10 136

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10 135.gif

10_135

10 137.gif

10_137

10 136.gif Visit 10 136's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

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10 136 Quick Notes


10 136 Further Notes and Views

Knot presentations

Planar diagram presentation X1425 X5,10,6,11 X3948 X9,3,10,2 X14,8,15,7 X18,12,19,11 X20,15,1,16 X16,19,17,20 X12,18,13,17 X6,14,7,13
Gauss code -1, 4, -3, 1, -2, -10, 5, 3, -4, 2, 6, -9, 10, -5, 7, -8, 9, -6, 8, -7
Dowker-Thistlethwaite code 4 8 10 -14 2 -18 -6 -20 -12 -16
Conway Notation [22,22,2-]

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-3][-6]
Hyperbolic Volume 7.74627
A-Polynomial See Data:10 136/A-polynomial

[edit Notes for 10 136'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{ 2 }[/math]
Rasmussen s-Invariant 0

[edit Notes for 10 136's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^2+4 t-5+4 t^{-1} - t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ 1-z^4 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 15, 2 }
Jones polynomial [math]\displaystyle{ -q^4+2 q^3-2 q^2+3 q-2+2 q^{-1} -2 q^{-2} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^4+a^2 z^2+2 z^2 a^{-2} -3 z^2+a^2+3 a^{-2} - a^{-4} -2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^8 a^{-2} +z^8+2 a z^7+3 z^7 a^{-1} +z^7 a^{-3} +a^2 z^6-4 z^6 a^{-2} -3 z^6-9 a z^5-14 z^5 a^{-1} -5 z^5 a^{-3} -4 a^2 z^4+2 z^4 a^{-2} -2 z^4+9 a z^3+16 z^3 a^{-1} +7 z^3 a^{-3} +3 a^2 z^2+4 z^2 a^{-2} +z^2 a^{-4} +6 z^2-2 a z-4 z a^{-1} -2 z a^{-3} -a^2-3 a^{-2} - a^{-4} -2 }[/math]
The A2 invariant [math]\displaystyle{ q^{10}-q^2+ q^{-4} +2 q^{-6} + q^{-8} + q^{-10} - q^{-12} - q^{-14} }[/math]
The G2 invariant [math]\displaystyle{ q^{46}-q^{44}+2 q^{42}-2 q^{40}+q^{38}-2 q^{34}+6 q^{32}-4 q^{30}+3 q^{28}-2 q^{24}+3 q^{22}-2 q^{20}-q^{18}+3 q^{16}-3 q^{14}+3 q^{10}-6 q^8+6 q^6-7 q^4+3-5 q^{-2} +4 q^{-4} -3 q^{-6} +3 q^{-8} + q^{-12} - q^{-14} +2 q^{-18} + q^{-20} + q^{-24} +3 q^{-26} +4 q^{-30} -6 q^{-32} +6 q^{-34} -2 q^{-36} +4 q^{-40} -7 q^{-42} +6 q^{-44} -2 q^{-48} + q^{-50} -2 q^{-52} -2 q^{-54} +3 q^{-56} -3 q^{-58} + q^{-60} + q^{-62} -3 q^{-64} +3 q^{-66} -3 q^{-68} + q^{-70} + q^{-72} -2 q^{-74} + q^{-76} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_136.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^7-q^5+ q^{-1} + q^{-3} + q^{-7} - q^{-9} }[/math]
2 [math]\displaystyle{ q^{22}-q^{20}-2 q^{18}+2 q^{16}+q^{14}-q^{12}+q^8+q^6-2 q^4+2- q^{-2} +2 q^{-6} - q^{-8} +2 q^{-12} + q^{-14} - q^{-16} - q^{-18} + q^{-20} -2 q^{-24} + q^{-28} }[/math]
