8 3: Difference between revisions

Revision as of 21:03, 29 August 2005

8_2

8_4

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

Visit 8 3's page at Knotilus!

Visit 8 3's page at the original Knot Atlas!

8 3 Quick Notes

8 3 Further Notes and Views

Knot presentations

Planar diagram presentation	X₆₂₇₁ X_14,10,15,9 X_10,5,11,6 X_12,3,13,4 X_4,11,5,12 X_2,13,3,14 X_16,8,1,7 X_8,16,9,15
Gauss code	1, -6, 4, -5, 3, -1, 7, -8, 2, -3, 5, -4, 6, -2, 8, -7
Dowker-Thistlethwaite code	6 12 10 16 14 4 2 8
Conway Notation	[44]

Minimum Braid Representative:

Length is 10, width is 5.

Braid index is 5.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type	Fully amphicheiral
Unknotting number	2
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 \{4,6\}}
Nakanishi index	1
Maximal Thurston-Bennequin number	[-5][-5]
Hyperbolic Volume	5.23868
A-Polynomial	See Data:8 3/A-polynomial

[edit Notes for 8 3'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 1}
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 1}
Concordance genus	$1$
Rasmussen s-Invariant	0

[edit Notes for 8 3's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-4t+9-4t^{-1}$
Conway polynomial	$1-4z^{2}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 17, 0 }
Jones polynomial	$q^{4}-q^{3}+2q^{2}-3q+3-3q^{-1}+2q^{-2}-q^{-3}+q^{-4}$
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^4-z^2 a^2-2 z^2-1-z^2 a^{-2} + 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 z^7+z^7 a^{-1} +a^2 z^6+z^6 a^{-2} +2 z^6+a^3 z^5-4 a z^5-4 z^5 a^{-1} +z^5 a^{-3} +a^4 z^4-2 a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} -6 z^4-2 a^3 z^3+8 a z^3+8 z^3 a^{-1} -2 z^3 a^{-3} -3 a^4 z^2+a^2 z^2+z^2 a^{-2} -3 z^2 a^{-4} +8 z^2-4 a z-4 z a^{-1} +a^4+ a^{-4} -1}
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^{14}+q^{12}+q^8-q^4-1- q^{-4} + q^{-8} + q^{-12} + q^{-14} }
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^{66}+q^{62}-q^{60}+q^{58}+q^{56}-q^{54}+2 q^{52}-q^{50}+2 q^{48}-q^{46}+q^{42}-2 q^{40}+4 q^{38}-3 q^{36}+q^{34}+q^{32}-2 q^{30}+3 q^{28}-2 q^{26}+q^{24}+2 q^{22}-2 q^{20}+q^{18}-2 q^{14}+4 q^{12}-4 q^{10}+q^8-3 q^4+3 q^2-5+3 q^{-2} -3 q^{-4} + q^{-8} -4 q^{-10} +4 q^{-12} -2 q^{-14} + q^{-18} -2 q^{-20} +2 q^{-22} + q^{-24} -2 q^{-26} +3 q^{-28} -2 q^{-30} + q^{-32} + q^{-34} -3 q^{-36} +4 q^{-38} -2 q^{-40} + q^{-42} - q^{-46} +2 q^{-48} - q^{-50} +2 q^{-52} - q^{-54} + q^{-56} + q^{-58} - q^{-60} + q^{-62} + q^{-66} }

Further Quantum Invariants

Further quantum knot invariants for 8_3.

