{62}+121q^{60}+223q^{58}+241q^{56}+128q^{54}-24q^{52}-158q^{50}-175q^{48}-111q^{46}-24q^{44}+76q^{42}+118q^{40}+100q^{38}+26q^{36}-28q^{34}-59q^{32}-64q^{30}-33q^{28}+5q^{26}+34q^{24}+31q^{22}+21q^{20}+8q^{18}-8q^{16}-15q^{14}-11q^{12}-3q^{10}+q^{8}+3q^{6}+3q^{4}+3q^{2}+1$

A2 Invariants.

Weight	Invariant
1,0	$q^{30}+q^{26}-2q^{22}-q^{20}-3q^{18}+q^{12}+3q^{10}+q^{8}+2q^{6}$
1,1	$q^{84}-2q^{82}+4q^{80}-8q^{78}+11q^{76}-14q^{74}+18q^{72}-24q^{70}+29q^{68}-30q^{66}+34q^{64}-36q^{62}+28q^{60}-16q^{58}-2q^{56}+20q^{54}-44q^{52}+62q^{50}-72q^{48}+82q^{46}-74q^{44}+72q^{42}-50q^{40}+38q^{38}-17q^{36}-10q^{34}+18q^{32}-44q^{30}+38q^{28}-52q^{26}+40q^{24}-30q^{22}+26q^{20}-10q^{18}+12q^{16}+4q^{14}+4q^{12}+2q^{10}$
2,0	$q^{76}-q^{68}-q^{66}-q^{64}+q^{62}-q^{60}-3q^{58}+2q^{54}+2q^{50}+6q^{48}+5q^{46}+q^{44}-4q^{38}-6q^{36}-2q^{34}-4q^{32}-5q^{30}-q^{28}-q^{24}+5q^{20}+4q^{18}+2q^{16}+2q^{14}+4q^{12}+q^{10}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{68}-q^{66}+q^{62}-3q^{60}+3q^{56}-4q^{54}-q^{52}+5q^{50}-3q^{48}+q^{46}+5q^{44}+3q^{42}+3q^{40}+2q^{38}+q^{36}-5q^{34}-10q^{32}-5q^{30}-6q^{28}-8q^{26}+4q^{24}+5q^{22}+2q^{20}+7q^{18}+5q^{16}+q^{14}+3q^{12}$
1,0,0	$q^{39}+2q^{35}+q^{31}-2q^{29}-2q^{27}-4q^{25}-3q^{23}-q^{21}+3q^{17}+2q^{15}+4q^{13}+q^{11}+2q^{9}$
1,0,1	$q^{110}-2q^{108}+3q^{106}-3q^{104}-q^{102}+6q^{100}-9q^{98}+9q^{96}-4q^{94}-6q^{92}+16q^{90}-19q^{88}+13q^{86}+6q^{84}-28q^{82}+38q^{80}-30q^{78}+5q^{76}+24q^{74}-41q^{72}+39q^{70}-30q^{68}+q^{66}+3q^{64}-16q^{62}+6q^{60}+7q^{58}+11q^{56}+4q^{54}+50q^{52}-29q^{50}+48q^{48}-15q^{46}-26q^{44}+18q^{42}-70q^{40}+8q^{38}-37q^{36}-15q^{34}+11q^{32}-11q^{30}+25q^{28}+11q^{26}+14q^{24}+16q^{22}+7q^{20}+5q^{18}+2q^{16}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{86}+2q^{80}-4q^{76}-q^{74}-5q^{70}-5q^{68}+q^{64}-q^{62}+5q^{60}+12q^{58}+9q^{56}+9q^{54}+14q^{52}+5q^{50}-5q^{48}-7q^{46}-12q^{44}-22q^{42}-19q^{40}-12q^{38}-7q^{36}-4q^{34}+5q^{32}+12q^{30}+9q^{28}+10q^{26}+9q^{24}+6q^{22}+2q^{20}+3q^{18}$
1,0,0,0	$q^{48}+2q^{44}+q^{42}+q^{40}+q^{38}-2q^{36}-2q^{34}-5q^{32}-4q^{30}-4q^{28}-q^{26}+3q^{22}+4q^{20}+3q^{18}+4q^{16}+q^{14}+2q^{12}$

B2 Invariants.

