5 2: Difference between revisions

Latest revision as of 06:15, 21 January 2008

5_1

6_1

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 5 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 5 2 at Knotilus!

[edit 5 2 Quick Notes]

5_2 is also known as the 3-twist knot.

[edit 5 2 Further Notes and Views]

3D depiction

Simple square depiction

Lissajous curve x=cos(2t+0.2), y=cos(3t+0.7), z=cos(7t); 2 crossings can be removed

Knot presentations

Planar diagram presentation	X₁₄₂₅ X₃₈₄₉ X_5,10,6,1 X_9,6,10,7 X₇₂₈₃
Gauss code	-1, 5, -2, 1, -3, 4, -5, 2, -4, 3
Dowker-Thistlethwaite code	4 8 10 2 6
Conway Notation	[32]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 6, width is 3,

Braid index is 3

[{7, 4}, {3, 5}, {4, 2}, {1, 3}, {2, 6}, {5, 7}, {6, 1}]

[edit Notes on presentations of 5 2]

Knot 5_2.

A graph, knot 5_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["5 2"];

In[4]:= PD[K]

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

Out[4]= X₁₄₂₅ X₃₈₄₉ X_5,10,6,1 X_9,6,10,7 X₇₂₈₃

In[5]:= GaussCode[K]

Out[5]= -1, 5, -2, 1, -3, 4, -5, 2, -4, 3

In[6]:= DTCode[K]

Out[6]= 4 8 10 2 6

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:= ConwayNotation[K]

Out[8]= [32]

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

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, 6, 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[{7, 4}, {3, 5}, {4, 2}, {1, 3}, {2, 6}, {5, 7}, {6, 1}]

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	1
Bridge index	2
Super bridge index	$\{3,4\}$
Nakanishi index	1
Maximal Thurston-Bennequin number	[-8][1]
Hyperbolic Volume	2.82812
A-Polynomial	See Data:5 2/A-polynomial

[edit Notes for 5 2's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	$1$
Topological 4 genus	$1$
Concordance genus	${\textrm {ConcordanceGenus}}({\textrm {Knot}}(5,2))$
Rasmussen s-Invariant	-2

[edit Notes for 5 2's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t+2t^{-1}-3$
Conway polynomial	$2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 7, -2 }
Jones polynomial	$-q^{-6}+q^{-5}-q^{-4}+2q^{-3}-q^{-2}+q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$-a^{6}+a^{4}z^{2}+a^{4}+a^{2}z^{2}+a^{2}$
Kauffman polynomial (db, data sources)	$a^{7}z^{3}-2a^{7}z+a^{6}z^{4}-2a^{6}z^{2}+a^{6}+2a^{5}z^{3}-2a^{5}z+a^{4}z^{4}-a^{4}z^{2}+a^{4}+a^{3}z^{3}+a^{2}z^{2}-a^{2}$
The A2 invariant	$-q^{20}-q^{18}+q^{12}+q^{10}+q^{8}+q^{6}+q^{2}$
The G2 invariant	$q^{100}+q^{96}-q^{94}-q^{92}+q^{90}-q^{88}-q^{84}-q^{82}-q^{78}-q^{76}-q^{74}-q^{72}-q^{68}-q^{66}+q^{64}+q^{60}+q^{56}+q^{54}+2q^{50}-q^{48}+2q^{46}+q^{44}+q^{40}+q^{34}+2q^{24}+q^{20}+q^{14}+q^{10}$

Further Quantum Invariants

Further quantum knot invariants for 5_2.

A1 Invariants.

