5 1: Difference between revisions

Revision as of 15:03, 1 September 2005

4_1

5_2

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 5 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 5 1 at Knotilus!

[edit 5 1 Quick Notes]

An interlaced pentagram, this is known variously as the "Cinquefoil Knot", after certain herbs and shrubs of the rose family which have 5-lobed leaves and 5-petaled flowers (see e.g. [4]), as the "Pentafoil Knot" (visit Bert Jagers' pentafoil page), as the "Double Overhand Knot", as 5_1, or finally as the torus knot T(5,2).

When taken off the post the strangle knot (hitch) of practical knot tying deforms to 5_1

[edit 5 1 Further Notes and Views]

A kolam of a 2x3 dot array	The VISA Interlink Logo [1]	Version of the US bicentennial emblem
A pentagonal table by Bob Mackay [2]	The Utah State Parks logo	As impossible object ("Penrose" pentagram)
Folded ribbon which is single-sided (more complex version of Möbius Strip).	Non-pentagonal shape.	Pentagram of circles.
Alternate pentagram of intersecting circles.	3D-looking rendition.	Partial view of US bicentennial logo on a shirt seen in Lisboa [3]
Non-prime knot with two 5_1 configurations on a closed loop.	Knotted epitrochoid	Sum of two 5_1s, Vienna, orthodox church

This sentence was last edited by Dror. Sometime later, Scott added this sentence.

Knot presentations

Planar diagram presentation	X₁₆₂₇ X₃₈₄₉ X_5,10,6,1 X₇₂₈₃ X_9,4,10,5
Gauss code	-1, 4, -2, 5, -3, 1, -4, 2, -5, 3
Dowker-Thistlethwaite code	6 8 10 2 4
Conway Notation	[5]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 5, width is 2,

Braid index is 2

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

[edit Notes on presentations of 5 1]

Knot 5_1.

A graph, knot 5_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.org/wiki/KnotTheory.

In[3]:= K = Knot["5 1"];

In[4]:= PD[K]

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

Out[4]= X₁₆₂₇ X₃₈₄₉ X_5,10,6,1 X₇₂₈₃ X_9,4,10,5

In[5]:= GaussCode[K]

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

In[6]:= DTCode[K]

Out[6]= 6 8 10 2 4

(The path below may be different on your system)

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

In[8]:= ConwayNotation[K]

Out[8]= [5]

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

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]= { 2, 5, 2 }

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

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	2
Bridge index	2
Super bridge index	3
Nakanishi index	1
Maximal Thurston-Bennequin number	[-10][3]
Hyperbolic Volume	Not hyperbolic
A-Polynomial	See Data:5 1/A-polynomial

[edit Notes for 5 1's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 5 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{2}+t^{-2}-t-t^{-1}+1$
Conway polynomial	$z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 5, -4 }
Jones polynomial	$-q^{-7}+q^{-6}-q^{-5}+q^{-4}+q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$a^{6}\left(-z^{2}\right)-2a^{6}+a^{4}z^{4}+4a^{4}z^{2}+3a^{4}$
Kauffman polynomial (db, data sources)	$a^{9}z+a^{8}z^{2}+a^{7}z^{3}-a^{7}z+a^{6}z^{4}-3a^{6}z^{2}+2a^{6}+a^{5}z^{3}-2a^{5}z+a^{4}z^{4}-4a^{4}z^{2}+3a^{4}$
The A2 invariant	$-q^{22}-q^{20}-q^{18}+q^{14}+q^{12}+2q^{10}+q^{8}+q^{6}$
The G2 invariant	$q^{120}-q^{100}-q^{98}-q^{92}-q^{90}-q^{88}-q^{82}-q^{80}-q^{78}-q^{72}+q^{58}+q^{56}+q^{52}+2q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+2q^{40}+q^{38}+q^{34}+q^{32}+q^{30}$

Further Quantum Invariants

Further quantum knot invariants for 5_1.

A1 Invariants.

