6 1: Difference between revisions

Revision as of 13:36, 16 December 2008

ricorsitr

5_2

6_2

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 6 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 6 1 at Knotilus!

[edit 6 1 Quick Notes]

6_1 is also known as "Stevedore's Knot" (see e.g. [1]), and as the pretzel knot P(5,-1,-1).

[edit 6 1 Further Notes and Views]

A Kolam of a 3x3 dot array	3D depiction	Polygonal depiction
Simple square depiction	An other one	Necklace

Knot presentations

Planar diagram presentation	X₁₄₂₅ X_7,10,8,11 X₃₉₄₈ X_9,3,10,2 X_5,12,6,1 X_11,6,12,7
Gauss code	-1, 4, -3, 1, -5, 6, -2, 3, -4, 2, -6, 5
Dowker-Thistlethwaite code	4 8 12 10 2 6
Conway Notation	[42]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 7, width is 4,

Braid index is 4

[{8, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 1}, {7, 2}, {6, 8}, {1, 7}]

[edit Notes on presentations of 6 1]

knot 6_1.

A graph, knot 6_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["6 1"];

In[4]:= PD[K]

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

Out[4]= X₁₄₂₅ X_7,10,8,11 X₃₉₄₈ X_9,3,10,2 X_5,12,6,1 X_11,6,12,7

In[5]:= GaussCode[K]

Out[5]= -1, 4, -3, 1, -5, 6, -2, 3, -4, 2, -6, 5

In[6]:= DTCode[K]

Out[6]= 4 8 12 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]= [42]

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]= ${\textrm {BR}}(4,\{-1,-1,-2,1,3,-2,3\})$

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]= { 4, 7, 4 }

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[{8, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 1}, {7, 2}, {6, 8}, {1, 7}]

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	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 \{3,4\}}
Nakanishi index	1
Maximal Thurston-Bennequin number	[-5][-3]
Hyperbolic Volume	3.16396
A-Polynomial	See Data:6 1/A-polynomial

[edit Notes for 6 1's three dimensional invariants]
6_1 is a ribbon knot (drawings by Yoko Mizuma):

a ribbon diagram

isotopy to a ribbon

6_1 has two slice disks, by Scott Carter

Scott Carter notes that 6_1 bounds two distinct slice disks. He says: "this was spoken of in Fox's Example 10, 11, and 12 in a Quick Trip through Knot Theory ... BTW, the cover of Carter and Saito's Knotted Surfaces and Their Diagrams contains an illustration of such a slice disk". A picture is on the right.

Four dimensional invariants

Smooth 4 genus	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 0}
Topological 4 genus	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 0}
Concordance genus	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 0}
Rasmussen s-Invariant	0

[edit Notes for 6 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial	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 t+5-2 t^{-1} }
Conway polynomial	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 1-2 z^2}
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 9, 0 }
Jones polynomial	$q^{2}-q+2-2q^{-1}+q^{-2}-q^{-3}+q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$a^{4}-z^{2}a^{2}-a^{2}-z^{2}+a^{-2}$
Kauffman polynomial (db, data sources)	$a^{3}z^{5}+az^{5}+a^{4}z^{4}+2a^{2}z^{4}+z^{4}-3a^{3}z^{3}-2az^{3}+z^{3}a^{-1}-3a^{4}z^{2}-4a^{2}z^{2}+z^{2}a^{-2}+2a^{3}z+2az+a^{4}+a^{2}-a^{-2}$
The A2 invariant	$q^{14}+q^{12}-q^{6}-q^{4}+q^{-2}+q^{-6}+q^{-8}$
The G2 invariant	$q^{66}+q^{62}-q^{60}+q^{56}-q^{54}+2q^{52}+q^{46}+q^{42}-q^{38}+q^{32}-2q^{28}+q^{26}+q^{24}-2q^{20}-2q^{18}+q^{16}-q^{14}+q^{12}-2q^{10}-q^{8}+2q^{6}-q^{4}-1+q^{-4}+q^{-10}+2q^{-14}-q^{-18}+q^{-20}+q^{-24}+q^{-28}+q^{-34}+q^{-38}$

Further Quantum Invariants

Further quantum knot invariants for 6_1.