Weight	Invariant
1	$-q^{13}+q^{7}+q^{5}+q$
2	$q^{36}-q^{32}-q^{26}-q^{20}+q^{14}+q^{10}+2q^{8}+q^{2}$
3	$-q^{69}+q^{65}+q^{63}-q^{59}+q^{55}-q^{51}+q^{47}+q^{45}-q^{43}-q^{37}-2q^{35}-q^{33}-q^{31}+q^{27}+q^{21}+2q^{19}-q^{15}+q^{13}+2q^{11}+q^{9}+q^{3}$
4	$q^{112}-q^{108}-q^{106}-q^{104}+q^{102}+q^{100}+q^{98}-2q^{94}+q^{90}+q^{88}-2q^{84}-q^{82}+2q^{78}+q^{76}-q^{74}+q^{70}+2q^{68}+q^{66}-q^{64}-q^{56}-2q^{54}-q^{52}-q^{50}-q^{48}+q^{44}-q^{42}-q^{40}-q^{38}+2q^{34}-q^{30}-q^{28}+q^{26}+4q^{24}+q^{22}-q^{18}+2q^{14}+q^{12}+q^{10}+q^{4}$
5	$-q^{165}+q^{161}+q^{159}+q^{157}-q^{153}-2q^{151}-q^{149}+q^{145}+2q^{143}+q^{141}-q^{139}-2q^{137}-q^{135}+q^{133}+2q^{131}+2q^{129}-2q^{125}-3q^{123}-q^{121}+q^{119}+2q^{117}+q^{115}-2q^{113}-2q^{111}-q^{109}+q^{107}+3q^{105}+q^{103}-q^{101}-q^{99}+q^{95}+2q^{93}+q^{91}-q^{89}+q^{85}+q^{83}+q^{81}-q^{77}-q^{73}-q^{71}-q^{69}-q^{63}-2q^{61}-2q^{59}-2q^{57}+2q^{53}+q^{51}-q^{49}-2q^{47}-2q^{45}+q^{43}+3q^{41}+3q^{39}-3q^{35}-2q^{33}+3q^{29}+3q^{27}+2q^{25}-q^{21}+q^{17}+q^{15}+q^{13}+q^{11}+q^{5}$
6	$q^{228}-q^{224}-q^{222}-q^{220}+2q^{214}+2q^{212}+q^{210}-q^{206}-2q^{204}-3q^{202}+q^{198}+2q^{196}+2q^{194}+q^{192}-q^{190}-4q^{188}-2q^{186}+2q^{182}+3q^{180}+4q^{178}+q^{176}-3q^{174}-3q^{172}-3q^{170}+2q^{166}+4q^{164}+2q^{162}-2q^{160}-3q^{158}-4q^{156}-q^{154}+2q^{152}+4q^{150}+2q^{148}-q^{146}-2q^{144}-3q^{142}-q^{140}+q^{138}+3q^{136}+q^{134}-2q^{132}-2q^{130}-2q^{128}+q^{124}+2q^{122}+q^{120}-q^{118}+q^{116}+q^{114}+2q^{112}+2q^{110}+2q^{108}+q^{106}-q^{98}-q^{96}-q^{94}+q^{90}-q^{88}-q^{86}-2q^{84}-2q^{82}-2q^{80}+3q^{76}+q^{74}-2q^{70}-4q^{68}-4q^{66}-q^{64}+4q^{62}+3q^{60}+2q^{58}-2q^{56}-4q^{54}-5q^{52}-q^{50}+5q^{48}+4q^{46}+4q^{44}+q^{42}-2q^{40}-4q^{38}-2q^{36}+2q^{34}+2q^{32}+3q^{30}+2q^{28}+q^{26}-q^{24}+q^{20}+q^{16}+q^{14}+q^{12}+q^{6}$

A2 Invariants.

Weight	Invariant
0,1	$-q^{20}-q^{18}+q^{12}+q^{10}+q^{8}+q^{6}+q^{2}$
0,2	$q^{50}+q^{48}+q^{46}-q^{44}-q^{42}-q^{40}-q^{38}-q^{36}-q^{34}-q^{30}-q^{28}-q^{26}+2q^{20}+2q^{18}+2q^{16}+2q^{14}+2q^{12}+q^{10}+q^{4}$
1,0	$-q^{20}-q^{18}+q^{12}+q^{10}+q^{8}+q^{6}+q^{2}$
1,1	$q^{52}+2q^{48}-2q^{46}-2q^{42}+2q^{34}-2q^{32}-5q^{28}-4q^{24}+2q^{22}+q^{20}+2q^{18}+4q^{16}+2q^{14}+4q^{12}+2q^{8}+q^{4}$
2,0	$q^{50}+q^{48}+q^{46}-q^{44}-q^{42}-q^{40}-q^{38}-q^{36}-q^{34}-q^{30}-q^{28}-q^{26}+2q^{20}+2q^{18}+2q^{16}+2q^{14}+2q^{12}+q^{10}+q^{4}$
3,0	$-q^{90}-q^{88}-q^{86}+2q^{82}+2q^{80}+2q^{78}-q^{66}+q^{62}+2q^{60}+q^{58}-q^{54}-q^{52}-3q^{50}-4q^{48}-5q^{46}-4q^{44}-3q^{42}-2q^{40}+2q^{36}+3q^{34}+2q^{32}+3q^{30}+2q^{28}+3q^{26}+2q^{24}+q^{22}+q^{20}+2q^{18}+3q^{16}+2q^{14}+q^{6}$

A3 Invariants.

