8 1

7_7

8_2

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8 1 Quick Notes

8 1 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X5,16,6,1 X7,14,8,15 X13,8,14,9 X15,6,16,7
Gauss code	-1, 4, -3, 1, -5, 8, -6, 7, -2, 3, -4, 2, -7, 6, -8, 5
Dowker-Thistlethwaite code	4 10 16 14 12 2 8 6
Conway Notation	[62]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 1 Bridge index 2 Super bridge index [math]\displaystyle{ \{4,5\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [-7][-3] Hyperbolic Volume 3.42721 A-Polynomial See Data:8 1/A-polynomial

[edit Notes for 8 1's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 1 }[/math] Topological 4 genus [math]\displaystyle{ 1 }[/math] Concordance genus [math]\displaystyle{ 1 }[/math] Rasmussen s-Invariant 0

[edit Notes for 8 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -3 t+7-3 t^{-1} }[/math]
Conway polynomial	[math]\displaystyle{ 1-3 z^2 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 13, 0 }
Jones polynomial	[math]\displaystyle{ q^2-q+2-2 q^{-1} +2 q^{-2} -2 q^{-3} + q^{-4} - q^{-5} + q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^6-z^2 a^4-a^4-z^2 a^2-z^2+ a^{-2} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^5 z^7+a^3 z^7+a^6 z^6+2 a^4 z^6+a^2 z^6-5 a^5 z^5-4 a^3 z^5+a z^5-5 a^6 z^4-8 a^4 z^4-2 a^2 z^4+z^4+7 a^5 z^3+5 a^3 z^3-a z^3+z^3 a^{-1} +6 a^6 z^2+7 a^4 z^2+z^2 a^{-2} -3 a^5 z-3 a^3 z-a^6-a^4- a^{-2} }[/math]
The A2 invariant	[math]\displaystyle{ q^{20}+q^{18}-q^{12}-q^{10}+ q^{-2} + q^{-6} + q^{-8} }[/math]
The G2 invariant	[math]\displaystyle{ q^{94}+q^{90}-q^{88}+2 q^{80}-2 q^{78}+q^{76}+q^{74}+q^{70}-q^{68}+q^{64}+q^{54}-q^{52}-q^{46}-q^{42}+q^{40}-q^{38}+q^{36}-q^{34}-2 q^{32}+q^{30}-q^{28}-q^{22}+q^{18}-q^{12}+q^8+ q^{-2} - q^{-6} + q^{-10} + q^{-14} + q^{-20} + q^{-24} + q^{-28} + q^{-34} + q^{-38} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 8_1.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{13}-q^7+ q^{-1} + q^{-5} }[/math]
2	[math]\displaystyle{ q^{38}-q^{34}-q^{28}+q^{24}+q^{12}+q^{10}-1+ q^{-8} + q^{-14} }[/math]
