5 1

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4 1.gif

4_1

5 2.gif

5_2

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

Visit 5 1's page at Knotilus!

Visit 5 1's page at the original Knot Atlas!

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




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 X5,10,6,1 X7283 X9,4,10,5
Gauss code -1, 4, -2, 5, -3, 1, -4, 2, -5, 3
Dowker-Thistlethwaite code 6 8 10 2 4
Conway Notation [5]

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

[edit Notes for 5 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 t^2+ t^{-2} -t- t^{-1} +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 z^4+3 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 { 5, -4 }
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^{-7} + q^{-6} - q^{-5} + q^{-4} + q^{-2} }
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 a^6 \left(-z^2\right)-2 a^6+a^4 z^4+4 a^4 z^2+3 a^4}
Kauffman 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 a^9 z+a^8 z^2+a^7 z^3-a^7 z+a^6 z^4-3 a^6 z^2+2 a^6+a^5 z^3-2 a^5 z+a^4 z^4-4 a^4 z^2+3 a^4}
The A2 invariant 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^{22}-q^{20}-q^{18}+q^{14}+q^{12}+2 q^{10}+q^8+q^6}
The G2 invariant 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^{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}+2 q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+2 q^{40}+q^{38}+q^{34}+q^{32}+q^{30}}
Further Quantum Invariants
Further quantum knot invariants for 5_1.

A1 Invariants.

Weight Invariant
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^{15}+q^7+q^5+q^3}
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^{40}-q^{32}-q^{30}-q^{28}+q^{14}+q^{12}+q^{10}+q^8+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^{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 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^{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 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^{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 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^{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 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^{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 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^{22}-q^{20}-q^{18}+q^{14}+q^{12}+2 q^{10}+q^8+q^6}
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^{60}-2 q^{36}-2 q^{34}-4 q^{32}-4 q^{30}-3 q^{28}+2 q^{24}+4 q^{22}+5 q^{20}+4 q^{18}+4 q^{16}+2 q^{14}+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^{54}+q^{52}+2 q^{50}+q^{48}-2 q^{44}-3 q^{42}-3 q^{40}-3 q^{38}-2 q^{36}-q^{34}+q^{28}+q^{26}+2 q^{24}+2 q^{22}+3 q^{20}+2 q^{18}+2 q^{16}+q^{14}+q^{12}}
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^{96}-q^{94}-2 q^{92}-2 q^{90}-q^{88}+q^{86}+3 q^{84}+4 q^{82}+5 q^{80}+4 q^{78}+4 q^{76}+2 q^{74}+q^{72}-q^{70}-2 q^{68}-3 q^{66}-4 q^{64}-5 q^{62}-5 q^{60}-5 q^{58}-4 q^{56}-3 q^{54}-2 q^{52}-q^{50}+q^{42}+q^{40}+2 q^{38}+2 q^{36}+3 q^{34}+3 q^{32}+4 q^{30}+3 q^{28}+3 q^{26}+2 q^{24}+2 q^{22}+q^{20}+q^{18}}

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

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

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

B3 Invariants.

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

B4 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^{160}-q^{114}-q^{110}-2 q^{106}-q^{104}-2 q^{102}-q^{100}-2 q^{98}-q^{96}-2 q^{94}-q^{92}-2 q^{90}-q^{88}-q^{86}+q^{78}+2 q^{74}+q^{72}+3 q^{70}+q^{68}+3 q^{66}+q^{64}+3 q^{62}+q^{60}+3 q^{58}+q^{56}+2 q^{54}+2 q^{50}+q^{46}+q^{42}}

C3 Invariants.

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

C4 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^{90}-q^{64}-q^{62}-2 q^{60}-q^{58}-q^{56}-q^{54}-q^{52}-q^{50}-q^{48}+q^{44}+2 q^{42}+2 q^{40}+2 q^{38}+2 q^{36}+2 q^{34}+2 q^{32}+2 q^{30}+2 q^{28}+q^{26}+q^{24}}

D4 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^{120}-q^{100}-q^{98}-3 q^{96}-3 q^{94}-q^{92}-q^{90}+2 q^{88}+6 q^{86}+9 q^{84}+11 q^{82}+14 q^{80}+11 q^{78}+9 q^{76}+3 q^{74}-4 q^{72}-12 q^{70}-18 q^{68}-24 q^{66}-27 q^{64}-27 q^{62}-24 q^{60}-17 q^{58}-11 q^{56}+7 q^{52}+14 q^{50}+19 q^{48}+22 q^{46}+19 q^{44}+19 q^{42}+14 q^{40}+10 q^{38}+6 q^{36}+4 q^{34}+q^{32}+q^{30}}
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^{70}-q^{50}-q^{48}-3 q^{46}-3 q^{44}-3 q^{42}-3 q^{40}-2 q^{38}+q^{34}+3 q^{32}+4 q^{30}+4 q^{28}+4 q^{26}+3 q^{24}+2 q^{22}+q^{20}+q^{18}}