Weight	Invariant
1	$-q^{15}+q^{7}+q^{5}+q^{3}$
2	$q^{40}-q^{32}-q^{30}-q^{28}+q^{14}+q^{12}+q^{10}+q^{8}+q^{6}$
3	$-q^{75}+q^{67}+q^{65}+q^{63}-q^{49}-q^{47}-q^{45}-q^{43}-q^{41}+q^{21}+q^{19}+q^{17}+q^{15}+q^{13}+q^{11}+q^{9}$
4	$q^{120}-q^{112}-q^{110}-q^{108}+q^{94}+q^{92}+q^{90}+q^{88}+q^{86}-q^{66}-q^{64}-q^{62}-q^{60}-q^{58}-q^{56}-q^{54}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}+q^{12}$
5	$-q^{175}+q^{167}+q^{165}+q^{163}-q^{149}-q^{147}-q^{145}-q^{143}-q^{141}+q^{121}+q^{119}+q^{117}+q^{115}+q^{113}+q^{111}+q^{109}-q^{83}-q^{81}-q^{79}-q^{77}-q^{75}-q^{73}-q^{71}-q^{69}-q^{67}+q^{35}+q^{33}+q^{31}+q^{29}+q^{27}+q^{25}+q^{23}+q^{21}+q^{19}+q^{17}+q^{15}$
6	$q^{240}-q^{232}-q^{230}-q^{228}+q^{214}+q^{212}+q^{210}+q^{208}+q^{206}-q^{186}-q^{184}-q^{182}-q^{180}-q^{178}-q^{176}-q^{174}+q^{148}+q^{146}+q^{144}+q^{142}+q^{140}+q^{138}+q^{136}+q^{134}+q^{132}-q^{100}-q^{98}-q^{96}-q^{94}-q^{92}-q^{90}-q^{88}-q^{86}-q^{84}-q^{82}-q^{80}+q^{42}+q^{40}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18}$
8	$q^{400}-q^{392}-q^{390}-q^{388}+q^{374}+q^{372}+q^{370}+q^{368}+q^{366}-q^{346}-q^{344}-q^{342}-q^{340}-q^{338}-q^{336}-q^{334}+q^{308}+q^{306}+q^{304}+q^{302}+q^{300}+q^{298}+q^{296}+q^{294}+q^{292}-q^{260}-q^{258}-q^{256}-q^{254}-q^{252}-q^{250}-q^{248}-q^{246}-q^{244}-q^{242}-q^{240}+q^{202}+q^{200}+q^{198}+q^{196}+q^{194}+q^{192}+q^{190}+q^{188}+q^{186}+q^{184}+q^{182}+q^{180}+q^{178}-q^{134}-q^{132}-q^{130}-q^{128}-q^{126}-q^{124}-q^{122}-q^{120}-q^{118}-q^{116}-q^{114}-q^{112}-q^{110}-q^{108}-q^{106}+q^{56}+q^{54}+q^{52}+q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+q^{40}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{22}-q^{20}-q^{18}+q^{14}+q^{12}+2q^{10}+q^{8}+q^{6}$
1,1	$q^{60}-2q^{36}-2q^{34}-4q^{32}-4q^{30}-3q^{28}+2q^{24}+4q^{22}+5q^{20}+4q^{18}+4q^{16}+2q^{14}+q^{12}$
2,0	$q^{54}+q^{52}+2q^{50}+q^{48}-2q^{44}-3q^{42}-3q^{40}-3q^{38}-2q^{36}-q^{34}+q^{28}+q^{26}+2q^{24}+2q^{22}+3q^{20}+2q^{18}+2q^{16}+q^{14}+q^{12}$
3,0	$-q^{96}-q^{94}-2q^{92}-2q^{90}-q^{88}+q^{86}+3q^{84}+4q^{82}+5q^{80}+4q^{78}+4q^{76}+2q^{74}+q^{72}-q^{70}-2q^{68}-3q^{66}-4q^{64}-5q^{62}-5q^{60}-5q^{58}-4q^{56}-3q^{54}-2q^{52}-q^{50}+q^{42}+q^{40}+2q^{38}+2q^{36}+3q^{34}+3q^{32}+4q^{30}+3q^{28}+3q^{26}+2q^{24}+2q^{22}+q^{20}+q^{18}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{50}-q^{36}-2q^{34}-3q^{32}-3q^{30}-2q^{28}-q^{26}+2q^{24}+3q^{22}+4q^{20}+3q^{18}+3q^{16}+q^{14}+q^{12}$
1,0,0	$-q^{29}-q^{27}-2q^{25}-q^{23}+q^{19}+2q^{17}+2q^{15}+2q^{13}+q^{11}+q^{9}$
1,0,1	$q^{80}+q^{58}+q^{56}+3q^{54}+3q^{52}+2q^{50}-q^{48}-5q^{46}-9q^{44}-12q^{42}-12q^{40}-9q^{38}-3q^{36}+2q^{34}+7q^{32}+10q^{30}+11q^{28}+10q^{26}+7q^{24}+5q^{22}+2q^{20}+q^{18}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{64}+q^{62}+q^{60}+q^{58}+q^{56}-q^{50}-2q^{48}-4q^{46}-5q^{44}-6q^{42}-6q^{40}-4q^{38}-q^{36}+2q^{34}+4q^{32}+7q^{30}+6q^{28}+6q^{26}+4q^{24}+3q^{22}+q^{20}+q^{18}$
1,0,0,0	$-q^{36}-q^{34}-2q^{32}-2q^{30}-q^{28}+q^{24}+2q^{22}+3q^{20}+2q^{18}+2q^{16}+q^{14}+q^{12}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{50}-q^{36}-q^{32}-q^{30}+q^{26}+q^{22}+2q^{20}+q^{18}+q^{16}+q^{14}+q^{12}$
1,0	$q^{80}-q^{58}-q^{56}-q^{54}-q^{52}-2q^{50}-q^{48}-q^{46}-q^{44}+q^{38}+q^{36}+2q^{34}+q^{32}+2q^{30}+q^{28}+2q^{26}+q^{24}+q^{22}+q^{18}$