A1 Invariants.

Weight	Invariant
1	$q^{9}-q^{3}+q^{-1}+q^{-5}$
2	$q^{26}-q^{22}-q^{16}+q^{12}+q^{8}+q^{6}+q^{-2}-q^{-4}-q^{-6}+q^{-8}+q^{-14}$
3	$q^{51}-q^{47}-q^{45}+q^{41}-q^{37}+q^{33}+q^{31}-q^{27}+q^{25}+q^{23}-q^{19}-q^{13}-q^{11}+q^{7}+q^{3}+q^{-1}+2q^{-3}+q^{-5}-q^{-7}-q^{-9}+q^{-11}+q^{-13}-q^{-15}-q^{-17}+q^{-27}$
4	$q^{84}-q^{80}-q^{78}-q^{76}+q^{74}+q^{72}+q^{70}-2q^{66}+q^{62}+q^{60}+q^{58}-q^{56}-q^{54}-q^{52}+q^{50}+2q^{48}-q^{44}-2q^{42}+q^{38}-q^{34}-2q^{32}+2q^{28}+q^{26}+q^{24}+q^{20}+2q^{18}-q^{14}-q^{12}-1-q^{-2}+2q^{-4}+q^{-6}+q^{-8}-q^{-10}-2q^{-12}+q^{-14}+2q^{-16}+3q^{-18}-2q^{-22}+q^{-28}-q^{-32}-q^{-36}+q^{-44}$
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 q^{125}-q^{121}-q^{119}-q^{117}+q^{113}+2 q^{111}+q^{109}-q^{105}-2 q^{103}-q^{101}+q^{99}+2 q^{97}+q^{95}-q^{91}-2 q^{89}-q^{87}+2 q^{83}+2 q^{81}+q^{79}-q^{77}-3 q^{75}-2 q^{73}+2 q^{69}+2 q^{67}-2 q^{63}-2 q^{61}-q^{59}+2 q^{57}+4 q^{55}+2 q^{53}-q^{51}-q^{49}-q^{47}+q^{45}+2 q^{43}+q^{41}-2 q^{39}-3 q^{37}-2 q^{35}+q^{31}-q^{27}-q^{25}+q^{23}+2 q^{21}+q^{19}-q^{13}+q^{11}+q^9+2 q^5+2 q^3- q^{-1} - q^{-3} + q^{-7} +2 q^{-9} + q^{-11} - q^{-13} -4 q^{-15} -3 q^{-17} +3 q^{-21} +4 q^{-23} + q^{-25} -2 q^{-27} -4 q^{-29} - q^{-31} +2 q^{-33} +3 q^{-35} +2 q^{-37} - q^{-41} - q^{-43} + q^{-47} - q^{-55} - q^{-57} + q^{-65} }
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^{174}-q^{170}-q^{168}-q^{166}+2 q^{160}+2 q^{158}+q^{156}-q^{152}-2 q^{150}-3 q^{148}+q^{144}+2 q^{142}+2 q^{140}+q^{138}-3 q^{134}-2 q^{132}-q^{130}+q^{126}+3 q^{124}+3 q^{122}-q^{118}-3 q^{116}-3 q^{114}-3 q^{112}+q^{110}+4 q^{108}+3 q^{106}+2 q^{104}-q^{102}-3 q^{100}-4 q^{98}-q^{96}+2 q^{94}+4 q^{92}+5 q^{90}+2 q^{88}-2 q^{86}-5 q^{84}-3 q^{82}-q^{80}+2 q^{78}+4 q^{76}+2 q^{74}-q^{72}-5 q^{70}-4 q^{68}-2 q^{66}+q^{64}+4 q^{62}+3 q^{60}-3 q^{56}-2 q^{54}+2 q^{50}+4 q^{48}+3 q^{46}-3 q^{42}-q^{40}+q^{38}+q^{36}+q^{34}-q^{30}-2 q^{28}+q^{24}+q^{22}-q^{18}-q^{16}-q^{14}-q^{12}-q^{10}+q^6+2 q^4+2 q^2+2- q^{-2} -2 q^{-4} - q^{-8} +2 q^{-10} +4 q^{-12} +5 q^{-14} + q^{-16} -3 q^{-18} -4 q^{-20} -5 q^{-22} - q^{-24} +4 q^{-26} +8 q^{-28} +4 q^{-30} - q^{-32} -5 q^{-34} -8 q^{-36} -4 q^{-38} + q^{-40} +6 q^{-42} +4 q^{-44} + q^{-46} - q^{-48} -4 q^{-50} -3 q^{-52} +3 q^{-56} +2 q^{-58} + q^{-60} + q^{-62} - q^{-64} - q^{-66} + q^{-70} + q^{-76} - q^{-78} - q^{-80} - q^{-82} + q^{-90} }