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^9+q^5-q^3- q^{-3} + q^{-5} + q^{-9} }
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^{26}+q^{20}-q^{18}-2 q^{16}+q^{14}-2 q^{10}+2 q^8+q^6-q^4+q^2+1+ q^{-2} - q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} + q^{-14} -2 q^{-16} - q^{-18} + q^{-20} + q^{-26} }
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^{51}-q^{41}-q^{39}+q^{35}-q^{33}-2 q^{31}+3 q^{27}+3 q^{25}-2 q^{23}-2 q^{21}+2 q^{19}+4 q^{17}-2 q^{15}-3 q^{13}+q^{11}+3 q^9-2 q^5-2 q^{-5} +3 q^{-9} + q^{-11} -3 q^{-13} -2 q^{-15} +4 q^{-17} +2 q^{-19} -2 q^{-21} -2 q^{-23} +3 q^{-25} +3 q^{-27} -2 q^{-31} - q^{-33} + q^{-35} - q^{-39} - q^{-41} + q^{-51} }
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^{84}-q^{76}-q^{72}+q^{68}-q^{66}-2 q^{62}-q^{60}+3 q^{58}+2 q^{56}+3 q^{54}-2 q^{52}-3 q^{50}-q^{48}+q^{46}+7 q^{44}+2 q^{42}-3 q^{40}-6 q^{38}-5 q^{36}+6 q^{34}+6 q^{32}+q^{30}-6 q^{28}-9 q^{26}+4 q^{24}+6 q^{22}+3 q^{20}-3 q^{18}-7 q^{16}+q^{14}+3 q^{12}+3 q^{10}-2 q^6+q^4+q^2+1+ q^{-2} + q^{-4} -2 q^{-6} +3 q^{-10} +3 q^{-12} + q^{-14} -7 q^{-16} -3 q^{-18} +3 q^{-20} +6 q^{-22} +4 q^{-24} -9 q^{-26} -6 q^{-28} + q^{-30} +6 q^{-32} +6 q^{-34} -5 q^{-36} -6 q^{-38} -3 q^{-40} +2 q^{-42} +7 q^{-44} + q^{-46} - q^{-48} -3 q^{-50} -2 q^{-52} +3 q^{-54} +2 q^{-56} +3 q^{-58} - q^{-60} -2 q^{-62} - q^{-66} + q^{-68} - q^{-72} - q^{-76} + q^{-84} }
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^{117}-q^{115}+q^{107}-2 q^{103}-q^{101}+q^{97}+3 q^{95}+3 q^{93}-3 q^{89}-3 q^{87}-2 q^{85}+3 q^{83}+5 q^{81}+5 q^{79}+q^{77}-5 q^{75}-8 q^{73}-6 q^{71}+8 q^{67}+10 q^{65}+5 q^{63}-6 q^{61}-14 q^{59}-11 q^{57}+12 q^{53}+16 q^{51}+7 q^{49}-11 q^{47}-18 q^{45}-10 q^{43}+6 q^{41}+18 q^{39}+16 q^{37}-3 q^{35}-15 q^{33}-12 q^{31}+11 q^{27}+12 q^{25}+2 q^{23}-7 q^{21}-8 q^{19}-2 q^{17}+4 q^{15}+4 q^{13}+2 q^{11}-q^9-3 q^7-3 q^{-7} - q^{-9} +2 q^{-11} +4 q^{-13} +4 q^{-15} -2 q^{-17} -8 q^{-19} -7 q^{-21} +2 q^{-23} +12 q^{-25} +11 q^{-27} -12 q^{-31} -15 q^{-33} -3 q^{-35} +16 q^{-37} +18 q^{-39} +6 q^{-41} -10 q^{-43} -18 q^{-45} -11 q^{-47} +7 q^{-49} +16 q^{-51} +12 q^{-53} -11 q^{-57} -14 q^{-59} -6 q^{-61} +5 q^{-63} +10 q^{-65} +8 q^{-67} -6 q^{-71} -8 q^{-73} -5 q^{-75} + q^{-77} +5 q^{-79} +5 q^{-81} +3 q^{-83} -2 q^{-85} -3 q^{-87} -3 q^{-89} +3 q^{-93} +3 q^{-95} + q^{-97} - q^{-101} -2 q^{-103} + q^{-107} - q^{-115} - q^{-117} + q^{-125} }