B3 Invariants.

Weight	Invariant
1,0,0	$q^{120}-q^{86}-q^{82}-q^{80}-2q^{78}-q^{76}-2q^{74}-q^{72}-2q^{70}-q^{68}-q^{66}-q^{64}+q^{58}+q^{56}+2q^{54}+q^{52}+3q^{50}+q^{48}+3q^{46}+q^{44}+2q^{42}+q^{40}+2q^{38}+q^{34}+q^{30}$

B4 Invariants.

Weight	Invariant
1,0,0,0	$q^{160}-q^{114}-q^{110}-2q^{106}-q^{104}-2q^{102}-q^{100}-2q^{98}-q^{96}-2q^{94}-q^{92}-2q^{90}-q^{88}-q^{86}+q^{78}+2q^{74}+q^{72}+3q^{70}+q^{68}+3q^{66}+q^{64}+3q^{62}+q^{60}+3q^{58}+q^{56}+2q^{54}+2q^{50}+q^{46}+q^{42}$

C3 Invariants.

Weight	Invariant
1,0,0	$-q^{70}-q^{50}-q^{48}-q^{46}-q^{44}-q^{42}-q^{40}+q^{34}+q^{32}+2q^{30}+2q^{28}+2q^{26}+q^{24}+2q^{22}+q^{20}+q^{18}$

C4 Invariants.

Weight	Invariant
1,0,0,0	$-q^{90}-q^{64}-q^{62}-2q^{60}-q^{58}-q^{56}-q^{54}-q^{52}-q^{50}-q^{48}+q^{44}+2q^{42}+2q^{40}+2q^{38}+2q^{36}+2q^{34}+2q^{32}+2q^{30}+2q^{28}+q^{26}+q^{24}$

D4 Invariants.

Weight	Invariant
0,1,0,0	$q^{120}-q^{100}-q^{98}-3q^{96}-3q^{94}-q^{92}-q^{90}+2q^{88}+6q^{86}+9q^{84}+11q^{82}+14q^{80}+11q^{78}+9q^{76}+3q^{74}-4q^{72}-12q^{70}-18q^{68}-24q^{66}-27q^{64}-27q^{62}-24q^{60}-17q^{58}-11q^{56}+7q^{52}+14q^{50}+19q^{48}+22q^{46}+19q^{44}+19q^{42}+14q^{40}+10q^{38}+6q^{36}+4q^{34}+q^{32}+q^{30}$
1,0,0,0	$q^{70}-q^{50}-q^{48}-3q^{46}-3q^{44}-3q^{42}-3q^{40}-2q^{38}+q^{34}+3q^{32}+4q^{30}+4q^{28}+4q^{26}+3q^{24}+2q^{22}+q^{20}+q^{18}$

G2 Invariants.