Weight	Invariant
0,1	$q^{68}-q^{66}+2q^{64}-3q^{62}+3q^{60}-4q^{58}+5q^{56}-4q^{54}+3q^{52}-q^{50}-q^{48}+3q^{46}-5q^{44}+7q^{42}-9q^{40}+8q^{38}-9q^{36}+5q^{34}-6q^{32}+q^{30}-2q^{26}+4q^{24}-3q^{22}+6q^{20}-3q^{18}+5q^{16}-q^{14}+3q^{12}$
1,0	$q^{110}-q^{106}-q^{104}+q^{102}+2q^{100}-q^{98}-3q^{96}-2q^{94}+2q^{92}+4q^{90}-q^{88}-5q^{86}-2q^{84}+4q^{82}+4q^{80}-2q^{78}-3q^{76}+2q^{74}+5q^{72}+q^{70}-2q^{68}+q^{66}+5q^{64}+2q^{62}-2q^{60}-3q^{58}+q^{56}-4q^{52}-7q^{50}-2q^{48}+q^{46}-2q^{44}-6q^{42}-4q^{40}+4q^{38}+5q^{36}+q^{34}-3q^{32}+q^{30}+5q^{28}+5q^{26}+q^{20}+3q^{18}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{94}-q^{92}+q^{90}-2q^{88}+2q^{86}-3q^{84}+2q^{82}-3q^{80}+4q^{78}-4q^{76}+2q^{74}-2q^{72}+2q^{70}-2q^{66}+3q^{64}-q^{62}+9q^{60}-2q^{58}+10q^{56}-3q^{54}+9q^{52}-6q^{50}+q^{48}-12q^{46}-6q^{44}-10q^{42}-8q^{40}-6q^{38}-5q^{36}+4q^{34}+q^{32}+9q^{30}+3q^{28}+10q^{26}+2q^{24}+6q^{22}+3q^{18}$

G2 Invariants.

Weight

Invariant

1,0

q^{162}-q^{160}+2q^{158}-3q^{156}+q^{154}-3q^{150}+5q^{148}-6q^{146}+6q^{144}-5q^{142}+q^{140}+3q^{138}-8q^{136}+12q^{134}-11q^{132}+9q^{130}-3q^{128}-5q^{126}+13q^{124}-13q^{122}+12q^{120}-3q^{118}-5q^{116}+11q^{114}-8q^{112}+2q^{110}+9q^{108}-14q^{106}+16q^{104}-8q^{102}-5q^{100}+15q^{98}-22q^{96}+21q^{94}-16q^{92}+2q^{90}+7q^{88}-17q^{86}+17q^{84}-17q^{82}+5q^{80}-11q^{76}+9q^{74}-9q^{72}+7q^{68}-13q^{66}+10q^{64}-4q^{62}-7q^{60}+17q^{58}-17q^{56}+14q^{54}-3q^{52}-4q^{50}+14q^{48}-12q^{46}+13q^{44}-4q^{42}+q^{40}+6q^{38}-5q^{36}+5q^{34}+2q^{30}+q^{28}

.

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.

In[3]:= K = Knot["10 127"];

In[4]:= Alexander[K][t]

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

Out[4]= $-t^{3}+4t^{2}-6t+7-6t^{-1}+4t^{-2}-t^{-3}$

In[5]:= Conway[K][z]

Out[5]= $-z^{6}-2z^{4}+z^{2}+1$

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]= { 29, -4 }

In[8]:= Jones[K][q]

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

Out[8]= $2q^{-2}-2q^{-3}+4q^{-4}-5q^{-5}+5q^{-6}-5q^{-7}+3q^{-8}-2q^{-9}+q^{-10}$

In[9]:= HOMFLYPT[K][a, z]

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

Out[9]= $z^{4}a^{8}+3z^{2}a^{8}+2a^{8}-z^{6}a^{6}-5z^{4}a^{6}-9z^{2}a^{6}-6a^{6}+2z^{4}a^{4}+7z^{2}a^{4}+5a^{4}$

In[10]:= Kauffman[K][a, z]

