© | Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table:
10.121
10121
10.123
10123
    10.122
KnotPlot
This page is passe. Go here instead!

   The Alternating Knot 10122   

Visit 10122's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10122's page at Knotilus!

Acknowledgement

10.122
KnotPlot

PD Presentation: X1627 X7,15,8,14 X15,2,16,3 X5,12,6,13 X9,19,10,18 X3,11,4,10 X17,5,18,4 X19,9,20,8 X11,16,12,17 X13,1,14,20

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

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

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 3 3 / NotAvailable 2

Alexander Polynomial: - 2t-3 + 11t-2 - 24t-1 + 31 - 24t + 11t2 - 2t3

Conway Polynomial: 1 + 2z2 - z4 - 2z6

Other knots with the same Alexander/Conway Polynomial: {K11n185, ...}

Determinant and Signature: {105, 0}

Jones Polynomial: q-4 - 4q-3 + 8q-2 - 13q-1 + 17 - 17q + 17q2 - 13q3 + 9q4 - 5q5 + q6

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

A2 (sl(3)) Invariant: q-12 - 2q-10 + q-8 + q-6 - 4q-4 + 3q-2 - 2 + 2q2 + 3q4 + 5q8 - 3q10 - 3q16 + q18

HOMFLY-PT Polynomial: - 2a-4 + a-4z4 + 4a-2 + 3a-2z2 - a-2z4 - a-2z6 - 1 - 2z2 - 2z4 - z6 + a2z2 + a2z4

Kauffman Polynomial: - a-6z4 + a-6z6 + 2a-5z + 4a-5z3 - 11a-5z5 + 5a-5z7 - 2a-4 + 12a-4z4 - 20a-4z6 + 8a-4z8 + 2a-3z + 14a-3z3 - 25a-3z5 + 3a-3z7 + 4a-3z9 - 4a-2 + 24a-2z4 - 42a-2z6 + 18a-2z8 + 18a-1z3 - 32a-1z5 + 9a-1z7 + 4a-1z9 - 1 + 2z2 + 3z4 - 13z6 + 10z8 + 6az3 - 14az5 + 11az7 + 2a2z2 - 7a2z4 + 8a2z6 - 2a3z3 + 4a3z5 + a4z4

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

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 10122. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6
j = 13          1
j = 11         4 
j = 9        51 
j = 7       84  
j = 5      95   
j = 3     88    
j = 1    99     
j = -1   59      
j = -3  38       
j = -5 15        
j = -7 3         
j = -91          