Weight

Invariant

0,1

q^{240}-q^{198}-q^{192}+q^{178}+q^{176}+q^{172}+2q^{170}+q^{168}+q^{166}+q^{164}+q^{162}+2q^{160}+q^{158}+q^{154}+q^{152}-q^{148}-q^{146}-q^{144}-q^{142}-2q^{140}-3q^{138}-2q^{136}-2q^{134}-3q^{132}-4q^{130}-4q^{128}-3q^{126}-3q^{124}-4q^{122}-4q^{120}-3q^{118}-2q^{116}-2q^{114}-3q^{112}-2q^{110}+q^{102}+q^{100}+2q^{98}+3q^{96}+2q^{94}+2q^{92}+4q^{90}+3q^{88}+3q^{86}+4q^{84}+3q^{82}+3q^{80}+4q^{78}+2q^{76}+2q^{74}+3q^{72}+2q^{70}+q^{68}+2q^{66}+q^{64}+q^{62}+q^{60}+q^{54}

1,0

q^{120}-q^{100}-q^{98}-q^{92}-q^{90}-q^{88}-q^{82}-q^{80}-q^{78}-q^{72}+q^{58}+q^{56}+q^{52}+2q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+2q^{40}+q^{38}+q^{34}+q^{32}+q^{30}

.

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 1"];

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

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

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

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

Out[5]= $z^{4}+3z^{2}+1$

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]= { 5, -4 }

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

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

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

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

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

Out[9]= $a^{6}\left(-z^{2}\right)-2a^{6}+a^{4}z^{4}+4a^{4}z^{2}+3a^{4}$

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

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

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {[[10_132]], }

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {[[10_132]], }

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 1"];

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]= { $t^{2}+t^{-2}-t-t^{-1}+1$ , $-q^{-7}+q^{-6}-q^{-5}+q^{-4}+q^{-2}$ }

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]= {[[10_132]], }

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

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

Out[6]= {[[10_132]], }

Vassiliev invariants

V₂ and V₃:

(3, -5)

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

12

-40

72

174

26

-480

-{\frac {2512}{3}}

-{\frac {448}{3}}

-104

288

800

2088

312

{\frac {41151}{10}}

{\frac {2494}{15}}

{\frac {7634}{5}}

{\frac {43}{2}}

{\frac {1951}{10}}

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=

-4 is the signature of 5 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

0

χ

-3

1

-5

1

-7

1

-9

0

-11

1

0

-13

0

-15

1

-1

Integral Khovanov Homology

(db, data source)

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

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the

n+1

-dimensional representation of

sl(2)

)