Weight	Invariant
0,0,1	$-q^{27}-q^{25}-q^{23}+q^{17}+q^{15}+q^{13}+q^{11}+q^{9}+q^{7}+q^{3}$
0,1,0	$q^{42}+q^{38}-2q^{34}-2q^{32}-2q^{30}-2q^{28}-q^{26}+q^{24}+q^{22}+2q^{20}+q^{18}+2q^{16}+q^{14}+q^{12}+2q^{10}+q^{8}+q^{4}$
1,0,0	$-q^{27}-q^{25}-q^{23}+q^{17}+q^{15}+q^{13}+q^{11}+q^{9}+q^{7}+q^{3}$
1,0,1	$q^{68}+2q^{64}-q^{60}-3q^{56}+q^{54}-q^{52}+q^{50}+3q^{48}-q^{46}+q^{44}-2q^{42}-4q^{40}-4q^{38}-4q^{36}-5q^{34}-3q^{32}-q^{30}+5q^{26}+3q^{24}+7q^{22}+6q^{20}+3q^{18}+5q^{16}+q^{14}+2q^{12}+2q^{10}+q^{6}$

A4 Invariants.

Weight	Invariant
0,0,0,1	$-q^{34}-q^{32}-q^{30}-q^{28}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}+q^{12}+q^{10}+q^{8}+q^{4}$
0,1,0,0	$q^{56}+q^{54}+q^{52}+q^{50}+q^{48}-q^{46}-3q^{44}-4q^{42}-4q^{40}-4q^{38}-3q^{36}+q^{32}+2q^{30}+3q^{28}+3q^{26}+2q^{24}+2q^{22}+2q^{20}+2q^{18}+q^{16}+2q^{14}+2q^{12}+q^{10}+q^{6}$
1,0,0,0	$-q^{34}-q^{32}-q^{30}-q^{28}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}+q^{12}+q^{10}+q^{8}+q^{4}$

A5 Invariants.

Weight	Invariant
0,0,0,0,1	$-q^{41}-q^{39}-q^{37}-q^{35}-q^{33}+q^{27}+q^{25}+q^{23}+q^{21}+q^{19}+q^{17}+q^{15}+q^{13}+q^{11}+q^{9}+q^{5}$
1,0,0,0,0	$-q^{41}-q^{39}-q^{37}-q^{35}-q^{33}+q^{27}+q^{25}+q^{23}+q^{21}+q^{19}+q^{17}+q^{15}+q^{13}+q^{11}+q^{9}+q^{5}$

A6 Invariants.

Weight	Invariant
0,0,0,0,0,1	$-q^{48}-q^{46}-q^{44}-q^{42}-q^{40}-q^{38}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}+q^{12}+q^{10}+q^{6}$
1,0,0,0,0,0	$-q^{48}-q^{46}-q^{44}-q^{42}-q^{40}-q^{38}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}+q^{12}+q^{10}+q^{6}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{42}-q^{38}+q^{26}-q^{24}+q^{22}+q^{18}+q^{14}+q^{12}+q^{8}+q^{4}$
1,0	$q^{68}+q^{60}-q^{56}-q^{54}-q^{52}-q^{50}-q^{48}-q^{46}-q^{44}+q^{34}+q^{32}+q^{30}+q^{26}+q^{24}+q^{22}+q^{18}+q^{16}+q^{14}+q^{6}$

B3 Invariants.

Weight	Invariant
1,0,0	$q^{100}+q^{92}-q^{82}-q^{80}-q^{78}-q^{76}-q^{74}-q^{72}-q^{70}-q^{68}-q^{66}-q^{64}+q^{56}+q^{54}+q^{50}+q^{48}+q^{46}+q^{42}+q^{40}+q^{38}+q^{34}+q^{30}+q^{26}+q^{24}+q^{22}+q^{18}+q^{10}$

B4 Invariants.

Weight	Invariant
1,0,0,0	$q^{132}+q^{124}-q^{110}-q^{106}-q^{104}-q^{102}-q^{100}-q^{98}-q^{96}-q^{94}-q^{92}-q^{90}-q^{88}-q^{86}+q^{74}+q^{72}+q^{70}+q^{66}+q^{64}+q^{62}+q^{58}+q^{56}+q^{54}+q^{50}+q^{46}+q^{42}+q^{38}+q^{34}+q^{32}+q^{30}+q^{26}+q^{22}+q^{14}$

C3 Invariants.

Weight	Invariant
1,0,0	$-q^{58}-q^{54}-q^{48}+q^{30}+q^{26}+q^{22}+q^{20}+q^{18}+q^{16}+q^{12}+q^{10}+q^{6}$

C4 Invariants.

Weight	Invariant
1,0,0,0	$-q^{74}-q^{70}-q^{62}-q^{60}-q^{48}+q^{42}+q^{38}+q^{34}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{16}+q^{14}+q^{12}+q^{8}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{58}+q^{54}-q^{48}-2q^{46}-2q^{44}-2q^{42}-2q^{40}-2q^{38}+2q^{32}+q^{30}+2q^{28}+q^{26}+2q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+2q^{14}+q^{12}+q^{10}+q^{6}$

G2 Invariants.