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

Out[10]= $z^{4}a^{12}-2z^{2}a^{12}+2z^{5}a^{11}-4z^{3}a^{11}+za^{11}+2z^{6}a^{10}-3z^{4}a^{10}+z^{2}a^{10}+2z^{7}a^{9}-5z^{5}a^{9}+7z^{3}a^{9}-2za^{9}+z^{8}a^{8}-2z^{6}a^{8}+4z^{4}a^{8}-2z^{2}a^{8}+2a^{8}+3z^{7}a^{7}-10z^{5}a^{7}+16z^{3}a^{7}-8za^{7}+z^{8}a^{6}-4z^{6}a^{6}+11z^{4}a^{6}-14z^{2}a^{6}+6a^{6}+z^{7}a^{5}-3z^{5}a^{5}+5z^{3}a^{5}-5za^{5}+3z^{4}a^{4}-9z^{2}a^{4}+5a^{4}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_150, K11n51, ...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {...}

Vassiliev invariants

V₂ and V₃:

(1, 1)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

4

8

8

-{\frac {130}{3}}

{\frac {58}{3}}

32

{\frac {272}{3}}

-{\frac {160}{3}}

-88

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{32}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 32}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{520}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{232}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2191}{30}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{3782}{15}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{38822}{45}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1073}{18}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2671}{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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} -4 is the signature of 10 127. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-3