$n$	$J_{n}$
2	$q^{-4}+q^{-7}-q^{-9}+q^{-10}-q^{-12}+q^{-13}-2q^{-15}+q^{-16}-q^{-18}+q^{-19}$
3	$q^{-6}+q^{-10}-q^{-13}+q^{-14}-q^{-17}+q^{-18}-q^{-21}-q^{-25}+q^{-27}-q^{-29}+q^{-31}+q^{-35}-q^{-36}$
4	$q^{-8}+q^{-13}-q^{-17}+q^{-18}-q^{-22}+q^{-23}-q^{-27}+q^{-28}-q^{-29}-q^{-32}+q^{-33}-q^{-34}+q^{-36}-q^{-37}+q^{-38}-q^{-39}+q^{-41}-q^{-42}+q^{-43}-q^{-44}+q^{-45}+q^{-46}-q^{-47}+q^{-48}-q^{-49}+q^{-51}-q^{-52}+q^{-53}-q^{-54}-q^{-57}+q^{-58}$
5	$q^{-10}+q^{-16}-q^{-21}+q^{-22}-q^{-27}+q^{-28}-q^{-33}+q^{-34}-q^{-36}-q^{-39}+q^{-40}-q^{-42}+q^{-46}-q^{-48}+q^{-52}-q^{-54}+q^{-57}+q^{-58}-q^{-60}+q^{-63}-q^{-66}+q^{-69}-q^{-72}-q^{-73}+q^{-75}-q^{-79}+q^{-81}+q^{-84}-q^{-85}$
6	$q^{-12}+q^{-19}-q^{-25}+q^{-26}-q^{-32}+q^{-33}-q^{-39}+q^{-40}-q^{-43}-q^{-46}+q^{-47}-q^{-50}-q^{-53}+2q^{-54}-q^{-57}-q^{-60}+2q^{-61}-q^{-64}-q^{-67}+2q^{-68}+q^{-69}-q^{-71}-q^{-74}+2q^{-75}+q^{-76}-2q^{-78}-q^{-81}+2q^{-82}+q^{-83}-2q^{-85}-q^{-88}+2q^{-89}-2q^{-92}-q^{-95}+2q^{-96}+q^{-97}-2q^{-99}-q^{-102}+2q^{-103}+q^{-104}-q^{-106}-q^{-109}+2q^{-110}-q^{-113}-q^{-116}+q^{-117}$
7	$q^{-14}+q^{-22}-q^{-29}+q^{-30}-q^{-37}+q^{-38}-q^{-45}+q^{-46}-q^{-50}-q^{-53}+q^{-54}-q^{-58}-q^{-61}+q^{-62}+q^{-63}-q^{-66}-q^{-69}+q^{-70}+q^{-71}-q^{-74}-q^{-77}+q^{-78}+q^{-79}+q^{-81}-q^{-82}-q^{-85}+q^{-86}+q^{-87}+q^{-89}-q^{-90}-q^{-92}-q^{-93}+q^{-94}+q^{-95}+q^{-97}-q^{-98}-q^{-100}-q^{-101}+q^{-102}+q^{-103}+q^{-105}-q^{-106}-q^{-107}-q^{-108}-q^{-109}+q^{-110}+q^{-111}+q^{-113}-q^{-114}-q^{-115}-q^{-117}+q^{-118}+q^{-119}+q^{-121}-q^{-122}-q^{-123}-q^{-125}+q^{-126}+q^{-127}+q^{-128}+q^{-129}-q^{-130}-q^{-131}-q^{-133}+q^{-134}+q^{-136}+q^{-137}-q^{-138}-q^{-139}-q^{-141}+q^{-142}+q^{-145}-q^{-146}-q^{-147}+q^{-150}+q^{-153}-q^{-154}$

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.

