8 20: Difference between revisions

Revision as of 19:15, 28 August 2005

8_19

8_21

Visit 8 20's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 8 20's page at Knotilus!

Visit 8 20's page at the original Knot Atlas!

8_20 is also known as the pretzel knot P(3,-3,2).

Its complement contains no complete totally geodesic immersed surfaces.^{[citation needed]}

This appears to be the Ashley/oysterman stopper knot of practical knot tying.

The Oysterman's stopper[1]

Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₄₉₃ X_5,12,6,13 X_13,16,14,1 X_9,14,10,15 X_15,10,16,11 X_11,6,12,7 X₂₈₃₇
Gauss code	1, -8, 2, -1, -3, 7, 8, -2, -5, 6, -7, 3, -4, 5, -6, 4
Dowker-Thistlethwaite code	4 8 -12 2 -14 -6 -16 -10
Conway Notation	[3,21,2-]

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	2
Bridge index	3
Super bridge index	4
Nakanishi index	1
Maximal Thurston-Bennequin number	[-6][-2]
Hyperbolic Volume	4.1249
A-Polynomial	See Data:8 20/A-polynomial

[edit Notes for 8 20's three dimensional invariants]
8_20 ribbon diagram from A. Kawauchi's text.

Ribbon diagram for 8_20

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 8 20'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 t^2-2 t+3-2 t^{-1} + t^{-2} }
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 z^4+2 z^2+1}
2nd Alexander ideal (db, data sources)	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\}}
Determinant and Signature	{ 9, 0 }
Jones 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 -q+2- q^{-1} +2 q^{-2} - q^{-3} + q^{-4} - q^{-5} }
HOMFLY-PT polynomial (db, data sources)	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 -z^2 a^4-2 a^4+z^4 a^2+4 z^2 a^2+4 a^2-z^2-1}
Kauffman polynomial (db, data sources)	$a^{4}z^{6}+a^{2}z^{6}+a^{5}z^{5}+2a^{3}z^{5}+az^{5}-4a^{4}z^{4}-4a^{2}z^{4}-4a^{5}z^{3}-7a^{3}z^{3}-3az^{3}+4a^{4}z^{2}+6a^{2}z^{2}+2z^{2}+3a^{5}z+5a^{3}z+3az+za^{-1}-2a^{4}-4a^{2}-1$
The A2 invariant	$-q^{16}-q^{14}-q^{12}+2q^{8}+2q^{6}+2q^{4}+q^{2}-q^{-4}$
The G2 invariant	$q^{80}+q^{76}-q^{74}-2q^{68}+q^{66}-q^{64}-q^{62}-q^{60}-2q^{58}-3q^{52}-q^{50}-q^{48}+q^{44}-2q^{42}+q^{40}+q^{38}+2q^{36}+q^{34}+2q^{30}+2q^{28}+3q^{26}+2q^{22}+3q^{20}+q^{18}+q^{16}+q^{14}+3q^{10}-2q^{6}+q^{4}+1-q^{-2}-2q^{-4}-q^{-12}-q^{-14}-q^{-20}+q^{-24}$

