L8a21

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L8a20.gif

L8a20

L8n1.gif

L8n1

L8a21.gif Visit L8a21's page at Knotilus!

Visit L8a21's page at the original Knot Atlas!

L8a21 is a closed four-link chain. It is [math]\displaystyle{ 8^4_{1} }[/math] in the Rolfsen table of links.



Four linked squares.
Floor decoration in the Biblioteca Medicea Laurenziana, Florence, intended to contain four rings interlinked in this maner (but there is an interlacing error at upper right).
Ornamental circular arcs in square.


Knot presentations

Planar diagram presentation X6172 X2536 X16,11,13,12 X10,3,11,4 X4,9,1,10 X14,7,15,8 X8,13,5,14 X12,15,9,16
Gauss code {1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -8}, {7, -6, 8, -3}

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ \frac{t(2) t(1)-t(2) t(3) t(1)+t(3) t(1)-t(2) t(4) t(1)-t(3) t(4) t(1)+2 t(4) t(1)-t(1)-t(2)+2 t(2) t(3)-t(3)+t(2) t(4)-t(2) t(3) t(4)+t(3) t(4)-t(4)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} \sqrt{t(4)}} }[/math] (db)
Jones polynomial [math]\displaystyle{ \frac{4}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{1}{q^{17/2}}-\frac{5}{q^{15/2}}+\frac{4}{q^{13/2}}-\frac{7}{q^{11/2}} }[/math] (db)
Signature -3 (db)
HOMFLY-PT polynomial [math]\displaystyle{ a^{11} z^{-3} -3 a^9 z^{-3} -4 a^9 z^{-1} +3 a^7 z^{-3} +6 a^7 z+8 a^7 z^{-1} -3 a^5 z^3-a^5 z^{-3} -6 a^5 z-4 a^5 z^{-1} -a^3 z^3 }[/math] (db)
Kauffman polynomial [math]\displaystyle{ -z^5 a^{11}+4 z^3 a^{11}-6 z a^{11}+4 a^{11} z^{-1} -a^{11} z^{-3} -z^6 a^{10}+6 z^2 a^{10}+3 a^{10} z^{-2} -8 a^{10}-z^7 a^9-2 z^5 a^9+11 z^3 a^9-14 z a^9+9 a^9 z^{-1} -3 a^9 z^{-3} -5 z^6 a^8+5 z^4 a^8+12 z^2 a^8+6 a^8 z^{-2} -15 a^8-z^7 a^7-7 z^5 a^7+17 z^3 a^7-14 z a^7+9 a^7 z^{-1} -3 a^7 z^{-3} -4 z^6 a^6+2 z^4 a^6+6 z^2 a^6+3 a^6 z^{-2} -8 a^6-6 z^5 a^5+9 z^3 a^5-6 z a^5+4 a^5 z^{-1} -a^5 z^{-3} -3 z^4 a^4-z^3 a^3 }[/math] (db)

Vassiliev invariants

V2 and V3: (0, [math]\displaystyle{ -\frac{185}{3} }[/math])
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
Data:L8a21/V 2,1 Data:L8a21/V 3,1 Data:L8a21/V 4,1 Data:L8a21/V 4,2 Data:L8a21/V 4,3 Data:L8a21/V 5,1 Data:L8a21/V 5,2 Data:L8a21/V 5,3 Data:L8a21/V 5,4 Data:L8a21/V 6,1 Data:L8a21/V 6,2 Data:L8a21/V 6,3 Data:L8a21/V 6,4 Data:L8a21/V 6,5 Data:L8a21/V 6,6 Data:L8a21/V 6,7 Data:L8a21/V 6,8 Data:L8a21/V 6,9

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]-3 is the signature of L8a21. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -8-7-6-5-4-3-2-10χ
-2        11
-4       31-2
-6      3  3
-8     13  2
-10    63   3
-12   47    3
-14  1      1
-16  4      4
-1811       0
-201        1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math]
[math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

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[Link[8, Alternating, 21]]
Out[2]=  
8
In[3]:=
PD[Link[8, Alternating, 21]]
Out[3]=  
PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[16, 11, 13, 12], X[10, 3, 11, 4], 
  X[4, 9, 1, 10], X[14, 7, 15, 8], X[8, 13, 5, 14], X[12, 15, 9, 16]]
In[4]:=
GaussCode[Link[8, Alternating, 21]]
Out[4]=  
GaussCode[{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -8}, 
  {7, -6, 8, -3}]
In[5]:=
BR[Link[8, Alternating, 21]]
Out[5]=  
BR[Link[8, Alternating, 21]]
In[6]:=
alex = Alexander[Link[8, Alternating, 21]][t]
Out[6]=  
ComplexInfinity
In[7]:=
Conway[Link[8, Alternating, 21]][z]
Out[7]=  
ComplexInfinity
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{}
In[9]:=
{KnotDet[Link[8, Alternating, 21]], KnotSignature[Link[8, Alternating, 21]]}
Out[9]=  
{Infinity, -3}
In[10]:=
J=Jones[Link[8, Alternating, 21]][q]
Out[10]=  
  -(19/2)    -(17/2)     5       4       7      4      6      3

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



                       15/2    13/2    11/2    9/2    7/2    5/2
                      q       q       q       q      q      q

  -(3/2)


  q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{}
In[12]:=
A2Invariant[Link[8, Alternating, 21]][q]
Out[12]=  
 -32    4     6     8    13    12    11    10     6     6     2

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



       30    28    26    24    22    20    18    16    14    12
      q     q     q     q     q     q     q     q     q     q

  2     -8   2     -4
 --- + q   - -- + q
  10          6


  q           q
In[13]:=
Kauffman[Link[8, Alternating, 21]][a, z]
Out[13]=  
                         5      7      9    11      6      8      10

   6       8      10   a    3 a    3 a    a     3 a    6 a    3 a



-8 a  - 15 a  - 8 a   - -- - ---- - ---- - --- + ---- + ---- + ----- + 



                        3     3      3     3      2      2      2
                       z     z      z     z      z      z      z

    5      7      9      11
 4 a    9 a    9 a    4 a        5         7         9        11
 ---- + ---- + ---- + ----- - 6 a  z - 14 a  z - 14 a  z - 6 a   z + 
  z      z      z       z

    6  2       8  2      10  2    3  3      5  3       7  3
 6 a  z  + 12 a  z  + 6 a   z  - a  z  + 9 a  z  + 17 a  z  + 

     9  3      11  3      4  4      6  4      8  4      5  5
 11 a  z  + 4 a   z  - 3 a  z  + 2 a  z  + 5 a  z  - 6 a  z  - 

    7  5      9  5    11  5      6  6      8  6    10  6    7  7
 7 a  z  - 2 a  z  - a   z  - 4 a  z  - 5 a  z  - a   z  - a  z  - 

  9  7


  a  z
In[14]:=
{Vassiliev[2][Link[8, Alternating, 21]], Vassiliev[3][Link[8, Alternating, 21]]}
Out[14]=  
      185

{0, -(---)}


       3
In[15]:=
Kh[Link[8, Alternating, 21]][q, t]
Out[15]=  
 -4    -2     1        1        1        4        1        4

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



            20  8    18  8    18  7    16  6    14  6    12  5
           q   t    q   t    q   t    q   t    q   t    q   t

   7        6        3        1       3       3      3
 ------ + ------ + ------ + ----- + ----- + ----- + ----
  12  4    10  4    10  3    8  3    8  2    6  2    4


  q   t    q   t    q   t    q  t    q  t    q  t    q  t
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