3	[math]\displaystyle{ q^{75}-q^{71}-q^{69}+q^{65}-q^{61}+q^{57}+q^{55}-q^{51}+q^{37}+q^{35}-q^{31}-q^{25}-q^{23}-q^{17}+q^{13}+2 q^{11}+q^9+q^7-q^5+q+ q^{-1} - q^{-5} + q^{-9} - q^{-13} - q^{-15} + q^{-19} + q^{-27} }[/math]
4	[math]\displaystyle{ q^{124}-q^{120}-q^{118}-q^{116}+q^{114}+q^{112}+q^{110}-2 q^{106}+q^{102}+q^{100}+q^{98}-q^{96}-q^{94}-q^{92}+q^{88}+q^{74}+q^{72}-q^{68}-2 q^{66}+q^{62}-q^{58}-2 q^{56}+2 q^{52}+q^{50}-q^{46}+2 q^{42}+q^{40}+q^{32}-q^{28}-q^{26}+q^{22}-q^{18}-q^{16}-q^{14}-q^{12}+q^{10}+2 q^8+q^6-q^4-2 q^2+2 q^{-2} +4 q^{-4} + q^{-6} -2 q^{-8} - q^{-10} + q^{-12} +3 q^{-14} -2 q^{-18} - q^{-20} +2 q^{-24} - q^{-28} - q^{-30} - q^{-32} + q^{-34} + q^{-44} }[/math]
5	[math]\displaystyle{ q^{185}-q^{181}-q^{179}-q^{177}+q^{173}+2 q^{171}+q^{169}-q^{165}-2 q^{163}-q^{161}+q^{159}+2 q^{157}+q^{155}-q^{151}-2 q^{149}-q^{147}+q^{143}+q^{141}+q^{139}-q^{135}+q^{123}+q^{121}-q^{117}-2 q^{115}-2 q^{113}+2 q^{109}+2 q^{107}-2 q^{103}-2 q^{101}-q^{99}+2 q^{97}+4 q^{95}+2 q^{93}-q^{91}-2 q^{89}-2 q^{87}+2 q^{83}+2 q^{81}-2 q^{77}-2 q^{75}+q^{71}+2 q^{69}+q^{67}-q^{65}-2 q^{63}-q^{61}+q^{57}+q^{55}-q^{53}-2 q^{51}-q^{49}+q^{45}-q^{41}+q^{37}+3 q^{35}+2 q^{33}-q^{25}+q^{23}+q^{21}+q^{19}+q^{17}-3 q^{13}-3 q^{11}-q^9+q^7+3 q^5+2 q^3-3 q^{-1} -3 q^{-3} +3 q^{-7} +3 q^{-9} -2 q^{-13} -2 q^{-15} +3 q^{-19} +2 q^{-21} -2 q^{-25} +2 q^{-29} +2 q^{-31} + q^{-33} - q^{-35} -2 q^{-37} - q^{-39} + q^{-41} + q^{-43} - q^{-49} - q^{-51} + q^{-65} }[/math]
6	[math]\displaystyle{ q^{258}-q^{254}-q^{252}-q^{250}+2 q^{244}+2 q^{242}+q^{240}-q^{236}-2 q^{234}-3 q^{232}+q^{228}+2 q^{226}+2 q^{224}+q^{222}-3 q^{218}-2 q^{216}-q^{214}+q^{210}+2 q^{208}+2 q^{206}-q^{200}-q^{198}-q^{196}+q^{192}+q^{184}+q^{182}-q^{178}-2 q^{176}-2 q^{174}-2 q^{172}+q^{170}+3 q^{168}+3 q^{166}+2 q^{164}-q^{162}-3 q^{160}-4 q^{158}-q^{156}+2 q^{154}+4 q^{152}+5 q^{150}+2 q^{148}-2 q^{146}-5 q^{144}-4 q^{142}-2 q^{140}+q^{138}+4 q^{136}+3 q^{134}+q^{132}-2 q^{130}-3 q^{128}-3 