Weight

Invariant

0,1

q^{204}+q^{192}+q^{190}+q^{188}-q^{186}-q^{184}-q^{176}+q^{172}-2q^{168}-q^{166}-q^{156}+q^{154}+q^{152}+q^{142}+q^{140}+q^{138}+2q^{136}+q^{134}-q^{132}-2q^{130}-q^{122}-q^{120}-q^{118}-2q^{116}-2q^{114}-3q^{112}-2q^{110}-q^{108}-2q^{106}-2q^{104}-q^{102}-q^{100}-q^{98}-q^{96}-2q^{94}-2q^{92}+q^{88}+q^{86}+2q^{84}+q^{82}+q^{78}+2q^{74}+2q^{72}+2q^{70}+2q^{68}+3q^{66}+2q^{64}+q^{62}+q^{60}+2q^{58}+2q^{56}+q^{54}+q^{50}+3q^{48}+q^{46}+q^{36}+q^{34}+q^{32}+q^{30}+q^{18}

1,0

q^{100}+q^{96}-q^{94}-q^{92}+q^{90}-q^{88}-q^{84}-q^{82}-q^{78}-q^{76}-q^{74}-q^{72}-q^{68}-q^{66}+q^{64}+q^{60}+q^{56}+q^{54}+2q^{50}-q^{48}+2q^{46}+q^{44}+q^{40}+q^{34}+2q^{24}+q^{20}+q^{14}+q^{10}

.

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["5 2"];

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

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

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

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

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

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]= { 7, -2 }

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

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

Out[8]= $-q^{-6}+q^{-5}-q^{-4}+2q^{-3}-q^{-2}+q^{-1}$

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

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

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

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

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

Out[10]= $a^{7}z^{3}-2a^{7}z+a^{6}z^{4}-2a^{6}z^{2}+a^{6}+2a^{5}z^{3}-2a^{5}z+a^{4}z^{4}-a^{4}z^{2}+a^{4}+a^{3}z^{3}+a^{2}z^{2}-a^{2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11n57,}

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.

In[3]:= K = Knot["5 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]= { $2t+2t^{-1}-3$ , $-q^{-6}+q^{-5}-q^{-4}+2q^{-3}-q^{-2}+q^{-1}$ }

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]= {K11n57,}

Vassiliev invariants

V₂ and V₃:

(2, -3)

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

8

-24

32

{\frac {268}{3}}

{\frac {44}{3}}

-192

-368

-64

-56

{\frac {256}{3}}

288

{\frac {2144}{3}}

{\frac {352}{3}}

{\frac {22951}{15}}

-{\frac {28}{5}}

{\frac {29764}{45}}

{\frac {137}{9}}

{\frac {1351}{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=

-2 is the signature of 5 2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

0

χ

-1

1

-3

1

0

-5

1

-7

1

-9

1

0

-11

0

-13

1

-1

Integral Khovanov Homology

(db, data source)

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

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the

n+1

-dimensional representation of

sl(2)

)