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^8-q^4-1- q^{-4} + q^{-8} + q^{-12} + q^{-14} }
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}+4 q^{28}-2 q^{26}+4 q^{24}-6 q^{22}+5 q^{20}-6 q^{18}+4 q^{16}-6 q^{14}-q^{12}-6 q^8+8 q^6-8 q^4+16 q^2-6+16 q^{-2} -8 q^{-4} +8 q^{-6} -6 q^{-8} - q^{-12} -6 q^{-14} +4 q^{-16} -6 q^{-18} +5 q^{-20} -6 q^{-22} +4 q^{-24} -2 q^{-26} +4 q^{-28} -2 q^{-30} +2 q^{-32} + q^{-36} }
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^{28}+q^{26}-q^{24}-3 q^{22}-2 q^{20}-2 q^{14}+2 q^{10}+q^4+2 q^2+2+2 q^{-2} + q^{-4} +2 q^{-10} -2 q^{-14} -2 q^{-20} -3 q^{-22} - q^{-24} + q^{-26} + q^{-28} + 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^{22}+q^{18}+2 q^{16}-2 q^{14}-q^{12}-3 q^8-q^6+q^4+2 q^2+2+2 q^{-2} + q^{-4} - q^{-6} -3 q^{-8} - q^{-12} -2 q^{-14} +2 q^{-16} + q^{-18} + q^{-22} + q^{-24} + q^{-28} }
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^{11}-q^5-q- q^{-1} - q^{-5} + q^{-11} + q^{-15} + q^{-17} + q^{-19} }

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^{22}+2 q^{20}-q^{18}+2 q^{16}+q^{12}-q^8+q^6-3 q^4+2 q^2-4+2 q^{-2} -3 q^{-4} + q^{-6} - q^{-8} + q^{-12} +2 q^{-16} - q^{-18} +2 q^{-20} - q^{-22} + q^{-24} + q^{-28} }

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^{36}-q^{32}+q^{28}+2 q^{26}-q^{24}-2 q^{22}-q^{20}+q^{18}-2 q^{14}-q^{12}+q^8+q^2+3+ q^{-2} + q^{-8} - q^{-12} -2 q^{-14} + q^{-18} - q^{-20} -2 q^{-22} - q^{-24} +2 q^{-26} + q^{-28} - q^{-32} + q^{-36} + q^{-38} + q^{-46} }

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^{58}+q^{56}-q^{54}+2 q^{52}-q^{50}+2 q^{48}-q^{46}+q^{42}-2 q^{40}+4 q^{38}-3 q^{36}+q^{34}+q^{32}-2 q^{30}+3 q^{28}-2 q^{26}+q^{24}+2 q^{22}-2 q^{20}+q^{18}-2 q^{14}+4 q^{12}-4 q^{10}+q^8-3 q^4+3 q^2-5+3 q^{-2} -3 q^{-4} + q^{-8} -4 q^{-10} +4 q^{-12} -2 q^{-14} + q^{-18} -2 q^{-20} +2 q^{-22} + q^{-24} -2 q^{-26} +3 q^{-28} -2 q^{-30} + q^{-32} + q^{-34} -3 q^{-36} +4 q^{-38} -2 q^{-40} + q^{-42} - q^{-46} +2 q^{-48} - q^{-50} +2 q^{-52} - q^{-54} + q^{-56} + q^{-58} - q^{-60} + q^{-62} + q^{-66} }

.

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 3"];

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

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

Out[4]= $-4t+9-4t^{-1}$

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

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

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]= { 17, 0 }

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

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

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

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^4-z^2 a^2-2 z^2-1-z^2 a^{-2} + 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 z^7+z^7 a^{-1} +a^2 z^6+z^6 a^{-2} +2 z^6+a^3 z^5-4 a z^5-4 z^5 a^{-1} +z^5 a^{-3} +a^4 z^4-2 a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} -6 z^4-2 a^3 z^3+8 a z^3+8 z^3 a^{-1} -2 z^3 a^{-3} -3 a^4 z^2+a^2 z^2+z^2 a^{-2} -3 z^2 a^{-4} +8 z^2-4 a z-4 z a^{-1} +a^4+ a^{-4} -1}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_1, ...}

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}} ): {...}

Vassiliev invariants

V₂ and V₃:

(-4, 0)

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 -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 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 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{520}{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{200}{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 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 -\frac{2048}{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 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 -\frac{8320}{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{3200}{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{37502}{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{6728}{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{96968}{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{2174}{9}}

-{\frac {7262}{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 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 3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-4