4_1

5_2

@@ Line 48: / Line 48: @@
          <tr valign=top><td colspan=2><nowiki>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</nowiki></td></tr>
          </table>
-         <nowiki>{{InOut &#124;
+         <nowiki></nowiki>{{InOut &#124;
 n = 2 &#124;
 in =  <nowiki>PD[Knot[5, 1]]</nowiki> &#124;
 out = <nowiki>PD[X[1, 6, 2, 7], X[3, 8, 4, 9], X[5, 10, 6, 1], X[7, 2, 8, 3],
-  X[9, 4, 10, 5]]</nowiki> }}</nowiki>
+  X[9, 4, 10, 5]]</nowiki> }}<nowiki></nowiki>
-         <nowiki>{{InOut &#124;
+         <nowiki></nowiki>{{InOut &#124;
 n = 3 &#124;
 in =  <nowiki>GaussCode[Knot[5, 1]]</nowiki> &#124;
-out = <nowiki>GaussCode[-1, 4, -2, 5, -3, 1, -4, 2, -5, 3]</nowiki> }}</nowiki>
+out = <nowiki>GaussCode[-1, 4, -2, 5, -3, 1, -4, 2, -5, 3]</nowiki> }}<nowiki></nowiki>
-         <nowiki>{{InOut &#124;
+         <nowiki></nowiki>{{InOut &#124;
 n = 4 &#124;
 in =  <nowiki>DTCode[Knot[5, 1]]</nowiki> &#124;
-out = <nowiki>DTCode[6, 8, 10, 2, 4]</nowiki> }}</nowiki>
+out = <nowiki>DTCode[6, 8, 10, 2, 4]</nowiki> }}<nowiki></nowiki>
-         <nowiki>{{InOut &#124;
+         <nowiki></nowiki>{{InOut &#124;
 n = 5 &#124;
 in =  <nowiki>br = BR[Knot[5, 1]]</nowiki> &#124;
-out = <nowiki>BR[2, {-1, -1, -1, -1, -1}]</nowiki> }}</nowiki>
+out = <nowiki>BR[2, {-1, -1, -1, -1, -1}]</nowiki> }}<nowiki></nowiki>
-         <nowiki>{{InOut &#124;
+         <nowiki></nowiki>{{InOut &#124;
 n = 6 &#124;
 in =  <nowiki>{First[br], Crossings[br]}</nowiki> &#124;
-out = <nowiki>{2, 5}</nowiki> }}</nowiki>
+out = <nowiki>{2, 5}</nowiki> }}<nowiki></nowiki>
-         <nowiki>{{InOut &#124;
+         <nowiki></nowiki>{{InOut &#124;
 n = 7 &#124;
 in =  <nowiki>BraidIndex[Knot[5, 1]]</nowiki> &#124;
-out = <nowiki>2</nowiki> }}</nowiki>
+out = <nowiki>2</nowiki> }}<nowiki></nowiki>
-         <nowiki><tr  valign=top><td><pre style="color: blue; border: 0px; padding:  0em"><nowiki>In[8]:=</nowiki></pre></td><td><pre style="color: red;  border: 0px; padding:  0em"><nowiki>Show[DrawMorseLink[Knot[5, 1]]]</nowiki></pre></td></tr><tr><td></td><td  align=left>[[Image:5_1_ML.gif]]</td></tr><tr valign=top><td><tt><font  color=blue>Out[8]=</font></tt><td><tt><font  color=black>-Graphics-</font></tt></td></tr></nowiki>
+         <nowiki></nowiki><tr  valign=top><td><pre style="color: blue; border: 0px; padding:  0em"><nowiki>In[8]:=</nowiki></pre></td><td><pre style="color: red;  border: 0px; padding:  0em"><nowiki>Show[DrawMorseLink[Knot[5, 1]]]</nowiki></pre></td></tr><tr><td></td><td  align=left>[[Image:5_1_ML.gif]]</td></tr><tr valign=top><td><tt><font  color=blue>Out[8]=</font></tt><td><tt><font  color=black>-Graphics-</font></tt></td></tr><nowiki></nowiki>
-         <nowiki>{{InOut &#124;
+         <nowiki></nowiki>{{InOut &#124;
 n = 9 &#124;
 in =  <nowiki> (#[Knot[5, 1]]&) /@ {
@@ Line 81: / Line 81: @@
                  BridgeIndex, SuperBridgeIndex, NakanishiIndex
                 }</nowiki> &#124;
-out = <nowiki>{Reversible, 2, 2, 2, 3, 1}</nowiki> }}</nowiki>
+out = <nowiki>{Reversible, 2, 2, 2, 3, 1}</nowiki> }}<nowiki></nowiki>
-         <nowiki>{{InOut &#124;
+         <nowiki></nowiki>{{InOut &#124;
 n = 10 &#124;
 in =  <nowiki>alex = Alexander[Knot[5, 1]][t]</nowiki> &#124;
 out = <nowiki>     -2   1        2
 + t   - - - t + t
-          t</nowiki> }}</nowiki>
+          t</nowiki> }}<nowiki></nowiki>
-         <nowiki>{{InOut &#124;
+         <nowiki></nowiki>{{InOut &#124;
 n = 11 &#124;
 in =  <nowiki>Conway[Knot[5, 1]][z]</nowiki> &#124;
