© | Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table:
6.2
62 		7.1
71
    6.3
KnotPlot 		This page is passe. Go here instead!

   The Alternating Knot 63   

Visit 63's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 63's page at Knotilus!

Acknowledgement

6.3
KnotPlot

PD Presentation: X4251 X8493 X12,9,1,10 X10,5,11,6 X6,11,7,12 X2837

Gauss Code: {1, -6, 2, -1, 4, -5, 6, -2, 3, -4, 5, -3}

DT (Dowker-Thistlethwaite) Code: 4 8 10 2 12 6

Minimum Braid Representative:


Length is 6, width is 3
Braid index is 3

A Morse Link Presentation:

3D Invariants:
Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index
FullyAmphicheiral 1 2 2 / 3--4 1

Alexander Polynomial: t-2 - 3t-1 + 5 - 3t + t2

Conway Polynomial: 1 + z2 + z4

Other knots with the same Alexander/Conway Polynomial: {K11n12, ...}

Determinant and Signature: {13, 0}

Jones Polynomial: - q-3 + 2q-2 - 2q-1 + 3 - 2q + 2q2 - q3

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

A2 (sl(3)) Invariant: - q-10 + 2q-2 + 1 + 2q2 - q10

HOMFLY-PT Polynomial: - a-2 - a-2z2 + 3 + 3z2 + z4 - a2 - a2z2

Kauffman Polynomial: - a-3z + a-3z3 + a-2 - 3a-2z2 + 2a-2z4 - 2a-1z + a-1z3 + a-1z5 + 3 - 6z2 + 4z4 - 2az + az3 + az5 + a2 - 3a2z2 + 2a2z4 - a3z + a3z3

V2 and V3, the type 2 and 3 Vassiliev invariants: {1, 0}

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 63. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.) 		  
trqj r = -3r = -2r = -1r = 0r = 1r = 2r = 3
j = 7      1
j = 5     1 
j = 3    11 
j = 1   21  
j = -1  12   
j = -3 11    
j = -5 1     
j = -71      