-3

-2

-1

0

1

2

3

4

χ

9

1

7

0

5

2

1

3

1

-1

1

2

0

-1

2

0

-3

1

-1

-5

1

2

1

-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}^{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}	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=0$	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}\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}^{2}\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=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^{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=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=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}}

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}	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_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^{12}-q^{11}+2 q^9-3 q^8-q^7+5 q^6-4 q^5-3 q^4+9 q^3-5 q^2-5 q+11-5 q^{-1} -5 q^{-2} +9 q^{-3} -3 q^{-4} -4 q^{-5} +5 q^{-6} - q^{-7} -3 q^{-8} +2 q^{-9} - 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^{24}-q^{23}+q^{20}-2 q^{19}+q^{17}+2 q^{16}-4 q^{15}-q^{14}+3 q^{13}+5 q^{12}-4 q^{11}-6 q^{10}+3 q^9+9 q^8-2 q^7-12 q^6+2 q^5+13 q^4-15 q^2+15-15 q^{-2} +13 q^{-4} +2 q^{-5} -12 q^{-6} -2 q^{-7} +9 q^{-8} +3 q^{-9} -6 q^{-10} -4 q^{-11} +5 q^{-12} +3 q^{-13} - q^{-14} -4 q^{-15} +2 q^{-16} + q^{-17} -2 q^{-19} + q^{-20} - 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^{40}-q^{39}-q^{36}+2 q^{35}-2 q^{34}+q^{33}+q^{32}-3 q^{31}+3 q^{30}-4 q^{29}+2 q^{28}+5 q^{27}-4 q^{26}+4 q^{25}-9 q^{24}+q^{23}+7 q^{22}-2 q^{21}+10 q^{20}-14 q^{19}-4 q^{18}+4 q^{17}-q^{16}+21 q^{15}-14 q^{14}-9 q^{13}-3 q^{12}-4 q^{11}+34 q^{10}-12 q^9-12 q^8-9 q^7-8 q^6+42 q^5-10 q^4-12 q^3-12 q^2-10 q+45-10 q^{-1} -12 q^{-2} -12 q^{-3} -10 q^{-4} +42 q^{-5} -8 q^{-6} -9 q^{-7} -12 q^{-8} -12 q^{-9} +34 q^{-10} -4 q^{-11} -3 q^{-12} -9 q^{-13} -14 q^{-14} +21 q^{-15} - q^{-16} +4 q^{-17} -4 q^{-18} -14 q^{-19} +10 q^{-20} -2 q^{-21} +7 q^{-22} + q^{-23} -9 q^{-24} +4 q^{-25} -4 q^{-26} +5 q^{-27} +2 q^{-28} -4 q^{-29} +3 q^{-30} -3 q^{-31} + q^{-32} + q^{-33} -2 q^{-34} +2 q^{-35} - q^{-36} - 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^{60}-q^{59}-q^{56}+2 q^{54}-q^{53}+q^{51}-2 q^{50}-2 q^{49}+3 q^{48}+q^{46}+3 q^{45}-2 q^{44}-5 q^{43}+2 q^{40}+8 q^{39}-5 q^{37}-4 q^{36}-6 q^{35}-q^{34}+10 q^{33}+6 q^{32}+3 q^{31}-2 q^{30}-11 q^{29}-12 q^{28}+2 q^{27}+9 q^{26}+14 q^{25}+10 q^{24}-7 