 out = <nowiki>       2    4
-+ 3 z  + z</nowiki> }}</nowiki>
++ 3 z  + z</nowiki> }}<nowiki></nowiki>
-         <nowiki>{{InOut &#124;
+         <nowiki></nowiki>{{InOut &#124;
 n = 12 &#124;
 in =  <nowiki>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki> &#124;
-out = <nowiki>{Knot[5, 1], Knot[10, 132]}</nowiki> }}</nowiki>
+out = <nowiki>{Knot[5, 1], Knot[10, 132]}</nowiki> }}<nowiki></nowiki>
-         <nowiki>{{InOut &#124;
+         <nowiki></nowiki>{{InOut &#124;
 n = 13 &#124;
 in =  <nowiki>{KnotDet[Knot[5, 1]], KnotSignature[Knot[5, 1]]}</nowiki> &#124;
-out = <nowiki>{5, -4}</nowiki> }}</nowiki>
+out = <nowiki>{5, -4}</nowiki> }}<nowiki></nowiki>
-         <nowiki>{{InOut &#124;
+         <nowiki></nowiki>{{InOut &#124;
 n = 14 &#124;
 in =  <nowiki>Jones[Knot[5, 1]][q]</nowiki> &#124;
 out = <nowiki>  -7    -6    -5    -4    -2
--q   + q   - q   + q   + q</nowiki> }}</nowiki>
+-q   + q   - q   + q   + q</nowiki> }}<nowiki></nowiki>
-         <nowiki>{{InOut &#124;
+         <nowiki></nowiki>{{InOut &#124;
 n = 15 &#124;
 in =  <nowiki>Select[AllKnots[], (J === Jones[#][q] &#124;&#124; (J /. q-> 1/q) === Jones[#][q])&]</nowiki> &#124;
-out = <nowiki>{Knot[5, 1], Knot[10, 132]}</nowiki> }}</nowiki>
+out = <nowiki>{Knot[5, 1], Knot[10, 132]}</nowiki> }}<nowiki></nowiki>
-         <nowiki>{{InOut &#124;
+         <nowiki></nowiki>{{InOut &#124;
 n = 16 &#124;
 in =  <nowiki>A2Invariant[Knot[5, 1]][q]</nowiki> &#124;
@@ Line 116: / Line 116: @@
 -q    - q    - q    + q    + q    + --- + q   + q
-                                    q</nowiki> }}</nowiki>
+                                    q</nowiki> }}<nowiki></nowiki>
-         <nowiki>{{InOut &#124;
+         <nowiki></nowiki>{{InOut &#124;
 n = 17 &#124;
 in =  <nowiki>HOMFLYPT[Knot[5, 1]][a, z]</nowiki> &#124;
 out = <nowiki>   4      6      4  2    6  2    4  4
-a  - 2 a  + 4 a  z  - a  z  + a  z</nowiki> }}</nowiki>
+a  - 2 a  + 4 a  z  - a  z  + a  z</nowiki> }}<nowiki></nowiki>
-         <nowiki>{{InOut &#124;
+         <nowiki></nowiki>{{InOut &#124;
 n = 18 &#124;
 in =  <nowiki>Kauffman[Knot[5, 1]][a, z]</nowiki> &#124;
@@ Line 129: / Line 129: @@
 3    7  3    4  4    6  4
-  a  z  + a  z  + a  z  + a  z</nowiki> }}</nowiki>
+  a  z  + a  z  + a  z  + a  z</nowiki> }}<nowiki></nowiki>
-         <nowiki>{{InOut &#124;
+         <nowiki></nowiki>{{InOut &#124;
 n = 19 &#124;
 in =  <nowiki>{Vassiliev[2][Knot[5, 1]], Vassiliev[3][Knot[5, 1]]}</nowiki> &#124;
-out = <nowiki>{3, -5}</nowiki> }}</nowiki>
+out = <nowiki>{3, -5}</nowiki> }}<nowiki></nowiki>
-         <nowiki>{{InOut &#124;
+         <nowiki></nowiki>{{InOut &#124;
 n = 20 &#124;
 in =  <nowiki>Kh[Knot[5, 1]][q, t]</nowiki> &#124;
@@ Line 140: / Line 140: @@
 q   + q   + ------ + ------ + ------ + -----
 5    11  4    11  3    7  2
-            q   t    q   t    q   t    q  t</nowiki> }}</nowiki>
+            q   t    q   t    q   t    q  t</nowiki> }}<nowiki></nowiki>
-         <nowiki>{{InOut &#124;
+         <nowiki></nowiki>{{InOut &#124;
 n = 21 &#124;
 in =  <nowiki>ColouredJones[Knot[5, 1], 2][q]</nowiki> &#124;
@@ Line 147: / Line 147: @@
 q    - q    + q    - --- + q    - q    + q    - q   + q   + q
-                     q</nowiki> }}</nowiki>  }}
+                     q</nowiki> }}<nowiki></nowiki>  }}

5 1: Difference between revisions

Revision as of 15:03, 1 September 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

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