$n$	$J_{n}$
2	$q^{-2}-q^{-3}+3q^{-5}-2q^{-6}-q^{-7}+4q^{-8}-3q^{-9}-q^{-10}+3q^{-11}-2q^{-12}-q^{-13}+2q^{-14}-q^{-15}-q^{-16}+q^{-17}$
3	$q^{-3}-q^{-4}+q^{-6}+2q^{-7}-2q^{-8}-2q^{-9}+2q^{-10}+4q^{-11}-3q^{-12}-3q^{-13}+2q^{-14}+5q^{-15}-4q^{-16}-4q^{-17}+2q^{-18}+4q^{-19}-3q^{-20}-3q^{-21}+2q^{-22}+3q^{-23}-q^{-24}-3q^{-25}+q^{-26}+2q^{-27}-2q^{-29}+q^{-31}+q^{-32}-q^{-33}$
4	$q^{-4}-q^{-5}+q^{-7}+2q^{-9}-3q^{-10}-q^{-11}+2q^{-12}+q^{-13}+5q^{-14}-6q^{-15}-3q^{-16}+2q^{-17}+2q^{-18}+7q^{-19}-8q^{-20}-4q^{-21}+2q^{-22}+2q^{-23}+9q^{-24}-9q^{-25}-5q^{-26}+2q^{-27}+2q^{-28}+8q^{-29}-8q^{-30}-4q^{-31}+2q^{-32}+2q^{-33}+7q^{-34}-6q^{-35}-3q^{-36}+q^{-37}+q^{-38}+6q^{-39}-4q^{-40}-2q^{-41}-q^{-42}+5q^{-44}-2q^{-45}-q^{-46}-q^{-47}-q^{-48}+3q^{-49}-q^{-52}-q^{-53}+q^{-54}$
5	$q^{-5}-q^{-6}+q^{-8}+q^{-11}-2q^{-12}-q^{-13}+2q^{-14}+2q^{-15}+q^{-16}+q^{-17}-5q^{-18}-3q^{-19}+q^{-20}+5q^{-21}+4q^{-22}+q^{-23}-7q^{-24}-6q^{-25}+q^{-26}+6q^{-27}+6q^{-28}+2q^{-29}-9q^{-30}-8q^{-31}+q^{-32}+6q^{-33}+7q^{-34}+3q^{-35}-9q^{-36}-9q^{-37}+q^{-38}+6q^{-39}+8q^{-40}+2q^{-41}-8q^{-42}-8q^{-43}+q^{-44}+6q^{-45}+7q^{-46}+q^{-47}-6q^{-48}-7q^{-49}+5q^{-51}+6q^{-52}+q^{-53}-4q^{-54}-5q^{-55}-2q^{-56}+3q^{-57}+5q^{-58}+q^{-59}-q^{-60}-4q^{-61}-3q^{-62}+q^{-63}+3q^{-64}+2q^{-65}+q^{-66}-2q^{-67}-3q^{-68}+q^{-70}+q^{-71}+2q^{-72}-2q^{-74}-q^{-75}+q^{-78}+q^{-79}-q^{-80}$
6	$q^{-6}-q^{-7}+q^{-9}-q^{-12}+2q^{-13}-2q^{-14}-q^{-15}+3q^{-16}+q^{-17}+q^{-18}-2q^{-19}+2q^{-20}-6q^{-21}-3q^{-22}+5q^{-23}+4q^{-24}+4q^{-25}-2q^{-26}+3q^{-27}-12q^{-28}-7q^{-29}+6q^{-30}+6q^{-31}+8q^{-32}-q^{-33}+4q^{-34}-17q^{-35}-10q^{-36}+6q^{-37}+8q^{-38}+10q^{-39}+6q^{-41}-20q^{-42}-12q^{-43}+6q^{-44}+8q^{-45}+11q^{-46}+8q^{-48}-21q^{-49}-13q^{-50}+6q^{-51}+8q^{-52}+12q^{-53}+8q^{-55}-20q^{-56}-12q^{-57}+6q^{-58}+8q^{-59}+11q^{-60}+6q^{-62}-18q^{-63}-11q^{-64}+5q^{-65}+7q^{-66}+9q^{-67}+6q^{-69}-15q^{-70}-9q^{-71}+3q^{-72}+5q^{-73}+7q^{-74}+q^{-75}+7q^{-76}-12q^{-77}-7q^{-78}+q^{-79}+2q^{-80}+4q^{-81}+2q^{-82}+8q^{-83}-8q^{-84}-5q^{-85}-q^{-86}+q^{-88}+2q^{-89}+8q^{-90}-4q^{-91}-2q^{-92}-2q^{-93}-q^{-94}-q^{-95}+6q^{-97}-q^{-98}-q^{-100}-q^{-101}-2q^{-102}-q^{-103}+3q^{-104}+q^{-106}-q^{-109}-q^{-110}+q^{-111}$
7	$q^{-7}-q^{-8}+q^{-10}-q^{-13}+2q^{-15}-2q^{-16}+2q^{-18}+q^{-19}+q^{-20}-2q^{-21}-2q^{-22}+2q^{-23}-5q^{-24}+4q^{-26}+4q^{-27}+5q^{-28}-2q^{-29}-4q^{-30}-2q^{-31}-10q^{-32}-2q^{-33}+6q^{-34}+6q^{-35}+12q^{-36}+2q^{-37}-6q^{-38}-5q^{-39}-17q^{-40}-5q^{-41}+6q^{-42}+9q^{-43}+18q^{-44}+5q^{-45}-6q^{-46}-7q^{-47}-22q^{-48}-9q^{-49}+7q^{-50}+10q^{-51}+21q^{-52}+7q^{-53}-5q^{-54}-6q^{-55}-25q^{-56}-11q^{-57}+7q^{-58}+10q^{-59}+23q^{-60}+7q^{-61}-4q^{-62}-5q^{-63}-26q^{-64}-12q^{-65}+7q^{-66}+9q^{-67}+24q^{-68}+7q^{-69}-3q^{-70}-6q^{-71}-25q^{-72}-11q^{-73}+7q^{-74}+9q^{-75}+23q^{-76}+7q^{-77}-4q^{-78}-7q^{-79}-23q^{-80}-10q^{-81}+6q^{-82}+8q^{-83}+21q^{-84}+7q^{-85}-5q^{-86}-6q^{-87}-20q^{-88}-8q^{-89}+4q^{-90}+6q^{-91}+18q^{-92}+7q^{-93}-3q^{-94}-4q^{-95}-16q^{-96}-7q^{-97}+2q^{-98}+2q^{-99}+14q^{-100}+8q^{-101}-q^{-102}-q^{-103}-12q^{-104}-6q^{-105}-q^{-106}-2q^{-107}+10q^{-108}+7q^{-109}+q^{-110}+2q^{-111}-6q^{-112}-5q^{-113}-3q^{-114}-5q^{-115}+6q^{-116}+5q^{-117}+q^{-118}+5q^{-119}-2q^{-120}-2q^{-121}-3q^{-122}-6q^{-123}+2q^{-124}+2q^{-125}+4q^{-127}+q^{-128}+q^{-129}-q^{-130}-5q^{-131}-q^{-134}+2q^{-135}+q^{-136}+2q^{-137}+q^{-138}-2q^{-139}-q^{-140}-q^{-142}+q^{-145}+q^{-146}-q^{-147}$