q^{126}-q^{124}+3 q^{122}+3 q^{120}+3 q^{118}+q^{116}-q^{114}-3 q^{112}-4 q^{110}-q^{108}+q^{106}+2 q^{104}+2 q^{102}+q^{100}-2 q^{98}-4 q^{96}-2 q^{94}+q^{92}+3 q^{90}+3 q^{88}+2 q^{86}-q^{84}-4 q^{82}-q^{80}+2 q^{78}+3 q^{76}+3 q^{74}+2 q^{72}-q^{70}-3 q^{68}-q^{66}+q^{64}+q^{62}-q^{58}-2 q^{56}-2 q^{54}+q^{50}-q^{46}-2 q^{44}-2 q^{42}-q^{40}+q^{38}+2 q^{36}+3 q^{34}+2 q^{32}+q^{30}-q^{26}-3 q^{24}-3 q^{22}+3 q^{18}+4 q^{16}+4 q^{14}+3 q^{12}-2 q^{10}-4 q^8-4 q^6-q^4+2 q^2+5+6 q^{-2} + q^{-4} -2 q^{-6} -5 q^{-8} -4 q^{-10} -2 q^{-12} +2 q^{-14} +4 q^{-16} + q^{-18} -2 q^{-22} - q^{-24} - q^{-26} + q^{-30} - q^{-32} + q^{-34} + q^{-36} +2 q^{-38} - q^{-42} - q^{-44} -2 q^{-46} + q^{-48} +2 q^{-50} +3 q^{-52} + q^{-54} - q^{-58} -2 q^{-60} + q^{-66} - q^{-74} - q^{-78} + q^{-90} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{20}+q^{18}-q^{12}-q^{10}+ q^{-2} + q^{-6} + q^{-8} }[/math]
1,1	[math]\displaystyle{ q^{52}+2 q^{48}-2 q^{46}+2 q^{44}-4 q^{42}+2 q^{40}-2 q^{38}-2 q^{32}+2 q^{30}-3 q^{28}+4 q^{26}-2 q^{24}+4 q^{22}-2 q^{20}+4 q^{18}+2 q^{14}+2 q^{12}-2 q^6-2 q^4-2 q^2-2+ q^{-4} +2 q^{-8} +2 q^{-12} +2 q^{-16} + q^{-20} }[/math]
2,0	[math]\displaystyle{ 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^{18}+2 q^{16}+q^{14}+q^{12}+q^{10}-q^4-2 q^2-2- q^{-2} + q^{-4} + q^{-10} + q^{-12} + q^{-16} + q^{-18} + q^{-20} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{40}+q^{36}-q^{32}-q^{30}-q^{28}+q^{24}+2 q^{22}+q^{20}+2 q^{18}-q^{14}-q^{12}-q^{10}-q^8-q^6+ q^{-4} + q^{-8} +2 q^{-10} + q^{-12} + q^{-16} }[/math]
1,0,0	[math]\displaystyle{ q^{27}+q^{25}+q^{23}-q^{17}-q^{15}-q^{13}+ q^{-3} + q^{-7} + q^{-9} + q^{-11} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{40}+q^{36}+q^{32}-q^{30}+q^{28}-q^{24}-q^{20}-2 q^{16}+q^{14}-q^{12}+q^{10}-q^8+q^6+ q^{-4} + q^{-8} + q^{-12} + q^{-16} }[/math]
1,0	[math]\displaystyle{ q^{66}+q^{58}-q^{54}-q^{52}-q^{46}-q^{44}+q^{40}+q^{38}+q^{36}+q^{32}+q^{30}+q^{28}-q^{18}-q^{16}-q^{10}-q^8+ q^{-6} + q^{-14} + q^{-16} + q^{-18} + q^{-26} }[/math]