Further Quantum Invariants

Further quantum knot invariants for 8_20.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+q^{5}+q^{3}+q+q^{-1}-q^{-3}$
2	$q^{32}-q^{28}-q^{22}-q^{20}+q^{12}+q^{10}+q^{6}+q^{4}+q^{2}+q^{-2}-q^{-6}$
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^{63}+q^{59}+q^{57}-q^{53}+q^{49}+q^{47}-q^{43}-q^{41}-q^{35}-2 q^{33}+q^{29}+q^{27}-q^{25}+q^{21}-q^{17}+q^{15}+q^{13}+q^5+2 q^3+2 q+2 q^{-7} - q^{-9} -2 q^{-11} - q^{-13} + q^{-17} }
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^{104}-q^{100}-q^{98}-q^{96}+q^{94}+q^{92}+q^{90}-2 q^{86}-q^{84}+q^{80}+2 q^{78}+q^{76}-q^{72}-q^{70}+q^{68}+2 q^{66}+q^{64}-q^{62}-3 q^{60}-2 q^{58}+2 q^{54}+q^{52}-2 q^{50}-2 q^{48}+2 q^{44}+2 q^{42}-q^{40}-2 q^{38}+q^{34}+q^{32}-q^{30}-q^{28}-q^{20}-q^{18}+q^{16}+q^{14}+q^{12}+q^{10}+q^8+q^4+3 q^2+4+2 q^{-2} - q^{-4} -4 q^{-6} -2 q^{-8} +3 q^{-10} +2 q^{-12} - q^{-14} -4 q^{-16} -3 q^{-18} + q^{-20} +2 q^{-22} +2 q^{-24} - q^{-28} }
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^{155}+q^{151}+q^{149}+q^{147}-q^{143}-2 q^{141}-q^{139}+q^{135}+2 q^{133}+2 q^{131}-2 q^{127}-2 q^{125}-2 q^{123}-q^{121}+q^{119}+2 q^{117}+2 q^{115}+q^{113}-q^{111}-3 q^{109}-2 q^{107}+3 q^{103}+4 q^{101}+3 q^{99}-3 q^{95}-4 q^{93}-2 q^{91}+2 q^{89}+4 q^{87}+3 q^{85}-q^{83}-4 q^{81}-5 q^{79}-2 q^{77}+3 q^{75}+4 q^{73}+2 q^{71}-q^{69}-4 q^{67}-2 q^{65}+q^{63}+4 q^{61}+3 q^{59}-3 q^{55}-3 q^{53}+2 q^{49}+2 q^{47}-3 q^{43}-2 q^{41}+q^{37}-2 q^{33}-2 q^{31}+q^{27}+q^{25}+q^{15}+2 q^{13}+3 q^{11}+3 q^9+2 q^7-q^5-2 q^3+3 q^{-1} +5 q^{-3} +5 q^{-5} - q^{-7} -7 q^{-9} -7 q^{-11} -2 q^{-13} +3 q^{-15} +7 q^{-17} +3 q^{-19} -3 q^{-21} -6 q^{-23} -4 q^{-25} + q^{-27} +3 q^{-29} +3 q^{-31} + q^{-33} - q^{-35} - q^{-37} }
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^{216}-q^{212}-q^{210}-q^{208}+2 q^{202}+2 q^{200}+q^{198}-q^{194}-2 q^{192}-3 q^{190}-q^{188}+2 q^{184}+3 q^{182}+3 q^{180}+2 q^{178}-q^{176}-2 q^{174}-3 q^{172}-3 q^{170}-2 q^{168}+3 q^{164}+4 q^{162}+3 q^{160}+q^{158}-2 q^{156}-5 q^{154}-6 q^{152}-3 q^{150}+q^{148}+4 q^{146}+6 q^{144}+6 q^{142}+2 q^{140}-4 q^{138}-6 q^{136}-6 q^{134}-3 q^{132}+3 q^{130}+8 q^{128}+8 q^{126}+4 q^{124}-q^{122}-7 q^{120}-9 q^{118}-5 q^{116}+q^{114}+6 q^{112}+7 q^{110}+5 q^{108}-2 q^{106}-8 q^{104}-8 q^{102}-4 q^{100}+2 q^{98}+7 q^{96}+9 q^{94}+4 q^{92}-3 q^{90}-7 q^{88}-6 q^{86}-q^{84}+4 q^{82}+8 q^{80}+5 q^{78}-2 q^{76}-6 q^{74}-5 q^{72}-q^{70}+3 q^{68}+6 q^{66}+3 q^{64}-3 q^{62}-5 q^{60}-4 q^{58}-q^{56}+2 q^{54}+2 q^{52}-3 q^{48}-2 q^{46}+q^{42}+q^{40}-q^{36}-2 q^{34}-q^{32}+q^{30}+2 q^{28}+2 q^{26}+2 q^{24}+q^{22}-q^{18}-2 q^{16}-q^{14}+q^{12}+4 q^{10}+6 q^8+6 q^6+2 q^4-3 q^2-6-5 q^{-2} -2 q^{-4} +6 q^{-6} +11 q^{-8} +9 q^{-10} + q^{-12} -8 q^{-14} -12 q^{-16} -13 q^{-18} -2 q^{-20} +10 q^{-22} +12 q^{-24} +7 q^{-26} - q^{-28} -7 q^{-30} -11 q^{-32} -6 q^{-34} + q^{-36} +5 q^{-38} +5 q^{-40} +4 q^{-42} +2 q^{-44} -2 q^{-46} - q^{-48} - q^{-50} - q^{-52} - q^{-54} + q^{-58} }

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^{16}-q^{14}-q^{12}+2 q^8+2 q^6+2 q^4+q^2- q^{-4} }
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^{44}+2 q^{40}-2 q^{38}+2 q^{36}-2 q^{34}-2 q^{30}-4 q^{28}-4 q^{24}+2 q^{22}-3 q^{20}+4 q^{18}+4 q^{14}+q^{12}+2 q^{10}+4 q^8+5 q^4+2- q^{-4} -2 q^{-6} -2 q^{-8} + q^{-12} }
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^{42}+q^{40}+q^{38}-q^{32}-2 q^{30}-4 q^{28}-4 q^{26}-3 q^{24}-q^{22}+2 q^{20}+3 q^{18}+5 q^{16}+3 q^{14}+3 q^{12}+q^{10}+q^8+q^4- q^{-8} }