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.

5_1

6_1

@@ Line 22: / Line 22: @@
 <tr align=center>
 <td width=20.%><table cellpadding=0 cellspacing=0>
-  <tr><td>\</td><td>
+  <tr><td>\</td><td>&nbsp;</td><td>r</td></tr>
+<tr><td>&nbsp;</td><td>&nbsp;\&nbsp;</td><td>&nbsp;</td></tr>
+<tr><td>j</td><td>&nbsp;</td><td>\</td></tr>
+</table></td>
+  <td width=10.%>-5</td    ><td width=10.%>-4</td    ><td width=10.%>-3</td    ><td width=10.%>-2</td    ><td width=10.%>-1</td    ><td width=10.%>0</td    ><td width=20.%>&chi;</td></tr>
+<tr align=center><td>-1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td>1</td></tr>
+<tr align=center><td>-3</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td>0</td></tr>
+<tr align=center><td>-5</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
+<tr align=center><td>-7</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
+<tr align=center><td>-9</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>0</td></tr>
+<tr align=center><td>-11</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>0</td></tr>
+<tr align=center><td>-13</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-1</td></tr>
+</table> |
+coloured_jones_2 = <math> q^{-2} - q^{-3} +3 q^{-5} -2 q^{-6} - q^{-7} +4 q^{-8} -3 q^{-9} - q^{-10} +3 q^{-11} -2 q^{-12} - q^{-13} +2 q^{-14} - q^{-15} - q^{-16} + q^{-17} </math> |
+coloured_jones_3 = <math> q^{-3} - q^{-4} + q^{-6} +2 q^{-7} -2 q^{-8} -2 q^{-9} +2 q^{-10} +4 q^{-11} -3 q^{-12} -3 q^{-13} +2 q^{-14} +5 q^{-15} -4 q^{-16} -4 q^{-17} +2 q^{-18} +4 q^{-19} -3 q^{-20} -3 q^{-21} +2 q^{-22} +3 q^{-23} - q^{-24} -3 q^{-25} + q^{-26} +2 q^{-27} -2 q^{-29} + q^{-31} + q^{-32} - q^{-33} </math> |
+coloured_jones_4 = <math> q^{-4} - q^{-5} + q^{-7} +2 q^{-9} -3 q^{-10} - q^{-11} +2 q^{-12} + q^{-13} +5 q^{-14} -6 q^{-15} -3 q^{-16} +2 q^{-17} +2 q^{-18} +7 q^{-19} -8 q^{-20} -4 q^{-21} +2 q^{-22} +2 q^{-23} +9 q^{-24} -9 q^{-25} -5 q^{-26} +2 q^{-27} +2 q^{-28} +8 q^{-29} -8 q^{-30} -4 q^{-31} +2 q^{-32} +2 q^{-33} +7 q^{-34} -6 q^{-35} -3 q^{-36} + q^{-37} + q^{-38} +6 q^{-39} -4 q^{-40} -2 q^{-41} - q^{-42} +5 q^{-44} -2 q^{-45} - q^{-46} - q^{-47} - q^{-48} +3 q^{-49} - q^{-52} - q^{-53} + q^{-54} </math> |
+coloured_jones_5 = <math> q^{-5} - q^{-6} + q^{-8} + q^{-11} -2 q^{-12} - q^{-13} +2 q^{-14} +2 q^{-15} + q^{-16} + q^{-17} -5 q^{-18} -3 q^{-19} + q^{-20} +5 q^{-21} +4 q^{-22} + q^{-23} -7 q^{-24} -6 q^{-25} + q^{-26} +6 q^{-27} +6 q^{-28} +2 q^{-29} -9 q^{-30} -8 q^{-31} + q^{-32} +6 q^{-33} +7 q^{-34} +3 q^{-35} -9 q^{-36} -9 q^{-37} + q^{-38} +6 q^{-39} +8 q^{-40} +2 q^{-41} -8 q^{-42} -8 q^{-43} + q^{-44} +6 q^{-45} +7 q^{-46} + q^{-47} -6 q^{-48} -7 q^{-49} +5 q^{-51} +6 q^{-52} + q^{-53} -4 q^{-54} -5 q^{-55} -2 q^{-56} +3 q^{-57} +5 q^{-58} + q^{-59} - q^{-60} -4 q^{-61} -3 q^{-62} + q^{-63} +3 q^{-64} +2 q^{-65} + q^{-66} -2 q^{-67} -3 q^{-68} + q^{-70} + q^{-71} +2 q^{-72} -2 q^{-74} - q^{-75} + q^{-78} + q^{-79} - q^{-80} </math> |
+coloured_jones_6 = <math> q^{-6} - q^{-7} + q^{-9} - q^{-12} +2 q^{-13} -2 q^{-14} - q^{-15} +3 q^{-16} + q^{-17} + q^{-18} -2 q^{-19} +2 q^{-20} -6 q^{-21} -3 q^{-22} +5 q^{-23} +4 q^{-24} +4 q^{-25} -2 q^{-26} +3 q^{-27} -12 q^{-28} -7 q^{-29} +6 q^{-30} +6 q^{-31} +8 q^{-32} - q^{-33} +4 q^{-34} -17 q^{-35} -10 q^{-36} +6 q^{-37} +8 q^{-38} +10 q^{-39} +6 q^{-41} -20 q^{-42} -12 q^{-43} +6 q^{-44} +8 q^{-45} +11 q^{-46} +8 q^{-48} -21 q^{-49} -13 q^{-50} +6 q^{-51} +8 q^{-52} +12 q^{-53} +8 q^{-55} -20 q^{-56} -12 q^{-57} +6 q^{-58} +8 q^{-59} +11 q^{-60} +6 q^{-62} -18 q^{-63} -11 q^{-64} +5 q^{-65} +7 q^{-66} +9 q^{-67} +6 q^{-69} -15 q^{-70} -9 q^{-71} +3 q^{-72} +5 q^{-73} +7 q^{-74} + q^{-75} +7 q^{-76} -12 q^{-77} -7 q^{-78} + q^{-79} +2 q^{-80} +4 q^{-81} +2 q^{-82} +8 q^{-83} -8 q^{-84} -5 q^{-85} - q^{-86} + q^{-88} +2 q^{-89} +8 q^{-90} -4 q^{-91} -2 q^{-92} -2 q^{-93} - q^{-94} - q^{-95} +6 q^{-97} - q^{-98} - q^{-100} - q^{-101} -2 q^{-102} - q^{-103} +3 q^{-104} + q^{-106} - q^{-109} - q^{-110} + q^{-111} </math> |