 n  Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q-9 - 2q-8 - q-7 + 5q-6 - 4q-5 - 3q-4 + 9q-3 - 5q-2 - 5q-1 + 11 - 5q - 5q2 + 9q3 - 3q4 - 4q5 + 5q6 - q7 - 2q8 + q9
3 - q-18 + 2q-17 + q-16 - 2q-15 - 4q-14 + 3q-13 + 7q-12 - 4q-11 - 10q-10 + 3q-9 + 14q-8 - 3q-7 - 16q-6 + q-5 + 21q-4 - 2q-3 - 20q-2 - q-1 + 23 - q - 20q2 - 2q3 + 21q4 + q5 - 16q6 - 3q7 + 14q8 + 3q9 - 10q10 - 4q11 + 7q12 + 3q13 - 4q14 - 2q15 + q16 + 2q17 - q18
4 q-30 - 2q-29 - q-28 + 2q-27 + q-26 + 5q-25 - 7q-24 - 5q-23 + 2q-22 + 3q-21 + 17q-20 - 12q-19 - 14q-18 - 3q-17 + 4q-16 + 34q-15 - 12q-14 - 22q-13 - 13q-12 + 2q-11 + 52q-10 - 9q-9 - 28q-8 - 23q-7 - q-6 + 64q-5 - 6q-4 - 30q-3 - 29q-2 - 4q-1 + 69 - 4q - 29q2 - 30q3 - 6q4 + 64q5 - q6 - 23q7 - 28q8 - 9q9 + 52q10 + 2q11 - 13q12 - 22q13 - 12q14 + 34q15 + 4q16 - 3q17 - 14q18 - 12q19 + 17q20 + 3q21 + 2q22 - 5q23 - 7q24 + 5q25 + q26 + 2q27 - q28 - 2q29 + q30
5 - q-45 + 2q-44 + q-43 - 2q-42 - q-41 - 2q-40 - q-39 + 5q-38 + 7q-37 - 2q-36 - 6q-35 - 9q-34 - 5q-33 + 9q-32 + 18q-31 + 10q-30 - 10q-29 - 25q-28 - 19q-27 + 6q-26 + 33q-25 + 32q-24 - 2q-23 - 41q-22 - 43q-21 - 6q-20 + 43q-19 + 61q-18 + 13q-17 - 49q-16 - 68q-15 - 25q-14 + 49q-13 + 84q-12 + 30q-11 - 53q-10 - 85q-9 - 41q-8 + 49q-7 + 100q-6 + 41q-5 - 53q-4 - 93q-3 - 50q-2 + 47q-1 + 105 + 47q - 50q2 - 93q3 - 53q4 + 41q5 + 100q6 + 49q7 - 41q8 - 85q9 - 53q10 + 30q11 + 84q12 + 49q13 - 25q14 - 68q15 - 49q16 + 13q17 + 61q18 + 43q19 - 6q20 - 43q21 - 41q22 - 2q23 + 32q24 + 33q25 + 6q26 - 19q27 - 25q28 - 10q29 + 10q30 + 18q31 + 9q32 - 5q33 - 9q34 - 6q35 - 2q36 + 7q37 + 5q38 - q39 - 2q40 - q41 - 2q42 + q43 + 2q44 - q45
6 q-63 - 2q-62 - q-61 + 2q-60 + q-59 + 2q-58 - 2q-57 + 3q-56 - 7q-55 - 7q-54 + 5q-53 + 5q-52 + 9q-51 + 9q-49 - 20q-48 - 22q-47 + 9q-45 + 24q-44 + 13q-43 + 34q-42 - 33q-41 - 51q-40 - 25q-39 - 3q-38 + 37q-37 + 40q-36 + 84q-35 - 29q-34 - 78q-33 - 69q-32 - 38q-31 + 32q-30 + 68q-29 + 153q-28 - 4q-27 - 89q-26 - 115q-25 - 88q-24 + 9q-23 + 85q-22 + 220q-21 + 31q-20 - 85q-19 - 150q-18 - 134q-17 - 20q-16 + 91q-15 + 271q-14 + 60q-13 - 75q-12 - 172q-11 - 164q-10 - 43q-9 + 90q-8 + 301q-7 + 77q-6 - 66q-5 - 180q-4 - 178q-3 - 57q-2 + 86q-1 + 311 + 86q - 57q2 - 178q3 - 180q4 - 66q5 + 77q6 + 301q7 + 90q8 - 43q9 - 164q10 - 172q11 - 75q12 + 60q13 + 271q14 + 91q15 - 20q16 - 134q17 - 150q18 - 85q19 + 31q20 + 220q21 + 85q22 + 9q23 - 88q24 - 115q25 - 89q26 - 4q27 + 153q28 + 68q29 + 32q30 - 38q31 - 69q32 - 78q33 - 29q34 + 84q35 + 40q36 + 37q37 - 3q38 - 25q39 - 51q40 - 33q41 + 34q42 + 13q43 + 24q44 + 9q45 - 22q47 - 20q48 + 9q49 + 9q51 + 5q52 + 5q53 - 7q54 - 7q55 + 3q56 - 2q57 + 2q58 + q59 + 2q60 - q61 - 2q62 + q63
7 - q-84 + 2q-83 + q-82 - 2q-81 - q-80 - 2q-79 + 2q-78 - q-76 + 7q-75 + 4q-74 - 4q-73 - 5q-72 - 11q-71 - q-70 + 3q-69 - q-68 + 21q-67 + 16q-66 + q-65 - 9q-64 - 34q-63 - 21q-62 - 8q-61 - 4q-60 + 44q-59 + 49q-58 + 30q-57 + 12q-56 - 59q-55 - 68q-54 - 55q-53 - 41q-52 + 58q-51 + 94q-50 + 98q-49 + 78q-48 - 50q-47 - 118q-46 - 138q-45 - 128q-44 + 28q-43 + 126q-42 + 181q-41 + 195q-40 + 7q-39 - 135q-38 - 224q-37 - 250q-36 - 50q-35 + 119q-34 + 256q-33 + 322q-32 + 101q-31 - 115q-30 - 281q-29 - 371q-28 - 149q-27 + 86q-26 + 299q-25 + 433q-24 + 191q-23 - 75q-22 - 312q-21 - 460q-20 - 232q-19 + 45q-18 + 319q-17 + 506q-16 + 260q-15 - 44q-14 - 321q-13 - 512q-12 - 284q-11 + 14q-10 + 322q-9 + 547q-8 + 298q-7 - 23q-6 - 320q-5 - 534q-4 - 309q-3 - 7q-2 + 316q-1 + 559 + 316q - 7q2 - 309q3 - 534q4 - 320q5 - 23q6 + 298q7 + 547q8 + 322q9 + 14q10 - 284q11 - 512q12 - 321q13 - 44q14 + 260q15 + 506q16 + 319q17 + 45q18 - 232q19 - 460q20 - 312q21 - 75q22 + 191q23 + 433q24 + 299q25 + 86q26 - 149q27 - 371q28 - 281q29 - 115q30 + 101q31 + 322q32 + 256q33 + 119q34 - 50q35 - 250q36 - 224q37 - 135q38 + 7q39 + 195q40 + 181q41 + 126q42 + 28q43 - 128q44 - 138q45 - 118q46 - 50q47 + 78q48 + 98q49 + 94q50 + 58q51 - 41q52 - 55q53 - 68q54 - 59q55 + 12q56 + 30q57 + 49q58 + 44q59 - 4q60 - 8q61 - 21q62 - 34q63 - 9q64 + q65 + 16q66 + 21q67 - q68 + 3q69 - q70 - 11q71 - 5q72 - 4q73 + 4q74 + 7q75 - q76 + 2q78 - 2q79 - q80 - 2q81 + q82 + 2q83 - q84

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

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 63
6.2
62 		7.1
71