q^{23}-21 q^{22}-16 q^{21}+2 q^{20}+22 q^{19}+26 q^{18}+5 q^{17}-23 q^{16}-35 q^{15}-10 q^{14}+25 q^{13}+38 q^{12}+16 q^{11}-22 q^{10}-45 q^9-19 q^8+24 q^7+44 q^6+22 q^5-22 q^4-47 q^3-22 q^2+22 q+47+22 q^{-1} -22 q^{-2} -47 q^{-3} -22 q^{-4} +22 q^{-5} +44 q^{-6} +24 q^{-7} -19 q^{-8} -45 q^{-9} -22 q^{-10} +16 q^{-11} +38 q^{-12} +25 q^{-13} -10 q^{-14} -35 q^{-15} -23 q^{-16} +5 q^{-17} +26 q^{-18} +22 q^{-19} +2 q^{-20} -16 q^{-21} -21 q^{-22} -7 q^{-23} +10 q^{-24} +14 q^{-25} +9 q^{-26} +2 q^{-27} -12 q^{-28} -11 q^{-29} -2 q^{-30} +3 q^{-31} +6 q^{-32} +10 q^{-33} - q^{-34} -6 q^{-35} -4 q^{-36} -5 q^{-37} +8 q^{-39} +2 q^{-40} -5 q^{-43} -2 q^{-44} +3 q^{-45} + q^{-46} +3 q^{-48} -2 q^{-49} -2 q^{-50} + q^{-51} - q^{-53} +2 q^{-54} - q^{-56} - 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^{84}-q^{83}-q^{80}+3 q^{77}-2 q^{76}+q^{74}-2 q^{73}-q^{72}-q^{71}+6 q^{70}-2 q^{69}+3 q^{67}-4 q^{66}-3 q^{65}-4 q^{64}+9 q^{63}-q^{62}+q^{61}+7 q^{60}-4 q^{59}-7 q^{58}-11 q^{57}+10 q^{56}-2 q^{55}+2 q^{54}+15 q^{53}+2 q^{52}-6 q^{51}-17 q^{50}+7 q^{49}-13 q^{48}-5 q^{47}+20 q^{46}+12 q^{45}+7 q^{44}-11 q^{43}+13 q^{42}-29 q^{41}-25 q^{40}+8 q^{39}+12 q^{38}+21 q^{37}+10 q^{36}+39 q^{35}-32 q^{34}-44 q^{33}-21 q^{32}-8 q^{31}+19 q^{30}+29 q^{29}+82 q^{28}-15 q^{27}-50 q^{26}-50 q^{25}-40 q^{24}+q^{23}+36 q^{22}+124 q^{21}+9 q^{20}-45 q^{19}-68 q^{18}-65 q^{17}-19 q^{16}+34 q^{15}+151 q^{14}+25 q^{13}-38 q^{12}-76 q^{11}-76 q^{10}-31 q^9+31 q^8+163 q^7+29 q^6-34 q^5-78 q^4-78 q^3-34 q^2+29 q+167+29 q^{-1} -34 q^{-2} -78 q^{-3} -78 q^{-4} -34 q^{-5} +29 q^{-6} +163 q^{-7} +31 q^{-8} -31 q^{-9} -76 q^{-10} -76 q^{-11} -38 q^{-12} +25 q^{-13} +151 q^{-14} +34 q^{-15} -19 q^{-16} -65 q^{-17} -68 q^{-18} -45 q^{-19} +9 q^{-20} +124 q^{-21} +36 q^{-22} + q^{-23} -40 q^{-24} -50 q^{-25} -50 q^{-26} -15 q^{-27} +82 q^{-28} +29 q^{-29} +19 q^{-30} -8 q^{-31} -21 q^{-32} -44 q^{-33} -32 q^{-34} +39 q^{-35} +10 q^{-36} +21 q^{-37} +12 q^{-38} +8 q^{-39} -25 q^{-40} -29 q^{-41} +13 q^{-42} -11 q^{-43} +7 q^{-44} +12 q^{-45} +20 q^{-46} -5 q^{-47} -13 q^{-48} +7 q^{-49} -17 q^{-50} -6 q^{-51} +2 q^{-52} +15 q^{-53} +2 q^{-54} -2 q^{-55} +10 q^{-56} -11 q^{-57} -7 q^{-58} -4 q^{-59} +7 q^{-60} + q^{-61} - q^{-62} +9 q^{-63} -4 q^{-64} -3 q^{-65} -4 q^{-66} +3 q^{-67} -2 q^{-69} +6 q^{-70} - q^{-71} - q^{-72} -2 q^{-73} + q^{-74} -2 q^{-76} +3 q^{-77} - q^{-80} - q^{-83} + q^{-84} }
7	Not Available