+coloured_jones_7 = <math> q^{-7} - q^{-8} + q^{-10} - q^{-13} +2 q^{-15} -2 q^{-16} +2 q^{-18} + q^{-19} + q^{-20} -2 q^{-21} -2 q^{-22} +2 q^{-23} -5 q^{-24} +4 q^{-26} +4 q^{-27} +5 q^{-28} -2 q^{-29} -4 q^{-30} -2 q^{-31} -10 q^{-32} -2 q^{-33} +6 q^{-34} +6 q^{-35} +12 q^{-36} +2 q^{-37} -6 q^{-38} -5 q^{-39} -17 q^{-40} -5 q^{-41} +6 q^{-42} +9 q^{-43} +18 q^{-44} +5 q^{-45} -6 q^{-46} -7 q^{-47} -22 q^{-48} -9 q^{-49} +7 q^{-50} +10 q^{-51} +21 q^{-52} +7 q^{-53} -5 q^{-54} -6 q^{-55} -25 q^{-56} -11 q^{-57} +7 q^{-58} +10 q^{-59} +23 q^{-60} +7 q^{-61} -4 q^{-62} -5 q^{-63} -26 q^{-64} -12 q^{-65} +7 q^{-66} +9 q^{-67} +24 q^{-68} +7 q^{-69} -3 q^{-70} -6 q^{-71} -25 q^{-72} -11 q^{-73} +7 q^{-74} +9 q^{-75} +23 q^{-76} +7 q^{-77} -4 q^{-78} -7 q^{-79} -23 q^{-80} -10 q^{-81} +6 q^{-82} +8 q^{-83} +21 q^{-84} +7 q^{-85} -5 q^{-86} -6 q^{-87} -20 q^{-88} -8 q^{-89} +4 q^{-90} +6 q^{-91} +18 q^{-92} +7 q^{-93} -3 q^{-94} -4 q^{-95} -16 q^{-96} -7 q^{-97} +2 q^{-98} +2 q^{-99} +14 q^{-100} +8 q^{-101} - q^{-102} - q^{-103} -12 q^{-104} -6 q^{-105} - q^{-106} -2 q^{-107} +10 q^{-108} +7 q^{-109} + q^{-110} +2 q^{-111} -6 q^{-112} -5 q^{-113} -3 q^{-114} -5 q^{-115} +6 q^{-116} +5 q^{-117} + q^{-118} +5 q^{-119} -2 q^{-120} -2 q^{-121} -3 q^{-122} -6 q^{-123} +2 q^{-124} +2 q^{-125} +4 q^{-127} + q^{-128} + q^{-129} - q^{-130} -5 q^{-131} - q^{-134} +2 q^{-135} + q^{-136} +2 q^{-137} + q^{-138} -2 q^{-139} - q^{-140} - q^{-142} + q^{-145} + q^{-146} - q^{-147} </math> |
+computer_talk =
+         <table>
+         <tr valign=top>
+         <td><pre style="color: blue; border: 0px; padding: 0em">In[1]:=&nbsp;&nbsp;&nbsp;&nbsp;</pre></td>
+         <td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
+         </tr>
+         <tr valign=top><td colspan=2><nowiki>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</nowiki></td></tr>
+         </table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[2]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[Knot[5, 2]]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[2]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 10, 6, 1], X[9, 6, 10, 7],
+  X[7, 2, 8, 3]]</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[3]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[Knot[5, 2]]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[3]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[-1, 5, -2, 1, -3, 4, -5, 2, -4, 3]</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[4]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[Knot[5, 2]]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[4]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[4, 8, 10, 2, 6]</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[5]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>br = BR[Knot[5, 2]]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[5]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>BR[3, {-1, -1, -1, -2, 1, -2}]</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[6]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{First[br], Crossings[br]}</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[6]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{3, 6}</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[7]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>BraidIndex[Knot[5, 2]]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[7]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>3</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[8]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Show[DrawMorseLink[Knot[5, 2]]]</nowiki></code></td></tr>
+<tr align=left><td></td><td>[[Image:5_2_ML.gif]]</td></tr><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[8]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>-Graphics-</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[9]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> (#[Knot[5, 2]]&) /@ {
+                 SymmetryType, UnknottingNumber, ThreeGenus,
+                 BridgeIndex, SuperBridgeIndex, NakanishiIndex