A2 Invariants.

Weight	Invariant
1,0	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 q^{14}+q^{12}-q^6-q^4+ q^{-2} + q^{-6} + q^{-8} }
1,1	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 q^{36}+2 q^{32}-2 q^{30}+2 q^{28}-2 q^{26}-2 q^{22}-2 q^{20}-2 q^{16}+4 q^{14}+q^{12}+6 q^{10}+4 q^6-2 q^4-2-2 q^{-2} - q^{-4} -2 q^{-6} +2 q^{-8} +2 q^{-12} +2 q^{-16} + q^{-20} }
2,0	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 q^{36}+q^{34}+q^{32}-q^{30}-q^{28}-q^{26}-q^{24}-q^{22}-q^{20}+q^{18}+q^{16}+q^{14}+q^{12}+2 q^{10}+q^8+q^6+q^4- q^{-2} - q^{-4} -2 q^{-6} - q^{-8} + q^{-10} + q^{-12} + q^{-16} + q^{-18} + q^{-20} }
3,0	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 q^{66}+q^{64}+q^{62}-2 q^{58}-2 q^{56}-2 q^{54}+2 q^{42}+2 q^{40}+2 q^{38}+q^{32}+2 q^{30}+q^{28}-q^{24}-q^{20}-3 q^{18}-3 q^{16}-3 q^{14}-q^{12}-q^{10}+q^8+2 q^6+3 q^4+3 q^2+3+3 q^{-2} +2 q^{-4} +2 q^{-6} - q^{-8} - q^{-10} + q^{-14} -3 q^{-18} -3 q^{-20} - q^{-22} + q^{-26} + q^{-30} + q^{-32} + q^{-34} + q^{-36} }

A3 Invariants.

Weight	Invariant
0,1,0	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 q^{28}+q^{24}-q^{20}+q^{12}+2 q^{10}-q^4-q^2-1- q^{-2} + q^{-4} + q^{-8} +2 q^{-10} + q^{-12} + q^{-16} }
1,0,0	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 q^{19}+q^{17}+q^{15}-q^9-q^7-q^5+ q^{-3} + q^{-7} + q^{-9} + q^{-11} }

A4 Invariants.

Weight

Invariant

0,1,0,0

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 q^{38}+q^{36}+q^{34}+q^{32}+q^{30}-q^{28}-2 q^{26}-2 q^{24}-q^{22}-q^{20}+3 q^{16}+3 q^{14}+3 q^{12}+2 q^{10}+q^8-q^6-2 q^4-2 q^2-2-2 q^{-2} - q^{-4} + q^{-6} + q^{-10} +2 q^{-12} +2 q^{-14} + q^{-16} + q^{-18} + q^{-20} + q^{-22} }

1,0,0,0

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 q^{24}+q^{22}+q^{20}+q^{18}-q^{12}-q^{10}-q^8-q^6+ q^{-4} + q^{-8} + q^{-10} + q^{-12} + q^{-14} }

B2 Invariants.