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^{34}+q^{30}-q^{26}-2 q^{24}-4 q^{22}-4 q^{20}-3 q^{18}+2 q^{14}+6 q^{12}+6 q^{10}+6 q^8+3 q^6+q^4-q^2-2-2 q^{-2} - q^{-4} - q^{-6} + q^{-10} }
1,0,0	$-q^{21}-q^{19}-2q^{17}-q^{15}+2q^{11}+3q^{9}+3q^{7}+2q^{5}+q^{3}-q^{-1}-q^{-5}$
1,0,1	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^{56}+2 q^{52}+q^{48}-q^{44}+q^{42}-2 q^{40}+q^{38}-4 q^{36}-2 q^{34}-6 q^{32}-5 q^{30}-5 q^{28}-6 q^{26}-2 q^{22}+6 q^{20}+5 q^{18}+10 q^{16}+9 q^{14}+9 q^{12}+8 q^{10}+2 q^8+3 q^6-2 q^4-3- q^{-2} - q^{-4} -2 q^{-6} - q^{-8} - q^{-10} + q^{-16} }

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^{44}+q^{42}+2 q^{40}+2 q^{38}+q^{36}-q^{34}-3 q^{32}-6 q^{30}-9 q^{28}-9 q^{26}-7 q^{24}-3 q^{22}+q^{20}+8 q^{18}+12 q^{16}+13 q^{14}+12 q^{12}+9 q^{10}+3 q^8-q^6-4 q^4-5 q^2-5-3 q^{-2} - q^{-4} + q^{-10} + q^{-12} }

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^{26}-q^{24}-2 q^{22}-2 q^{20}-q^{18}+2 q^{14}+3 q^{12}+4 q^{10}+3 q^8+2 q^6+q^4-1- q^{-2} - q^{-6} }

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^{34}-q^{30}-q^{26}+q^{18}+2 q^{14}+2 q^{10}+q^6+q^4+q^2+ q^{-4} - q^{-6} - q^{-10} }
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^{56}+q^{48}-q^{44}-q^{42}-q^{38}-2 q^{36}-2 q^{34}-2 q^{32}-q^{30}-q^{28}+2 q^{22}+2 q^{20}+3 q^{18}+2 q^{16}+3 q^{14}+3 q^{12}+2 q^{10}+q^6-1- q^{-2} - q^{-4} - q^{-8} - q^{-10} + q^{-16} }

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^{46}+q^{42}+q^{38}-q^{36}-2 q^{34}-3 q^{32}-4 q^{30}-4 q^{28}-4 q^{26}-2 q^{24}-q^{22}+3 q^{20}+4 q^{18}+7 q^{16}+6 q^{14}+7 q^{12}+4 q^{10}+3 q^8+q^6-q^4-2 q^2-2-2 q^{-2} -2 q^{-4} - q^{-8} + q^{-14} }

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^{80}+q^{76}-q^{74}-2 q^{68}+q^{66}-q^{64}-q^{62}-q^{60}-2 q^{58}-3 q^{52}-q^{50}-q^{48}+q^{44}-2 q^{42}+q^{40}+q^{38}+2 q^{36}+q^{34}+2 q^{30}+2 q^{28}+3 q^{26}+2 q^{22}+3 q^{20}+q^{18}+q^{16}+q^{14}+3 q^{10}-2 q^6+q^4+1- q^{-2} -2 q^{-4} - q^{-12} - q^{-14} - q^{-20} + q^{-24} }

.

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["8 20"];

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 t^2-2 t+3-2 t^{-1} + t^{-2} }

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 z^4+2 z^2+1}

In[6]:= Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]= 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\}}

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 9, 0 }

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

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

Out[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 -q+2- q^{-1} +2 q^{-2} - q^{-3} + q^{-4} - q^{-5} }

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

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

Out[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 -z^2 a^4-2 a^4+z^4 a^2+4 z^2 a^2+4 a^2-z^2-1}

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

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

Out[10]= $a^{4}z^{6}+a^{2}z^{6}+a^{5}z^{5}+2a^{3}z^{5}+az^{5}-4a^{4}z^{4}-4a^{2}z^{4}-4a^{5}z^{3}-7a^{3}z^{3}-3az^{3}+4a^{4}z^{2}+6a^{2}z^{2}+2z^{2}+3a^{5}z+5a^{3}z+3az+za^{-1}-2a^{4}-4a^{2}-1$

Vassiliev invariants

V₂ and V₃:

(2, -2)

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

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}

-16

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{172}{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{20}{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 -128}

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{640}{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 -48}

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}}

128

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{1376}{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{160}{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{11911}{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{524}{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{15484}{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{233}{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{631}{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

r

). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 0 is the signature of 8 20. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