+                }</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[9]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Reversible, 1, 1, 2, {3, 4}, 1}</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[10]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>alex = Alexander[Knot[5, 2]][t]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[10]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>     2
+-3 + - + 2 t
+     t</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[11]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Conway[Knot[5, 2]][z]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[11]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>       2
++ 2 z</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[12]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[12]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Knot[5, 2]}</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[13]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{KnotDet[Knot[5, 2]], KnotSignature[Knot[5, 2]]}</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[13]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{7, -2}</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[14]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Jones[Knot[5, 2]][q]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[14]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>  -6    -5    -4   2     -2   1
+-q   + q   - q   + -- - q   + -
+q
+                   q</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[15]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[15]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Knot[5, 2], Knot[11, NonAlternating, 57]}</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[16]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>A2Invariant[Knot[5, 2]][q]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[16]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>  -20    -18    -12    -10    -8    -6    -2
+-q    - q    + q    + q    + q   + q   + q</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[17]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>HOMFLYPT[Knot[5, 2]][a, z]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[17]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2    4    6    2  2    4  2
+a  + a  - a  + a  z  + a  z</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[18]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kauffman[Knot[5, 2]][a, z]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[18]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>  2    4    6      5        7      2  2    4  2      6  2    3  3
+-a  + a  + a  - 2 a  z - 2 a  z + a  z  - a  z  - 2 a  z  + a  z  +
+3    7  3    4  4    6  4
+a  z  + a  z  + a  z  + a  z</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[19]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Vassiliev[2][Knot[5, 2]], Vassiliev[3][Knot[5, 2]]}</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[19]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{2, -3}</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[20]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kh[Knot[5, 2]][q, t]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[20]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -3   1     1        1       1       1       1      1
+q   + - + ------ + ----- + ----- + ----- + ----- + ----
+      q    13  5    9  4    9  3    7  2    5  2    3
+          q   t    q  t    q  t    q  t    q  t    q  t</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[21]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>ColouredJones[Knot[5, 2], 2][q]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[21]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -17    -16    -15    2     -13    2     3     -10   3    4     -7
+q    - q    - q    + --- - q    - --- + --- - q    - -- + -- - q   -
+12    11           9    8
+                     q            q     q            q    q
+3     -3    -2
+  -- + -- - q   + q
+5
+  q    q</nowiki></code></td></tr>
+</table>  }}

5 2: Difference between revisions

Latest revision as of 06:15, 21 January 2008

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Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

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