2

-5

1

0

-7

3

1

2

-9

2

1

-1

-11

3

0

-13

2

0

-15

1

3

-2

-17

1

2

1

-19

1

-1

-21

1

Integral Khovanov Homology

(db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-5}	$i=-3$
$r=-8$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-7}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-6}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-5}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-4}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-3} +2 q^{-4} -4 q^{-5} + q^{-6} +8 q^{-7} -9 q^{-8} -4 q^{-9} +17 q^{-10} -12 q^{-11} -11 q^{-12} +25 q^{-13} -12 q^{-14} -17 q^{-15} +28 q^{-16} -9 q^{-17} -17 q^{-18} +22 q^{-19} -4 q^{-20} -12 q^{-21} +11 q^{-22} -6 q^{-24} +4 q^{-25} -2 q^{-27} + q^{-28} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 q^{-4} -6 q^{-7} +4 q^{-8} +7 q^{-9} +3 q^{-10} -16 q^{-11} -4 q^{-12} +14 q^{-13} +19 q^{-14} -22 q^{-15} -23 q^{-16} +12 q^{-17} +40 q^{-18} -12 q^{-19} -45 q^{-20} +56 q^{-22} +6 q^{-23} -61 q^{-24} -14 q^{-25} +65 q^{-26} +22 q^{-27} -68 q^{-28} -26 q^{-29} +65 q^{-30} +32 q^{-31} -62 q^{-32} -31 q^{-33} +49 q^{-34} +36 q^{-35} -42 q^{-36} -29 q^{-37} +25 q^{-38} +27 q^{-39} -16 q^{-40} -19 q^{-41} +7 q^{-42} +13 q^{-43} -3 q^{-44} -8 q^{-45} +2 q^{-46} +4 q^{-47} - q^{-48} -3 q^{-49} +2 q^{-50} + q^{-51} -2 q^{-53} + q^{-54} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-4} +2 q^{-5} -4 q^{-7} -2 q^{-8} -3 q^{-9} +9 q^{-10} +12 q^{-11} -5 q^{-12} -8 q^{-13} -25 q^{-14} +6 q^{-15} +33 q^{-16} +11 q^{-17} +8 q^{-18} -57 q^{-19} -27 q^{-20} +32 q^{-21} +34 q^{-22} +63 q^{-23} -62 q^{-24} -69 q^{-25} -13 q^{-26} +26 q^{-27} +138 q^{-28} -24 q^{-29} -85 q^{-30} -80 q^{-31} -24 q^{-32} +196 q^{-33} +35 q^{-34} -68 q^{-35} -136 q^{-36} -89 q^{-37} +229 q^{-38} +84 q^{-39} -41 q^{-40} -170 q^{-41} -140 q^{-42} +242 q^{-43} +115 q^{-44} -15 q^{-45} -187 q^{-46} -172 q^{-47} +234 q^{-48} +131 q^{-49} +13 q^{-50} -176 q^{-51} -189 q^{-52} +187 q^{-53} +125 q^{-54} +51 q^{-55} -128 q^{-56} -181 q^{-57} +109 q^{-58} +85 q^{-59} +75 q^{-60} -54 q^{-61} -133 q^{-62} +38 q^{-63} +25 q^{-64} +64 q^{-65} + q^{-66} -68 q^{-67} +11 q^{-68} -14 q^{-69} +31 q^{-70} +14 q^{-71} -24 q^{-72} +11 q^{-73} -17 q^{-74} +8 q^{-75} +6 q^{-76} -9 q^{-77} +11 q^{-78} -7 q^{-79} + q^{-80} + q^{-81} -5 q^{-82} +5 q^{-83} - q^{-84} + q^{-85} -2 q^{-87} + q^{-88} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 q^{-4} +2 q^{-6} -2 q^{-7} -6 q^{-8} -6 q^{-9} +6 q^{-10} +4 q^{-11} +15 q^{-12} +12 q^{-13} -14 q^{-14} -28 q^{-15} -15 q^{-16} -8 q^{-17} +30 q^{-18} +56 q^{-19} +23 q^{-20} -34 q^{-21} -52 q^{-22} -70 q^{-23} -11 q^{-24} +79 q^{-25} +97 q^{-26} +45 q^{-27} -26 q^{-28} -125 q^{-29} -125 q^{-30} -5 q^{-31} +102 q^{-32} +158 q^{-33} +124 q^{-34} -64 q^{-35} -206 q^{-36} -183 q^{-37} -43 q^{-38} +178 q^{-39} +304 q^{-40} +146 q^{-41} -152 q^{-42} -335 q^{-43} -284 q^{-44} +51 q^{-45} +400 q^{-46} +399 q^{-47} +31 q^{-48} -389 q^{-49} -509 q^{-50} -149 q^{-51} +393 q^{-52} +594 q^{-53} +237 q^{-54} -365 q^{-55} -660 q^{-56} -321 q^{-57} +344 q^{-58} +707 q^{-59} +388 q^{-60} -326 q^{-61} -741 q^{-62} -433 q^{-63} +301 q^{-64} +767 