 n  Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q-12 - 4q-11 + 4q-10 + 7q-9 - 25q-8 + 25q-7 + 23q-6 - 88q-5 + 66q-4 + 74q-3 - 183q-2 + 85q-1 + 152 - 246q + 59q2 + 207q3 - 239q4 + 7q5 + 212q6 - 174q7 - 41q8 + 164q9 - 83q10 - 57q11 + 84q12 - 15q13 - 33q14 + 20q15 + 3q16 - 5q17 + q18
3 q-24 - 4q-23 + 4q-22 + 3q-21 - 5q-20 - 7q-19 + 15q-18 + 10q-17 - 50q-16 + q-15 + 111q-14 + 6q-13 - 243q-12 - 39q-11 + 438q-10 + 152q-9 - 683q-8 - 360q-7 + 910q-6 + 677q-5 - 1080q-4 - 1050q-3 + 1145q-2 + 1404q-1 - 1066 - 1730q + 929q2 + 1920q3 - 682q4 - 2048q5 + 443q6 + 2034q7 - 139q8 - 1978q9 - 125q10 + 1801q11 + 413q12 - 1578q13 - 631q14 + 1254q15 + 799q16 - 897q17 - 856q18 + 534q19 + 788q20 - 208q21 - 638q22 - 9q23 + 429q24 + 121q25 - 241q26 - 125q27 + 92q28 + 97q29 - 29q30 - 42q31 + 14q33 + 3q34 - 5q35 + q36
4 q-40 - 4q-39 + 4q-38 + 3q-37 - 9q-36 + 13q-35 - 17q-34 + 12q-33 - 2q-32 - 43q-31 + 97q-30 - 13q-29 - 5q-28 - 141q-27 - 215q-26 + 451q-25 + 342q-24 + 41q-23 - 882q-22 - 1214q-21 + 1092q-20 + 1994q-19 + 1225q-18 - 2335q-17 - 4606q-16 + 477q-15 + 5148q-14 + 5657q-13 - 2413q-12 - 10460q-11 - 3926q-10 + 7028q-9 + 13052q-8 + 1703q-7 - 15187q-6 - 11417q-5 + 4533q-4 + 19206q-3 + 8955q-2 - 15351q-1 - 17650 - 1299q + 20794q2 + 15253q3 - 11705q4 - 19850q5 - 7057q6 + 18492q7 + 18435q8 - 6782q9 - 18735q10 - 11277q11 + 14190q12 + 19134q13 - 1524q14 - 15584q15 - 14277q16 + 8401q17 + 17807q18 + 4000q19 - 10334q20 - 15524q21 + 1429q22 + 13620q23 + 8247q24 - 3251q25 - 13248q26 - 4473q27 + 6684q28 + 8585q29 + 2969q30 - 7414q31 - 6159q32 + 171q33 + 4858q34 + 4943q35 - 1547q36 - 3644q37 - 2369q38 + 747q39 + 2997q40 + 880q41 - 673q42 - 1468q43 - 701q44 + 741q45 + 586q46 + 281q47 - 285q48 - 370q49 + 14q50 + 73q51 + 126q52 + 8q53 - 56q54 - 9q55 - 6q56 + 14q57 + 3q58 - 5q59 + q60


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, 122]]
Out[2]=   
PD[X[1, 6, 2, 7], X[7, 15, 8, 14], X[15, 2, 16, 3], X[5, 12, 6, 13], 
 
>   X[9, 19, 10, 18], X[3, 11, 4, 10], X[17, 5, 18, 4], X[19, 9, 20, 8], 
 
>   X[11, 16, 12, 17], X[13, 1, 14, 20]]
In[3]:=
GaussCode[Knot[10, 122]]
Out[3]=   
GaussCode[-1, 3, -6, 7, -4, 1, -2, 8, -5, 6, -9, 4, -10, 2, -3, 9, -7, 5, -8, 
 