Weight	Invariant
0,1	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 q^{28}+q^{24}+q^{20}-q^{12}-2 q^8-q^4+q^2-1+ q^{-2} + q^{-4} + q^{-8} + q^{-12} + q^{-16} }
1,0	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 q^{46}+q^{38}-q^{34}-q^{32}+q^{20}+q^{18}+q^{16}+q^{12}-q^6-q^4- q^{-2} - q^{-4} + q^{-6} + q^{-14} + q^{-16} + q^{-18} + q^{-26} }

D4 Invariants.

Weight	Invariant
1,0,0,0	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 q^{38}+q^{34}+q^{30}+q^{16}+q^{12}-q^{10}-q^6-2 q^2- q^{-2} + q^{-6} + q^{-10} + q^{-12} +2 q^{-14} + q^{-16} + q^{-18} + q^{-22} }

G2 Invariants.

Weight

Invariant

1,0

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 q^{66}+q^{62}-q^{60}+q^{56}-q^{54}+2 q^{52}+q^{46}+q^{42}-q^{38}+q^{32}-2 q^{28}+q^{26}+q^{24}-2 q^{20}-2 q^{18}+q^{16}-q^{14}+q^{12}-2 q^{10}-q^8+2 q^6-q^4-1+ q^{-4} + q^{-10} +2 q^{-14} - q^{-18} + q^{-20} + q^{-24} + q^{-28} + q^{-34} + q^{-38} }

.

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["6 1"];

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

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

Out[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 -2 t+5-2 t^{-1} }

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

Out[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 1-2 z^2}

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]= { 9, 0 }

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

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

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

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

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

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

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

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

Out[10]= $a^{3}z^{5}+az^{5}+a^{4}z^{4}+2a^{2}z^{4}+z^{4}-3a^{3}z^{3}-2az^{3}+z^{3}a^{-1}-3a^{4}z^{2}-4a^{2}z^{2}+z^{2}a^{-2}+2a^{3}z+2az+a^{4}+a^{2}-a^{-2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_46, K11n67, K11n97, K11n139,}

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

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

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

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]= {9_46, K11n67, K11n97, K11n139,}

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

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

Out[6]= {}

Vassiliev invariants

V₂ and V₃:

(-2, 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

-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 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 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{116}{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{52}{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 -64}

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{304}{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{64}{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 -24}

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{256}{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{928}{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{416}{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{2791}{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{884}{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{10084}{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{343}{9}}

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{871}{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 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 t^rq^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

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=} 0 is the signature of 6 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-4

-3

-2

-1

0

1

2

χ

5

1

3

0

1

2

1

-1

1

0

-3

1

-1

-5

1

0

-7

0

-9

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=-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 i=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 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}}
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}_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=-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=-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}}
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}}
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}_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}}

