© | Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table:
10.129
10129
10.131
10131
    10.130
KnotPlot
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   The Non Alternating Knot 10130   

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Acknowledgement

10.130
KnotPlot

PD Presentation: X4251 X8493 X5,14,6,15 X15,20,16,1 X9,16,10,17 X19,10,20,11 X11,18,12,19 X17,12,18,13 X13,6,14,7 X2837

Gauss Code: {1, -10, 2, -1, -3, 9, 10, -2, -5, 6, -7, 8, -9, 3, -4, 5, -8, 7, -6, 4}

DT (Dowker-Thistlethwaite) Code: 4 8 -14 2 -16 -18 -6 -20 -12 -10

Minimum Braid Representative:


Length is 11, width is 4
Braid index is 4

A Morse Link Presentation:

3D Invariants:
Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index
Reversible 2 2 3 / NotAvailable 1

Alexander Polynomial: 2t-2 - 4t-1 + 5 - 4t + 2t2

Conway Polynomial: 1 + 4z2 + 2z4

Other knots with the same Alexander/Conway Polynomial: {75, ...}

Determinant and Signature: {17, 0}

Jones Polynomial: - q-7 + q-6 - 2q-5 + 3q-4 - 2q-3 + 3q-2 - 2q-1 + 2 - q

Other knots (up to mirrors) with the same Jones Polynomial: {K11n61, ...}

A2 (sl(3)) Invariant: - q-22 - q-20 - q-18 - q-16 + q-14 + q-12 + 2q-10 + 2q-8 + q-6 + q-4 - q4

HOMFLY-PT Polynomial: - 1 - z2 + 2a2 + 3a2z2 + a2z4 + 2a4 + 3a4z2 + a4z4 - 2a6 - a6z2

Kauffman Polynomial: a-1z - 1 + 2z2 + az - 2az3 + az5 - 2a2 + 6a2z2 - 7a2z4 + 2a2z6 - 3a3z + 8a3z3 - 8a3z5 + 2a3z7 + 2a4 - 3a4z6 + a4z8 - 9a5z + 21a5z3 - 15a5z5 + 3a5z7 + 2a6 - 4a6z2 + 7a6z4 - 5a6z6 + a6z8 - 6a7z + 11a7z3 - 6a7z5 + a7z7

V2 and V3, the type 2 and 3 Vassiliev invariants: {4, -6}

Khovanov Homology:
(The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=0 is the signature of 10130. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1
j = 3        1
j = 1       1 
j = -1      22 
j = -3     1   
j = -5    12   
j = -7   21    
j = -9   1     
j = -11 12      
j = -13         
j = -151        

 n  Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q-21 - q-20 - q-19 + 3q-18 - q-17 - 4q-16 + 4q-15 - 5q-13 + 4q-12 + q-11 - 4q-10 + 2q-9 + 2q-8 - 2q-7 + 2q-5 + q-4 - 2q-3 + q-2 + q-1 - 1
3 - q-42 + q-41 + q-40 - 3q-38 + 3q-36 + 3q-35 - 4q-34 - 3q-33 + 2q-32 + 5q-31 - 2q-30 - 3q-29 + 2q-27 - q-26 + 2q-24 - 3q-23 - 3q-22 + 3q-21 + 6q-20 - 6q-19 - 6q-18 + 4q-17 + 10q-16 - 7q-15 - 9q-14 + 5q-13 + 15q-12 - 9q-11 - 13q-10 + 6q-9 + 18q-8 - 8q-7 - 15q-6 + 4q-5 + 17q-4 - 4q-3 - 12q-2 + q-1 + 9 - q - 5q2 + q3 + 2q4 - 2q6 + q7
4 q-70 - q-69 - q-68 + 4q-65 - q-64 - 2q-63 - 2q-62 - 4q-61 + 7q-60 + 2q-59 + q-58 - 2q-57 - 9q-56 + 5q-55 + 4q-53 + 3q-52 - 7q-51 + 5q-50 - 5q-49 - 2q-48 + 3q-47 - 2q-46 + 14q-45 - 3q-44 - 11q-43 - 5q-42 - 5q-41 + 22q-40 + 9q-39 - 13q-38 - 12q-37 - 15q-36 + 20q-35 + 21q-34 - 7q-33 - 11q-32 - 24q-31 + 10q-30 + 27q-29 - 5q-27 - 29q-26 + 29q-24 + 5q-23 + q-22 - 33q-21 - 8q-20 + 32q-19 + 11q-18 + 4q-17 - 38q-16 - 17q-15 + 35q-14 + 19q-13 + 11q-12 - 41q-11 - 27q-10 + 30q-9 + 22q-8 + 18q-7 - 31q-6 - 28q-5 + 16q-4 + 13q-3 + 19q-2 - 15q-1 - 17 + 7q + q2 + 10q3 - 5q4 - 5q5 + 4q6 - 2q7 + 2q8 - 2q9 + 2q11 - q12


Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=    
<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=
PD[Knot[10, 130]]
Out[2]=   
PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 14, 6, 15], X[15, 20, 16, 1], 
 
>   X[9, 16, 10, 17], X[19, 10, 20, 11], X[11, 18, 12, 19], X[17, 12, 18, 13], 
 
>   X[13, 6, 14, 7], X[2, 8, 3, 7]]
In[3]:=
GaussCode[Knot[10, 130]]
Out[3]=   
GaussCode[1, -10, 2, -1, -3, 9, 10, -2, -5, 6, -7, 8, -9, 3, -4, 5, -8, 7, -6, 
 
>   4]
In[4]:=
DTCode[Knot[10, 130]]
Out[4]=   
DTCode[4, 8, -14, 2, -16, -18, -6, -20, -12, -10]
In[5]:=
br = BR[Knot[10, 130]]
Out[5]=   
BR[4, {1, 1, 1, -2, -1, -1, -2, -2, -3, 2, -3}]
In[6]:=
{First[br], Crossings[br]}
Out[6]=   
{4, 11}
In[7]:=
BraidIndex[Knot[10, 130]]
Out[7]=   
4
In[8]:=
Show[DrawMorseLink[Knot[10, 130]]]
Out[8]=   
 -Graphics- 
In[9]:=
#[Knot[10, 130]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=   
{Reversible, 2, 2, 3, NotAvailable, 1}
In[10]:=
alex = Alexander[Knot[10, 130]][t]
Out[10]=   
    2    4            2
5 + -- - - - 4 t + 2 t
     2   t
    t
In[11]:=
Conway[Knot[10, 130]][z]
Out[11]=   
       2      4
1 + 4 z  + 2 z
In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=   
{Knot[7, 5], Knot[10, 130]}
In[13]:=
{KnotDet[Knot[10, 130]], KnotSignature[Knot[10, 130]]}
Out[13]=   
{17, 0}
In[14]:=
Jones[Knot[10, 130]][q]
Out[14]=   
     -7    -6   2    3    2    3    2
2 - q   + q   - -- + -- - -- + -- - - - q
                 5    4    3    2   q
                q    q    q    q
In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=   
{Knot[10, 130], Knot[11, NonAlternating, 61]}
In[16]:=
A2Invariant[Knot[10, 130]][q]
Out[16]=   
  -22    -20    -18    -16    -14    -12    2    2     -6    -4    4
-q    - q    - q    - q    + q    + q    + --- + -- + q   + q   - q
                                            10    8
                                           q     q
In[17]:=
HOMFLYPT[Knot[10, 130]][a, z]
Out[17]=   
        2      4      6    2      2  2      4  2    6  2    2  4    4  4
-1 + 2 a  + 2 a  - 2 a  - z  + 3 a  z  + 3 a  z  - a  z  + a  z  + a  z
In[18]:=
Kauffman[Knot[10, 130]][a, z]
Out[18]=   
        2      4      6   z            3        5        7        2      2  2
-1 - 2 a  + 2 a  + 2 a  + - + a z - 3 a  z - 9 a  z - 6 a  z + 2 z  + 6 a  z  - 
                          a
 
       6  2        3      3  3       5  3       7  3      2  4      6  4
>   4 a  z  - 2 a z  + 8 a  z  + 21 a  z  + 11 a  z  - 7 a  z  + 7 a  z  + 
 
       5      3  5       5  5      7  5      2  6      4  6      6  6
>   a z  - 8 a  z  - 15 a  z  - 6 a  z  + 2 a  z  - 3 a  z  - 5 a  z  + 
 
       3  7      5  7    7  7    4  8    6  8
>   2 a  z  + 3 a  z  + a  z  + a  z  + a  z
In[19]:=
{Vassiliev[2][Knot[10, 130]], Vassiliev[3][Knot[10, 130]]}
Out[19]=   
{4, -6}
In[20]:=
Kh[Knot[10, 130]][q, t]
Out[20]=   
2         1        1        2        1       2       1       1       2
- + q + ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 
q        15  7    11  6    11  5    9  4    7  4    7  3    5  3    5  2
        q   t    q   t    q   t    q  t    q  t    q  t    q  t    q  t
 
      1      2     3
>   ----- + --- + q  t
     3  2   q t
    q  t
In[21]:=
ColouredJones[Knot[10, 130], 2][q]
Out[21]=   
      -21    -20    -19    3     -17    4     4     5     4     -11    4
-1 + q    - q    - q    + --- - q    - --- + --- - --- + --- + q    - --- + 
                           18           16    15    13    12           10
                          q            q     q     q     q            q
 
    2    2    2    2     -4   2     -2   1
>   -- + -- - -- + -- + q   - -- + q   + -
     9    8    7    5          3         q
    q    q    q    q          q


Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10130
10.129
10129
10.131
10131