G2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{94}+q^{90}-q^{88}+2 q^{80}-2 q^{78}+q^{76}+q^{74}+q^{70}-q^{68}+q^{64}+q^{54}-q^{52}-q^{46}-q^{42}+q^{40}-q^{38}+q^{36}-q^{34}-2 q^{32}+q^{30}-q^{28}-q^{22}+q^{18}-q^{12}+q^8+ q^{-2} - q^{-6} + q^{-10} + q^{-14} + q^{-20} + q^{-24} + q^{-28} + q^{-34} + q^{-38} }[/math]

.

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["8 1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -3 t+7-3 t^{-1} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 1-3 z^2 }[/math]

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]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 13, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^2-q+2-2 q^{-1} +2 q^{-2} -2 q^{-3} + q^{-4} - q^{-5} + q^{-6} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ a^6-z^2 a^4-a^4-z^2 a^2-z^2+ a^{-2} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^5 z^7+a^3 z^7+a^6 z^6+2 a^4 z^6+a^2 z^6-5 a^5 z^5-4 a^3 z^5+a z^5-5 a^6 z^4-8 a^4 z^4-2 a^2 z^4+z^4+7 a^5 z^3+5 a^3 z^3-a z^3+z^3 a^{-1} +6 a^6 z^2+7 a^4 z^2+z^2 a^{-2} -3 a^5 z-3 a^3 z-a^6-a^4- a^{-2} }[/math]

Vassiliev invariants

V2 and V3:

(-3, 3)

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 [math]\displaystyle{ -12 }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 66 }[/math] [math]\displaystyle{ 38 }[/math] [math]\displaystyle{ -288 }[/math] [math]\displaystyle{ -432 }[/math] [math]\displaystyle{ -96 }[/math] [math]\displaystyle{ -104 }[/math] [math]\displaystyle{ -288 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ -792 }[/math] [math]\displaystyle{ -456 }[/math] [math]\displaystyle{ \frac{1009}{10} }[/math] [math]\displaystyle{ \frac{2626}{15} }[/math] [math]\displaystyle{ -\frac{6182}{15} }[/math] [math]\displaystyle{ \frac{943}{6} }[/math] [math]\displaystyle{ -\frac{1551}{10} }[/math]

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s+1 }[/math], where [math]\displaystyle{ s= }[/math]0 is the signature of 8 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-6	-5	-4	-3	-2	-1	0	1	2	χ
5									1	1
3										0
1							2	1		1
-1						1	1			0
-3					1	1				0
-5				1	1					0
-7				1						-1
-9		1	1							0
-11										0
-13	1									1

Computer Talk

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

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math]

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[8, 1]]
Out[2]=	8
In[3]:=	PD[Knot[8, 1]]
Out[3]=	PD[X[1, 4, 2, 5], X[9, 12, 10, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], X[5, 16, 6, 1], X[7, 14, 8, 15], X[13, 8, 14, 9], X[15, 6, 16, 7]]
In[4]:=	GaussCode[Knot[8, 1]]
Out[4]=	GaussCode[-1, 4, -3, 1, -5, 8, -6, 7, -2, 3, -4, 2, -7, 6, -8, 5]
In[5]:=	BR[Knot[8, 1]]
Out[5]=	BR[5, {-1, -1, -2, 1, -2, -3, 2, 4, -3, 4}]
In[6]:=	alex = Alexander[Knot[8, 1]][t]
Out[6]=	3 7 - - - 3 t t
In[7]:=	Conway[Knot[8, 1]][z]
Out[7]=	2 1 - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 1]}
In[9]:=	{KnotDet[Knot[8, 1]], KnotSignature[Knot[8, 1]]}
Out[9]=	{13, 0}
In[10]:=	J=Jones[Knot[8, 1]][q]
Out[10]=	-6 -5 -4 2 2 2 2 2 + q - q + q - -- + -- - - - q + q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[8, 1], Knot[11, NonAlternating, 70]}
In[12]:=	A2Invariant[Knot[8, 1]][q]
Out[12]=	-20 -18 -12 -10 2 6 8 q + q - q - q + q + q + q
In[13]:=	Kauffman[Knot[8, 1]][a, z]
Out[13]=	2 3 -2 4 6 3 5 z 4 2 6 2 z 3 -a - a - a - 3 a z - 3 a z + -- + 7 a z + 6 a z + -- - a z + 2 a a 3 3 5 3 4 2 4 4 4 6 4 5 5 a z + 7 a z + z - 2 a z - 8 a z - 5 a z + a z - 3 5 5 5 2 6 4 6 6 6 3 7 5 7 4 a z - 5 a z + a z + 2 a z + a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[8, 1]], Vassiliev[3][Knot[8, 1]]}
Out[14]=	{0, 3}
In[15]:=	Kh[Knot[8, 1]][q, t]
Out[15]=	1 1 1 1 1 1 1 1 - + 2 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 9 5 9 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 1 1 5 2 ---- + --- + q t + q t 3 q t q t