q^{-65} +478 q^{-66} -289 q^{-67} -775 q^{-68} -511 q^{-69} +251 q^{-70} +779 q^{-71} +549 q^{-72} -217 q^{-73} -750 q^{-74} -570 q^{-75} +131 q^{-76} +709 q^{-77} +598 q^{-78} -70 q^{-79} -611 q^{-80} -579 q^{-81} -47 q^{-82} +502 q^{-83} +557 q^{-84} +107 q^{-85} -353 q^{-86} -469 q^{-87} -188 q^{-88} +218 q^{-89} +377 q^{-90} +201 q^{-91} -93 q^{-92} -258 q^{-93} -195 q^{-94} +6 q^{-95} +157 q^{-96} +152 q^{-97} +42 q^{-98} -73 q^{-99} -105 q^{-100} -54 q^{-101} +19 q^{-102} +59 q^{-103} +47 q^{-104} +8 q^{-105} -25 q^{-106} -33 q^{-107} -16 q^{-108} +9 q^{-109} +13 q^{-110} +14 q^{-111} +7 q^{-112} -9 q^{-113} -10 q^{-114} - q^{-115} -3 q^{-116} +2 q^{-117} +9 q^{-118} + q^{-119} -4 q^{-120} + q^{-121} -3 q^{-122} -3 q^{-123} +3 q^{-124} +2 q^{-125} - q^{-126} + q^{-127} -2 q^{-129} + q^{-130} }
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-3} +2 q^{-4} -2 q^{-7} -4 q^{-8} -8 q^{-9} -3 q^{-10} +9 q^{-11} +16 q^{-12} +13 q^{-13} +8 q^{-14} - q^{-15} -37 q^{-16} -41 q^{-17} -22 q^{-18} +21 q^{-19} +44 q^{-20} +62 q^{-21} +73 q^{-22} -19 q^{-23} -83 q^{-24} -122 q^{-25} -66 q^{-26} -20 q^{-27} +79 q^{-28} +207 q^{-29} +133 q^{-30} +30 q^{-31} -140 q^{-32} -168 q^{-33} -237 q^{-34} -128 q^{-35} +165 q^{-36} +258 q^{-37} +298 q^{-38} +117 q^{-39} +14 q^{-40} -348 q^{-41} -470 q^{-42} -225 q^{-43} +16 q^{-44} +368 q^{-45} +475 q^{-46} +590 q^{-47} -9 q^{-48} -527 q^{-49} -691 q^{-50} -629 q^{-51} -86 q^{-52} +489 q^{-53} +1213 q^{-54} +738 q^{-55} -28 q^{-56} -788 q^{-57} -1288 q^{-58} -950 q^{-59} -38 q^{-60} +1470 q^{-61} +1492 q^{-62} +844 q^{-63} -395 q^{-64} -1602 q^{-65} -1819 q^{-66} -878 q^{-67} +1303 q^{-68} +1954 q^{-69} +1696 q^{-70} +234 q^{-71} -1564 q^{-72} -2423 q^{-73} -1661 q^{-74} +952 q^{-75} +2121 q^{-76} +2292 q^{-77} +783 q^{-78} -1381 q^{-79} -2751 q^{-80} -2189 q^{-81} +659 q^{-82} +2147 q^{-83} +2629 q^{-84} +1119 q^{-85} -1231 q^{-86} -2915 q^{-87} -2479 q^{-88} +484 q^{-89} +2145 q^{-90} +2817 q^{-91} +1314 q^{-92} -1121 q^{-93} -3001 q^{-94} -2667 q^{-95} +308 q^{-96} +2095 q^{-97} +2937 q^{-98} +1528 q^{-99} -905 q^{-100} -2958 q^{-101} -2827 q^{-102} -46 q^{-103} +1822 q^{-104} +2907 q^{-105} +1814 q^{-106} -411 q^{-107} -2582 q^{-108} -2826 q^{-109} -580 q^{-110} +1161 q^{-111} +2490 q^{-112} +1970 q^{-113} +297 q^{-114} -1740 q^{-115} -2399 q^{-116} -1001 q^{-117} +269 q^{-118} +1601 q^{-119} +1699 q^{-120} +834 q^{-121} -702 q^{-122} -1524 q^{-123} -971 q^{-124} -381 q^{-125} +604 q^{-126} +1022 q^{-127} +867 q^{-128} +9 q^{-129} -620 q^{-130} -544 q^{-131} -504 q^{-132} -11 q^{-133} +358 q^{-134} +529 q^{-135} +196 q^{-136} -107 q^{-137} -125 q^{-138} -298 q^{-139} -157 q^{-140} +22 q^{-141} +204 q^{-142} +110 q^{-143} +26 q^{-144} +53 q^{-145} -104 q^{-146} -95 q^{-147} -48 q^{-148} +55 q^{-149} +23 q^{-150} +12 q^{-151} +67 q^{-152} -21 q^{-153} -32 q^{-154} -33 q^{-155} +14 q^{-156} -5 q^{-157} -7 q^{-158} +39 q^{-159} -4 q^{-161} -15 q^{-162} +5 q^{-163} -7 q^{-164} -11 q^{-165} +17 q^{-166} +2 q^{-167} +3 q^{-168} -5 q^{-169} +3 q^{-170} -3 q^{-171} -7 q^{-172} +5 q^{-173} +2 q^{-175} - q^{-176} + q^{-177} -2 q^{-179} + q^{-180} }
7	Not Available

