8 1

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7 7.gif

7_7

8 2.gif

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]

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 [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 \
 -6-5-4-3-2-1012χ
5        11
3         0
1      21 1
-1     11  0
-3    11   0
-5   11    0
-7   1     -1
-9 11      0
-11         0
-131        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
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