Computer Talk

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

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[8, 3]]
Out[2]=	PD[X[6, 2, 7, 1], X[14, 10, 15, 9], X[10, 5, 11, 6], X[12, 3, 13, 4], X[4, 11, 5, 12], X[2, 13, 3, 14], X[16, 8, 1, 7], X[8, 16, 9, 15]]
In[3]:=	GaussCode[Knot[8, 3]]
Out[3]=	GaussCode[1, -6, 4, -5, 3, -1, 7, -8, 2, -3, 5, -4, 6, -2, 8, -7]
In[4]:=	DTCode[Knot[8, 3]]
Out[4]=	DTCode[6, 12, 10, 16, 14, 4, 2, 8]
In[5]:=	br = BR[Knot[8, 3]]
Out[5]=	BR[5, {-1, -1, -2, 1, 3, -2, 3, 4, -3, 4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 10}
In[7]:=	BraidIndex[Knot[8, 3]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[8, 3]]]

`Out[8]=`	`-Graphics-`
In[9]:=	(#[Knot[8, 3]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{FullyAmphicheiral, 2, 1, 2, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[8, 3]][t]
Out[10]=	4 9 - - - 4 t t
In[11]:=	Conway[Knot[8, 3]][z]
Out[11]=	2 1 - 4 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 3], Knot[10, 1]}
In[13]:=	{KnotDet[Knot[8, 3]], KnotSignature[Knot[8, 3]]}
Out[13]=	{17, 0}
In[14]:=	Jones[Knot[8, 3]][q]
Out[14]=	-4 -3 2 3 2 3 4 3 + q - q + -- - - - 3 q + 2 q - q + q 2 q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[8, 3]}
In[16]:=	A2Invariant[Knot[8, 3]][q]
Out[16]=	-14 -12 -8 -4 4 8 12 14 -1 + q + q + q - q - q + q + q + q
In[17]:=	HOMFLYPT[Knot[8, 3]][a, z]
Out[17]=	2 -4 4 2 z 2 2 -1 + a + a - 2 z - -- - a z 2 a
In[18]:=	Kauffman[Knot[8, 3]][a, z]
Out[18]=	2 2 -4 4 4 z 2 3 z z 2 2 4 2 -1 + a + a - --- - 4 a z + 8 z - ---- + -- + a z - 3 a z - a 4 2 a a 3 3 4 4 2 z 8 z 3 3 3 4 z 2 z 2 4 4 4 ---- + ---- + 8 a z - 2 a z - 6 z + -- - ---- - 2 a z + a z + 3 a 4 2 a a a 5 5 6 7 z 4 z 5 3 5 6 z 2 6 z 7 -- - ---- - 4 a z + a z + 2 z + -- + a z + -- + a z 3 a 2 a a a
In[19]:=	{Vassiliev[2][Knot[8, 3]], Vassiliev[3][Knot[8, 3]]}
Out[19]=	{-4, 0}
In[20]:=	Kh[Knot[8, 3]][q, t]
Out[20]=	2 1 1 2 1 2 3 5 2 - + 2 q + ----- + ----- + ----- + ---- + --- + 2 q t + q t + 2 q t + q 9 4 5 3 5 2 3 q t q t q t q t q t 5 3 9 4 q t + q t
In[21]:=	ColouredJones[Knot[8, 3], 2][q]
Out[21]=	-12 -11 2 3 -7 5 4 3 9 5 5 11 + q - q + -- - -- - q + -- - -- - -- + -- - -- - - - 5 q - 9 8 6 5 4 3 2 q q q q q q q q 2 3 4 5 6 7 8 9 11 12 5 q + 9 q - 3 q - 4 q + 5 q - q - 3 q + 2 q - q + q

See/edit the Rolfsen_Splice_Template.

Back to the top.

8_2

8_4

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+<!-- This page was generated from the splice template "Rolfsen_Splice_Template". Please do not edit! -->
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 </table>
+{| width=100%
-See/edit the [[Rolfsen_Splice_Template]].
+|align=left|See/edit the [[Rolfsen_Splice_Template]].
+Back to the [[#top|top]].
+|align=right|{{Knot Navigation Links|ext=gif}}
+|}
  [[Category:Knot Page]]

8 3: Difference between revisions

Revision as of 21:03, 29 August 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

Computer Talk

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