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}	$J_{n}$
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^6-q^5+2 q^3-3 q^2+4-4 q^{-1} +4 q^{-3} -3 q^{-4} +3 q^{-6} -2 q^{-7} - q^{-8} +2 q^{-9} - q^{-10} - q^{-11} + q^{-12} }
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 q^{12}-q^{11}+q^8-2 q^7+2 q^5+q^4-4 q^3+4 q+2-5 q^{-1} - q^{-2} +5 q^{-3} + q^{-4} -4 q^{-5} -2 q^{-6} +4 q^{-7} + q^{-8} -3 q^{-9} -2 q^{-10} +3 q^{-11} +2 q^{-12} -2 q^{-13} -2 q^{-14} + q^{-15} +3 q^{-16} - q^{-17} -2 q^{-18} +2 q^{-20} - q^{-22} - q^{-23} + q^{-24} }
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^{20}-q^{19}-q^{16}+2 q^{15}-2 q^{14}+q^{13}+q^{12}-2 q^{11}+2 q^{10}-4 q^9+3 q^8+4 q^7-3 q^6+q^5-7 q^4+4 q^3+6 q^2-3 q+2-10 q^{-1} +4 q^{-2} +7 q^{-3} -3 q^{-4} +2 q^{-5} -10 q^{-6} +4 q^{-7} +6 q^{-8} -3 q^{-9} +3 q^{-10} -8 q^{-11} +3 q^{-12} +5 q^{-13} -2 q^{-14} +3 q^{-15} -7 q^{-16} + q^{-17} +3 q^{-18} - q^{-19} +4 q^{-20} -6 q^{-21} + q^{-23} +5 q^{-25} -4 q^{-26} - q^{-27} - q^{-28} +5 q^{-30} -2 q^{-31} - q^{-32} - q^{-33} - q^{-34} +3 q^{-35} - q^{-38} - q^{-39} + q^{-40} }
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 q^{30}-q^{29}-q^{26}+2 q^{24}-q^{23}+q^{21}-2 q^{20}-q^{19}+2 q^{18}+2 q^{16}+2 q^{15}-3 q^{14}-4 q^{13}-q^{12}+2 q^{11}+5 q^{10}+5 q^9-4 q^8-7 q^7-4 q^6+q^5+8 q^4+7 q^3-3 q^2-8 q-5+8 q^{-2} +8 q^{-3} - q^{-4} -8 q^{-5} -7 q^{-6} + q^{-7} +8 q^{-8} +6 q^{-9} -8 q^{-11} -6 q^{-12} +2 q^{-13} +7 q^{-14} +4 q^{-15} -7 q^{-17} -5 q^{-18} + q^{-19} +5 q^{-20} +3 q^{-21} + q^{-22} -4 q^{-23} -4 q^{-24} +3 q^{-26} +3 q^{-27} + q^{-28} - q^{-29} -2 q^{-30} -2 q^{-31} +2 q^{-33} + q^{-34} + q^{-35} -2 q^{-37} -2 q^{-38} +3 q^{-41} +2 q^{-42} - q^{-43} -2 q^{-44} -2 q^{-45} - q^{-46} +2 q^{-47} +3 q^{-48} - q^{-50} - q^{-51} -2 q^{-52} +2 q^{-54} + q^{-55} - q^{-58} - q^{-59} + q^{-60} }
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^{42}-q^{41}-q^{38}+3 q^{35}-2 q^{34}+q^{32}-2 q^{31}-q^{30}+5 q^{28}-2 q^{27}+q^{26}+2 q^{25}-5 q^{24}-4 q^{23}-q^{22}+8 q^{21}+4 q^{19}+4 q^{18}-10 q^{17}-9 q^{16}-5 q^{15}+11 q^{14}+4 q^{13}+9 q^{12}+8 q^{11}-14 q^{10}-14 q^9-9 q^8+12 q^7+5 q^6+13 q^5+12 q^4-15 q^3-16 q^2-12 q+13+3 q^{-1} +14 q^{-2} +15 q^{-3} -15 q^{-4} -16 q^{-5} -13 q^{-6} +12 q^{-7} +2 q^{-8} +14 q^{-9} +15 q^{-10} -15 q^{-11} -16 q^{-12} -12 q^{-13} +13 q^{-14} +2 q^{-15} +13 q^{-16} +13 q^{-17} -14 q^{-18} -15 q^{-19} -11 q^{-20} +13 q^{-21} +2 q^{-22} +11 q^{-23} +11 q^{-24} -11 q^{-25} -12 q^{-26} -10 q^{-27} +11 q^{-28} +9 q^{-30} +10 q^{-31} -8 q^{-32} -9 q^{-33} -9 