3 [math]\displaystyle{ q^{45}-q^{43}-2 q^{41}+3 q^{37}+3 q^{35}-3 q^{33}-3 q^{31}+2 q^{27}+2 q^{25}-2 q^{21}-4 q^{19}+q^{17}+6 q^{15}+2 q^{13}-6 q^{11}-4 q^9+5 q^7+6 q^5-3 q^3-5 q+2 q^{-1} +6 q^{-3} - q^{-5} -4 q^{-7} + q^{-9} +3 q^{-11} - q^{-13} -2 q^{-15} + q^{-17} + q^{-19} +2 q^{-21} -4 q^{-25} - q^{-27} +7 q^{-29} +5 q^{-31} -7 q^{-33} -10 q^{-35} +5 q^{-37} +8 q^{-39} -2 q^{-41} -7 q^{-43} - q^{-45} +5 q^{-47} +2 q^{-49} - q^{-51} - q^{-53} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{10}-q^2+ q^{-4} +2 q^{-6} + q^{-8} + q^{-10} - q^{-12} - q^{-14} }[/math]
2,0 [math]\displaystyle{ q^{28}-q^{24}-q^{22}+q^{18}+2 q^8+q^6+q^2+1-2 q^{-4} - q^{-6} - q^{-8} - q^{-10} +2 q^{-14} +4 q^{-16} +3 q^{-18} +4 q^{-20} - q^{-22} - q^{-24} -2 q^{-26} -2 q^{-28} -2 q^{-30} -2 q^{-32} + q^{-34} + q^{-36} + q^{-38} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{20}-q^{18}+q^{14}+q^{10}+q^6-q^4-q^2- q^{-2} + q^{-6} +3 q^{-8} + q^{-10} +3 q^{-12} + q^{-14} - q^{-18} - q^{-20} - q^{-24} }[/math]
1,0,0 [math]\displaystyle{ q^{13}+q^9-q^3-q- q^{-1} + q^{-5} +2 q^{-7} +3 q^{-9} + q^{-11} + q^{-13} - q^{-15} - q^{-17} - q^{-19} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{20}-q^{18}+2 q^{16}-q^{14}+2 q^{12}-q^{10}-q^6-q^4+q^2-4+3 q^{-2} -2 q^{-4} +3 q^{-6} - q^{-8} +3 q^{-10} + q^{-12} + q^{-14} +2 q^{-16} - q^{-18} + q^{-20} -2 q^{-22} + q^{-24} -2 q^{-26} }[/math]
1,0 [math]\displaystyle{ q^{34}-q^{30}-q^{28}+q^{26}+q^{24}-q^{22}-q^{20}+2 q^{18}+2 q^{16}-q^{14}-q^{12}+2 q^8-2 q^4-q^2+1- q^{-4} +2 q^{-10} + q^{-12} +3 q^{-18} +2 q^{-20} -2 q^{-24} + q^{-26} + q^{-28} - q^{-30} -2 q^{-32} + q^{-36} - q^{-40} - q^{-42} + q^{-44} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{46}-q^{44}+2 q^{42}-2 q^{40}+q^{38}-2 q^{34}+6 q^{32}-4 q^{30}+3 q^{28}-2 q^{24}+3 q^{22}-2 q^{20}-q^{18}+3 q^{16}-3 q^{14}+3 q^{10}-6 q^8+6 q^6-7 q^4+3-5 q^{-2} +4 q^{-4} -3 q^{-6} +3 q^{-8} + q^{-12} - q^{-14} +2 q^{-18} + q^{-20} + q^{-24} +3 q^{-26} +4 q^{-30} -6 q^{-32} +6 q^{-34} -2 q^{-36} +4 q^{-40} -7 q^{-42} +6 q^{-44} -2 q^{-48} + q^{-50} -2 q^{-52} -2 q^{-54} +3 q^{-56} -3 q^{-58} + q^{-60} + q^{-62} -3 q^{-64} +3 q^{-66} -3 q^{-68} + q^{-70} + q^{-72} -2 q^{-74} + q^{-76} }[/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["10 136"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^2+4 t-5+4 t^{-1} - t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 1-z^4 }[/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]= { 15, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^4+2 q^3-2 q^2+3 q-2+2 q^{-1} -2 q^{-2} + q^{-3} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -z^4+a^2 z^2+2 z^2 a^{-2} -3 z^2+a^2+3 a^{-2} - a^{-4} -2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^8 a^{-2} +z^8+2 a z^7+3 z^7 a^{-1} +z^7 a^{-3} +a^2 z^6-4 z^6 a^{-2} -3 z^6-9 a z^5-14 z^5 a^{-1} -5 z^5 a^{-3} -4 a^2 z^4+2 z^4 a^{-2} -2 z^4+9 a z^3+16 z^3 a^{-1} +7 z^3 a^{-3} +3 a^2 z^2+4 z^2 a^{-2} +z^2 a^{-4} +6 z^2-2 a z-4 z a^{-1} -2 z a^{-3} -a^2-3 a^{-2} - a^{-4} -2 }[/math]