>   10]
In[4]:=
DTCode[Knot[10, 122]]
Out[4]=   
DTCode[6, 10, 12, 14, 18, 16, 20, 2, 4, 8]
In[5]:=
br = BR[Knot[10, 122]]
Out[5]=   
BR[4, {1, 1, 2, -3, 2, -1, -3, 2, -3, 2, -3}]
In[6]:=
{First[br], Crossings[br]}
Out[6]=   
{4, 11}
In[7]:=
BraidIndex[Knot[10, 122]]
Out[7]=   
4
In[8]:=
Show[DrawMorseLink[Knot[10, 122]]]
Out[8]=   
 -Graphics- 
In[9]:=
#[Knot[10, 122]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=   
{Reversible, 2, 3, 3, NotAvailable, 2}
In[10]:=
alex = Alexander[Knot[10, 122]][t]
Out[10]=   
     2    11   24              2      3
31 - -- + -- - -- - 24 t + 11 t  - 2 t
      3    2   t
     t    t
In[11]:=
Conway[Knot[10, 122]][z]
Out[11]=   
       2    4      6
1 + 2 z  - z  - 2 z
In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=   
{Knot[10, 122], Knot[11, NonAlternating, 185]}
In[13]:=
{KnotDet[Knot[10, 122]], KnotSignature[Knot[10, 122]]}
Out[13]=   
{105, 0}
In[14]:=
Jones[Knot[10, 122]][q]
Out[14]=   
      -4   4    8    13              2       3      4      5    6
17 + q   - -- + -- - -- - 17 q + 17 q  - 13 q  + 9 q  - 5 q  + q
            3    2   q
           q    q
In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=   
{Knot[10, 122]}
In[16]:=
A2Invariant[Knot[10, 122]][q]
Out[16]=   
      -12    2     -8    -6   4    3       2      4      8      10      16    18
-2 + q    - --- + q   + q   - -- + -- + 2 q  + 3 q  + 5 q  - 3 q   - 3 q   + q
             10                4    2
            q                 q    q
In[17]:=
HOMFLYPT[Knot[10, 122]][a, z]
Out[17]=   
                         2                   4    4                 6
     2    4       2   3 z     2  2      4   z    z     2  4    6   z
-1 - -- + -- - 2 z  + ---- + a  z  - 2 z  + -- - -- + a  z  - z  - --
      4    2            2                    4    2                 2
     a    a            a                    a    a                 a
In[18]:=
Kauffman[Knot[10, 122]][a, z]
Out[18]=   
                                               3       3       3
     2    4    2 z   2 z      2      2  2   4 z    14 z    18 z         3
-1 - -- - -- + --- + --- + 2 z  + 2 a  z  + ---- + ----- + ----- + 6 a z  - 
      4    2    5     3                       5      3       a
     a    a    a     a                       a      a
 
                      4       4       4                         5       5
       3  3      4   z    12 z    24 z       2  4    4  4   11 z    25 z
>   2 a  z  + 3 z  - -- + ----- + ----- - 7 a  z  + a  z  - ----- - ----- - 
                      6     4       2                         5       3
                     a     a       a                         a       a
 
        5                                6       6       6                7
    32 z          5      3  5       6   z    20 z    42 z       2  6   5 z
>   ----- - 14 a z  + 4 a  z  - 13 z  + -- - ----- - ----- + 8 a  z  + ---- + 
      a                                  6     4       2                 5
                                        a     a       a                 a
 
       7      7                        8       8      9      9
    3 z    9 z          7       8   8 z    18 z    4 z    4 z
>   ---- + ---- + 11 a z  + 10 z  + ---- + ----- + ---- + ----
      3     a                         4      2       3     a
     a                               a      a       a
In[19]:=
{Vassiliev[2][Knot[10, 122]], Vassiliev[3][Knot[10, 122]]}
Out[19]=   
{2, 2}
In[20]:=
Kh[Knot[10, 122]][q, t]
Out[20]=   
9           1       3       1       5       3      8      5               3
- + 9 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 9 q t + 8 q  t + 
q          9  4    7  3    5  3    5  2    3  2    3     q t
          q  t    q  t    q  t    q  t    q  t    q  t
 
       3  2      5  2      5  3      7  3      7  4      9  4    9  5
>   8 q  t  + 9 q  t  + 5 q  t  + 8 q  t  + 4 q  t  + 5 q  t  + q  t  + 
 
       11  5    13  6
>   4 q   t  + q   t
In[21]:=
ColouredJones[Knot[10, 122], 2][q]
Out[21]=   
       -12    4     4    7    25   25   23   88   66   74   183   85
152 + q    - --- + --- + -- - -- + -- + -- - -- + -- + -- - --- + -- - 246 q + 
              11    10    9    8    7    6    5    4    3    2    q
             q     q     q    q    q    q    q    q    q    q
 
        2        3        4      5        6        7       8        9
>   59 q  + 207 q  - 239 q  + 7 q  + 212 q  - 174 q  - 41 q  + 164 q  - 
 
        10       11       12       13       14       15      16      17    18
>   83 q   - 57 q   + 84 q   - 15 q   - 33 q   + 20 q   + 3 q   - 5 q   + q


Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10122
10.121
10121
10.123
10123