G2 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^{240}-q^{198}-q^{192}+q^{178}+q^{176}+q^{172}+2 q^{170}+q^{168}+q^{166}+q^{164}+q^{162}+2 q^{160}+q^{158}+q^{154}+q^{152}-q^{148}-q^{146}-q^{144}-q^{142}-2 q^{140}-3 q^{138}-2 q^{136}-2 q^{134}-3 q^{132}-4 q^{130}-4 q^{128}-3 q^{126}-3 q^{124}-4 q^{122}-4 q^{120}-3 q^{118}-2 q^{116}-2 q^{114}-3 q^{112}-2 q^{110}+q^{102}+q^{100}+2 q^{98}+3 q^{96}+2 q^{94}+2 q^{92}+4 q^{90}+3 q^{88}+3 q^{86}+4 q^{84}+3 q^{82}+3 q^{80}+4 q^{78}+2 q^{76}+2 q^{74}+3 q^{72}+2 q^{70}+q^{68}+2 q^{66}+q^{64}+q^{62}+q^{60}+q^{54}}
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^{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}+2 q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+2 q^{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.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["5 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 t^2+ t^{-2} -t- t^{-1} +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 z^4+3 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]= { 5, -4 }
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^{-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]= 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 a^6 \left(-z^2\right)-2 a^6+a^4 z^4+4 a^4 z^2+3 a^4}
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= 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 a^9 z+a^8 z^2+a^7 z^3-a^7 z+a^6 z^4-3 a^6 z^2+2 a^6+a^5 z^3-2 a^5 z+a^4 z^4-4 a^4 z^2+3 a^4}

Vassiliev invariants

V2 and V3: (3, -5)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
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 12} 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 -40} 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 72} 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 174} 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 26} 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 -480} 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{2512}{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{448}{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 -104} 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 288} 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 800} 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 2088} 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 312} 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{41151}{10}} 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{2494}{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{7634}{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 \frac{43}{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 \frac{1951}{10}}

V2,1 through V6,9 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} -4 is the signature of 5 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 -5-4-3-2-10χ
-3     11
-5     11
-7   1  1
-9      0
-11 11   0
-13      0
-151     -1

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[5, 1]]
Out[2]=  
5
In[3]:=
PD[Knot[5, 1]]
Out[3]=  
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]]
In[4]:=
GaussCode[Knot[5, 1]]
Out[4]=  
GaussCode[-1, 4, -2, 5, -3, 1, -4, 2, -5, 3]
In[5]:=
BR[Knot[5, 1]]
Out[5]=  
BR[2, {-1, -1, -1, -1, -1}]
In[6]:=
alex = Alexander[Knot[5, 1]][t]
Out[6]=  
     -2   1        2

1 + t   - - - t + t


          t
In[7]:=
Conway[Knot[5, 1]][z]
Out[7]=  
       2    4
1 + 3 z  + z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[5, 1], Knot[10, 132]}
In[9]:=
{KnotDet[Knot[5, 1]], KnotSignature[Knot[5, 1]]}
Out[9]=  
{5, -4}
In[10]:=
J=Jones[Knot[5, 1]][q]
Out[10]=  
  -7    -6    -5    -4    -2
-q   + q   - q   + q   + q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[5, 1], Knot[10, 132]}
In[12]:=
A2Invariant[Knot[5, 1]][q]
Out[12]=  
  -22    -20    -18    -14    -12    2     -8    -6

-q    - q    - q    + q    + q    + --- + q   + q



                                    10


                                    q
In[13]:=
Kauffman[Knot[5, 1]][a, z]
Out[13]=  
   4      6      5      7      9        4  2      6  2    8  2

3 a  + 2 a  - 2 a  z - a  z + a  z - 4 a  z  - 3 a  z  + a  z  + 



  5  3    7  3    4  4    6  4


  a  z  + a  z  + a  z  + a  z
In[14]:=
{Vassiliev[2][Knot[5, 1]], Vassiliev[3][Knot[5, 1]]}
Out[14]=  
{0, -5}
In[15]:=
Kh[Knot[5, 1]][q, t]
Out[15]=  
 -5    -3     1        1        1        1

q   + q   + ------ + ------ + ------ + -----



            15  5    11  4    11  3    7  2


            q   t    q   t    q   t    q  t
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