Vassiliev invariants

V2 and V3: (0, 1)
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{ 0 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{80}{3} }[/math] [math]\displaystyle{ -\frac{64}{3} }[/math] [math]\displaystyle{ 40 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 88 }[/math] [math]\displaystyle{ -88 }[/math] [math]\displaystyle{ \frac{328}{3} }[/math] [math]\displaystyle{ \frac{88}{3} }[/math] [math]\displaystyle{ 24 }[/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]2 is the signature of 10 136. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-10123χ
9       1-1
7      1 1
5     11 0
3    21  1
1   12   1
-1  121   0
-3 11     0
-5 1      -1
-71       1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ i=3 }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/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[Knot[10, 136]]
Out[2]=  
10
In[3]:=
PD[Knot[10, 136]]
Out[3]=  
PD[X[1, 4, 2, 5], X[5, 10, 6, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], 

 X[14, 8, 15, 7], X[18, 12, 19, 11], X[20, 15, 1, 16], 



  X[16, 19, 17, 20], X[12, 18, 13, 17], X[6, 14, 7, 13]]
In[4]:=
GaussCode[Knot[10, 136]]
Out[4]=  
GaussCode[-1, 4, -3, 1, -2, -10, 5, 3, -4, 2, 6, -9, 10, -5, 7, -8, 9, 
  -6, 8, -7]
In[5]:=
BR[Knot[10, 136]]
Out[5]=  
BR[5, {1, -2, 1, -2, -3, 2, 2, 4, -3, 4}]
In[6]:=
alex = Alexander[Knot[10, 136]][t]
Out[6]=  
      -2   4          2

-5 - t   + - + 4 t - t


           t
In[7]:=
Conway[Knot[10, 136]][z]
Out[7]=  
     4
1 - z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[8, 21], Knot[10, 136]}
In[9]:=
{KnotDet[Knot[10, 136]], KnotSignature[Knot[10, 136]]}
Out[9]=  
{15, 2}
In[10]:=
J=Jones[Knot[10, 136]][q]
Out[10]=  
      -3   2    2            2      3    4

-2 + q   - -- + - + 3 q - 2 q  + 2 q  - q



           2   q


           q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[10, 136], Knot[11, NonAlternating, 92]}
In[12]:=
A2Invariant[Knot[10, 136]][q]
Out[12]=  
 -10    -2    4      6    8    10    12    14
q    - q   + q  + 2 q  + q  + q   - q   - q
In[13]:=
Kauffman[Knot[10, 136]][a, z]
Out[13]=  
                                                 2      2

     -4   3     2   2 z   4 z              2   z    4 z       2  2



-2 - a   - -- - a  - --- - --- - 2 a z + 6 z  + -- + ---- + 3 a  z  + 



           2         3     a                    4     2
          a         a                          a     a

    3       3                      4                5       5
 7 z    16 z         3      4   2 z       2  4   5 z    14 z
 ---- + ----- + 9 a z  - 2 z  + ---- - 4 a  z  - ---- - ----- - 
   3      a                       2                3      a
  a                              a                a

                    6            7      7                  8
      5      6   4 z     2  6   z    3 z         7    8   z
 9 a z  - 3 z  - ---- + a  z  + -- + ---- + 2 a z  + z  + --
                   2             3    a                    2


                   a             a                         a
In[14]:=
{Vassiliev[2][Knot[10, 136]], Vassiliev[3][Knot[10, 136]]}
Out[14]=  
{0, 1}
In[15]:=
Kh[Knot[10, 136]][q, t]
Out[15]=  
1            3     1       1       1       1      1      2    q

- + 2 q + 2 q  + ----- + ----- + ----- + ----- + ---- + --- + - + 
q                 7  4    5  3    3  3    3  2      2   q t   t



                q  t    q  t    q  t    q  t    q t

  3      5      5  2    7  2    9  3


  q  t + q  t + q  t  + q  t  + q  t
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