q^{-34} +8 q^{-35} -3 q^{-36} +7 q^{-37} +9 q^{-38} -4 q^{-39} -6 q^{-40} -7 q^{-41} +6 q^{-42} -6 q^{-43} +5 q^{-44} +7 q^{-45} - q^{-46} -2 q^{-47} -4 q^{-48} +5 q^{-49} -8 q^{-50} +2 q^{-51} +4 q^{-52} - q^{-55} +6 q^{-56} -7 q^{-57} - q^{-58} + q^{-62} +7 q^{-63} -4 q^{-64} - q^{-65} -2 q^{-66} - q^{-67} - q^{-68} +6 q^{-70} - q^{-71} - q^{-73} - q^{-74} -2 q^{-75} - q^{-76} +3 q^{-77} + q^{-79} - q^{-82} - q^{-83} + q^{-84} }
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 q^{56}-q^{55}-q^{52}+q^{49}+2 q^{48}-2 q^{47}+q^{45}-2 q^{44}-q^{42}+2 q^{41}+4 q^{40}-3 q^{39}+q^{37}-4 q^{36}-q^{35}-2 q^{34}+4 q^{33}+7 q^{32}-q^{31}-2 q^{29}-9 q^{28}-3 q^{27}-3 q^{26}+6 q^{25}+15 q^{24}+4 q^{23}+3 q^{22}-7 q^{21}-17 q^{20}-9 q^{19}-7 q^{18}+10 q^{17}+23 q^{16}+11 q^{15}+8 q^{14}-8 q^{13}-25 q^{12}-16 q^{11}-11 q^{10}+10 q^9+28 q^8+14 q^7+14 q^6-7 q^5-28 q^4-19 q^3-15 q^2+8 q+30+15 q^{-1} +16 q^{-2} -5 q^{-3} -28 q^{-4} -18 q^{-5} -17 q^{-6} +6 q^{-7} +29 q^{-8} +16 q^{-9} +17 q^{-10} -6 q^{-11} -28 q^{-12} -16 q^{-13} -16 q^{-14} +5 q^{-15} +29 q^{-16} +17 q^{-17} +15 q^{-18} -7 q^{-19} -28 q^{-20} -14 q^{-21} -15 q^{-22} +5 q^{-23} +28 q^{-24} +16 q^{-25} +13 q^{-26} -7 q^{-27} -26 q^{-28} -12 q^{-29} -14 q^{-30} +4 q^{-31} +24 q^{-32} +14 q^{-33} +12 q^{-34} -5 q^{-35} -22 q^{-36} -10 q^{-37} -12 q^{-38} + q^{-39} +19 q^{-40} +11 q^{-41} +13 q^{-42} - q^{-43} -17 q^{-44} -8 q^{-45} -12 q^{-46} -2 q^{-47} +14 q^{-48} +7 q^{-49} +12 q^{-50} +4 q^{-51} -11 q^{-52} -6 q^{-53} -11 q^{-54} -5 q^{-55} +9 q^{-56} +2 q^{-57} +10 q^{-58} +7 q^{-59} -6 q^{-60} -2 q^{-61} -8 q^{-62} -6 q^{-63} +4 q^{-64} -2 q^{-65} +6 q^{-66} +7 q^{-67} -3 q^{-68} +2 q^{-69} -3 q^{-70} -4 q^{-71} +2 q^{-72} -5 q^{-73} +2 q^{-74} +4 q^{-75} -3 q^{-76} +3 q^{-77} + q^{-78} +3 q^{-80} -5 q^{-81} - q^{-82} + q^{-83} -5 q^{-84} +2 q^{-85} + q^{-86} +2 q^{-87} +5 q^{-88} -2 q^{-89} - q^{-90} -4 q^{-92} - q^{-93} - q^{-94} + q^{-95} +5 q^{-96} + q^{-99} -2 q^{-100} - q^{-101} -2 q^{-102} - q^{-103} +2 q^{-104} + q^{-105} + q^{-107} - q^{-110} - q^{-111} + q^{-112} }

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_2

6_2

@@ Line 1: / Line 1: @@
+ricorsitr
 <!--                       WARNING! WARNING! WARNING!
 <!-- This page was generated from the splice base [[Rolfsen_Splice_Base]]. Please do not edit!

6 1: Difference between revisions

Revision as of 13:36, 16 December 2008

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

Modifying This Page

Navigation menu

Page actions

Page actions

Personal tools

Navigation

Search

Tools