0

1

χ

3

1

-1

1

-1

1

2

1

-3

1

-5

1

-7

1

0

-9

0

-11

1

-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=-5}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-4}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2}	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}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2}	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}	${\mathbb {Z} }_{2}$	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}^{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}_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}}

Computer Talk

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

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

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[8, 20]]
Out[2]=	8
In[3]:=	PD[Knot[8, 20]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 12, 6, 13], X[13, 16, 14, 1], X[9, 14, 10, 15], X[15, 10, 16, 11], X[11, 6, 12, 7], X[2, 8, 3, 7]]
In[4]:=	GaussCode[Knot[8, 20]]
Out[4]=	GaussCode[1, -8, 2, -1, -3, 7, 8, -2, -5, 6, -7, 3, -4, 5, -6, 4]
In[5]:=	BR[Knot[8, 20]]
Out[5]=	BR[3, {1, 1, 1, -2, -1, -1, -1, -2}]
In[6]:=	alex = Alexander[Knot[8, 20]][t]
Out[6]=	-2 2 2 3 + t - - - 2 t + t t
In[7]:=	Conway[Knot[8, 20]][z]
Out[7]=	2 4 1 + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 20], Knot[10, 140], Knot[11, NonAlternating, 73], Knot[11, NonAlternating, 74]}
In[9]:=	{KnotDet[Knot[8, 20]], KnotSignature[Knot[8, 20]]}
Out[9]=	{9, 0}
In[10]:=	J=Jones[Knot[8, 20]][q]
Out[10]=	-5 -4 -3 2 1 2 - q + q - q + -- - - - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[8, 20]}
In[12]:=	A2Invariant[Knot[8, 20]][q]
Out[12]=	-16 -14 -12 2 2 2 -2 4 -q - q - q + -- + -- + -- + q - q 8 6 4 q q q
In[13]:=	Kauffman[Knot[8, 20]][a, z]
Out[13]=	2 4 z 3 5 2 2 2 -1 - 4 a - 2 a + - + 3 a z + 5 a z + 3 a z + 2 z + 6 a z + a 4 2 3 3 3 5 3 2 4 4 4 5 4 a z - 3 a z - 7 a z - 4 a z - 4 a z - 4 a z + a z + 3 5 5 5 2 6 4 6 2 a z + a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[8, 20]], Vassiliev[3][Knot[8, 20]]}
Out[14]=	{0, -2}
In[15]:=	Kh[Knot[8, 20]][q, t]
Out[15]=	2 1 1 1 1 1 1 3 - + q + ------ + ----- + ----- + ----- + ----- + --- + q t q 11 5 7 4 7 3 5 2 3 2 q t q t q t q t q t q t

@@ Line 1: / Line 1: @@
 <!--  -->
+<!-- -->
+<!--  -->
+<!-- -->
 <!-- provide an anchor so we can return to the top of the page -->
 <span id="top"></span>
+<!-- -->
 <!-- this relies on transclusion for next and previous links -->
 {{Knot Navigation Links|ext=gif}}
+{{Rolfsen Knot Page Header|n=8|k=20|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-8,2,-1,-3,7,8,-2,-5,6,-7,3,-4,5,-6,4/goTop.html}}
-{| align=left
-|- valign=top
-|[[Image:{{PAGENAME}}.gif]]
-|{{Rolfsen Knot Site Links|n=8|k=20|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-8,2,-1,-3,7,8,-2,-5,6,-7,3,-4,5,-6,4/goTop.html}}
-|{{:{{PAGENAME}} Quick Notes}}
-|}
 <br style="clear:both" />
@@ Line 24: / Line 21: @@
 {{Vassiliev Invariants}}
-===[[Khovanov Homology]]===
+{{Khovanov Homology|table=<table border=1>
-The coefficients of the monomials <math>t^rq^j</math> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>). The squares with <font class=HLYellow>yellow</font> highlighting are those on the "critical diagonals", where <math>j-2r=s+1</math> or <math>j-2r=s+1</math>, where <math>s=</math>{{Data:{{PAGENAME}}/Signature}} is the signature of {{PAGENAME}}. Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>.
-<center><table border=1>
 <tr align=center>
 <td width=18.1818%><table cellpadding=0 cellspacing=0>
@@ Line 44: / Line 37: @@
 <tr align=center><td>-9</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>0</td></tr>
 <tr align=center><td>-11</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-1</td></tr>
-</table></center>
+</table>}}
 {{Computer Talk Header}}
@@ Line 108: / Line 100: @@
         q   t    q  t    q  t    q  t    q  t</nowiki></pre></td></tr>
 </table>
+ [[Category:Knot Page]]

8 20: Difference between revisions

Revision as of 19:15, 28 August 2005

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

Navigation menu

Page actions

Page actions

Personal tools

Navigation

Search

Tools