Computer Talk

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

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[10, 127]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[14, 6, 15, 5], X[15, 20, 16, 1], X[9, 16, 10, 17], X[11, 18, 12, 19], X[17, 10, 18, 11], X[19, 12, 20, 13], X[6, 14, 7, 13], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[10, 127]]
Out[3]=	GaussCode[-1, 10, -2, 1, 3, -9, -10, 2, -5, 7, -6, 8, 9, -3, -4, 5, -7, 6, -8, 4]
In[4]:=	DTCode[Knot[10, 127]]
Out[4]=	DTCode[4, 8, -14, 2, 16, 18, -6, 20, 10, 12]
In[5]:=	br = BR[Knot[10, 127]]
Out[5]=	BR[3, {-1, -1, -1, -1, -1, -2, 1, 1, -2, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 127]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 127]]]

`Out[8]=`	`-Graphics-`
In[9]:=	(#[Knot[10, 127]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 127]][t]
Out[10]=	-3 4 6 2 3 7 - t + -- - - - 6 t + 4 t - t 2 t t
In[11]:=	Conway[Knot[10, 127]][z]
Out[11]=	2 4 6 1 + z - 2 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 127], Knot[10, 150], Knot[11, NonAlternating, 51]}
In[13]:=	{KnotDet[Knot[10, 127]], KnotSignature[Knot[10, 127]]}
Out[13]=	{29, -4}
In[14]:=	Jones[Knot[10, 127]][q]
Out[14]=	-10 2 3 5 5 5 4 2 2 q - -- + -- - -- + -- - -- + -- - -- + -- 9 8 7 6 5 4 3 2 q q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 127]}
In[16]:=	A2Invariant[Knot[10, 127]][q]
Out[16]=	-30 -26 2 -20 3 -12 3 -8 2 q + q - --- - q - --- + q + --- + q + -- 22 18 10 6 q q q q
In[17]:=	HOMFLYPT[Knot[10, 127]][a, z]
Out[17]=	4 6 8 4 2 6 2 8 2 4 4 6 4 5 a - 6 a + 2 a + 7 a z - 9 a z + 3 a z + 2 a z - 5 a z + 8 4 6 6 a z - a z
In[18]:=	Kauffman[Knot[10, 127]][a, z]
Out[18]=	4 6 8 5 7 9 11 4 2 5 a + 6 a + 2 a - 5 a z - 8 a z - 2 a z + a z - 9 a z - 6 2 8 2 10 2 12 2 5 3 7 3 14 a z - 2 a z + a z - 2 a z + 5 a z + 16 a z + 9 3 11 3 4 4 6 4 8 4 10 4 7 a z - 4 a z + 3 a z + 11 a z + 4 a z - 3 a z + 12 4 5 5 7 5 9 5 11 5 6 6 a z - 3 a z - 10 a z - 5 a z + 2 a z - 4 a z - 8 6 10 6 5 7 7 7 9 7 6 8 8 8 2 a z + 2 a z + a z + 3 a z + 2 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 127]], Vassiliev[3][Knot[10, 127]]}
Out[19]=	{1, 1}
In[20]:=	Kh[Knot[10, 127]][q, t]
Out[20]=	-5 2 1 1 1 2 1 3 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + 3 21 8 19 7 17 7 17 6 15 6 15 5 q q t q t q t q t q t q t 2 2 3 3 2 1 3 1 ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 13 5 13 4 11 4 11 3 9 3 9 2 7 2 7 q t q t q t q t q t q t q t q t 1 ---- 5 q t
In[21]:=	ColouredJones[Knot[10, 127], 2][q]
Out[21]=	-28 2 4 6 11 12 4 22 17 9 28 q - --- + --- - --- + --- - --- - --- + --- - --- - --- + --- - 27 25 24 22 21 20 19 18 17 16 q q q q q q q q q q 17 12 25 11 12 17 4 9 8 -6 4 2 -3 --- - --- + --- - --- - --- + --- - -- - -- + -- + q - -- + -- + q 15 14 13 12 11 10 9 8 7 5 4 q q q q q q q q q q q

See/edit the Rolfsen_Splice_Template.

Back to the top.

10_126

10_128

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+<!-- This page was generated from the splice template "Rolfsen_Splice_Template". Please do not edit! -->
 <!--  -->
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 </table>
+{| width=100%
-See/edit the [[Rolfsen_Splice_Template]].
+|align=left|See/edit the [[Rolfsen_Splice_Template]].
+Back to the [[#top|top]].
+|align=right|{{Knot Navigation Links|ext=gif}}
+|}
  [[Category:Knot Page]]

10 127: Difference between revisions

Revision as of 20:03, 29 August 2005

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

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