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9.9
99 		9.11
911
    9.10
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   The Alternating Knot 910   

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Acknowledgement

9.10
KnotPlot

PD Presentation: X8291 X12,4,13,3 X18,10,1,9 X10,18,11,17 X16,8,17,7 X2,12,3,11 X4,16,5,15 X14,6,15,5 X6,14,7,13

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

DT (Dowker-Thistlethwaite) Code: 8 12 14 16 18 2 6 4 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--3 2 2 / 4--6 1

Alexander Polynomial: 4t-2 - 8t-1 + 9 - 8t + 4t2

Conway Polynomial: 1 + 8z2 + 4z4

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

Determinant and Signature: {33, 4}

Jones Polynomial: q2 - 2q3 + 4q4 - 5q5 + 6q6 - 5q7 + 5q8 - 3q9 + q10 - q11

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

A2 (sl(3)) Invariant: q6 - q8 + q10 + 2q16 + 2q20 + q22 + q24 + q26 - 2q28 - q30 - q32 - q34

HOMFLY-PT Polynomial: - 2a-10 - a-10z2 + a-8 + 2a-8z2 + a-8z4 + 2a-6 + 5a-6z2 + 2a-6z4 + 2a-4z2 + a-4z4

Kauffman Polynomial: 4a-13z - 4a-13z3 + a-13z5 - 2a-12z4 + a-12z6 - a-11z3 - a-11z5 + a-11z7 + 2a-10 - 11a-10z2 + 9a-10z4 - 3a-10z6 + a-10z8 - 4a-9z + 9a-9z3 - 7a-9z5 + 3a-9z7 + a-8 - 2a-8z2 + 3a-8z4 - a-8z6 + a-8z8 + 3a-7z3 - 3a-7z5 + 2a-7z7 - 2a-6 + 7a-6z2 - 7a-6z4 + 3a-6z6 - 3a-5z3 + 2a-5z5 - 2a-4z2 + a-4z4

V2 and V3, the type 2 and 3 Vassiliev invariants: {8, 22}

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=4 is the signature of 910. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.) 		  
trqj r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7r = 8r = 9
j = 23         1
j = 21          
j = 19       31 
j = 17      2   
j = 15     33   
j = 13    32    
j = 11   23     
j = 9  23      
j = 7  2       
j = 512        
j = 31         

 n  Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q4 - 2q5 + q6 + 5q7 - 8q8 + 15q10 - 16q11 - 5q12 + 28q13 - 22q14 - 11q15 + 35q16 - 22q17 - 13q18 + 32q19 - 15q20 - 14q21 + 22q22 - 7q23 - 10q24 + 10q25 - 2q26 - 4q27 + 3q28 - q30 + q31
3 q6 - 2q7 + q8 + 2q9 + q10 - 6q11 + 9q13 + 4q14 - 17q15 - 6q16 + 21q17 + 19q18 - 32q19 - 28q20 + 35q21 + 46q22 - 41q23 - 60q24 + 41q25 + 77q26 - 43q27 - 84q28 + 36q29 + 94q30 - 37q31 - 90q32 + 24q33 + 94q34 - 22q35 - 81q36 + 7q37 + 76q38 - q39 - 62q40 - 10q41 + 50q42 + 13q43 - 37q44 - 13q45 + 21q46 + 16q47 - 16q48 - 7q49 + 5q50 + 8q51 - 5q52 - q53 + 3q55 - 2q56 + q59 - q60
4 q8 - 2q9 + q10 + 2q11 - 2q12 + 3q13 - 7q14 + 3q15 + 8q16 - 7q17 + 8q18 - 20q19 + 5q20 + 23q21 - 10q22 + 16q23 - 51q24 - 2q25 + 48q26 + 8q27 + 44q28 - 108q29 - 43q30 + 66q31 + 55q32 + 113q33 - 168q34 - 120q35 + 50q36 + 108q37 + 220q38 - 203q39 - 203q40 + 5q41 + 139q42 + 318q43 - 204q44 - 252q45 - 44q46 + 138q47 + 373q48 - 180q49 - 261q50 - 80q51 + 114q52 + 379q53 - 137q54 - 232q55 - 107q56 + 69q57 + 349q58 - 79q59 - 172q60 - 125q61 + 8q62 + 284q63 - 14q64 - 94q65 - 122q66 - 49q67 + 193q68 + 27q69 - 22q70 - 83q71 - 72q72 + 97q73 + 28q74 + 20q75 - 35q76 - 55q77 + 34q78 + 7q79 + 21q80 - 5q81 - 25q82 + 11q83 - 6q84 + 9q85 + 2q86 - 8q87 + 6q88 - 5q89 + 2q90 + q91 - 3q92 + 3q93 - q94 - q97 + q98

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[9, 10]]
Out[2]=   
PD[X[8, 2, 9, 1], X[12, 4, 13, 3], X[18, 10, 1, 9], X[10, 18, 11, 17], 
 
>   X[16, 8, 17, 7], X[2, 12, 3, 11], X[4, 16, 5, 15], X[14, 6, 15, 5], 
 
>   X[6, 14, 7, 13]]
In[3]:=
GaussCode[Knot[9, 10]]
Out[3]=   
GaussCode[1, -6, 2, -7, 8, -9, 5, -1, 3, -4, 6, -2, 9, -8, 7, -5, 4, -3]
In[4]:=
DTCode[Knot[9, 10]]
Out[4]=   
DTCode[8, 12, 14, 16, 18, 2, 6, 4, 10]
In[5]:=
br = BR[Knot[9, 10]]
Out[5]=   
BR[4, {1, 1, 2, -1, 2, 2, 2, 2, 3, -2, 3}]
In[6]:=
{First[br], Crossings[br]}
Out[6]=   
{4, 11}
In[7]:=
BraidIndex[Knot[9, 10]]
Out[7]=   
4
In[8]:=
Show[DrawMorseLink[Knot[9, 10]]]
Out[8]=   
 -Graphics- 
In[9]:=
#[Knot[9, 10]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=   
{Reversible, {2, 3}, 2, 2, {4, 6}, 1}
In[10]:=
alex = Alexander[Knot[9, 10]][t]
Out[10]=   
    4    8            2
9 + -- - - - 8 t + 4 t
     2   t
    t
In[11]:=
Conway[Knot[9, 10]][z]
Out[11]=   
       2      4
1 + 8 z  + 4 z
In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=   
{Knot[9, 10]}
In[13]:=
{KnotDet[Knot[9, 10]], KnotSignature[Knot[9, 10]]}
Out[13]=   
{33, 4}
In[14]:=
Jones[Knot[9, 10]][q]
Out[14]=   
 2      3      4      5      6      7      8      9    10    11
q  - 2 q  + 4 q  - 5 q  + 6 q  - 5 q  + 5 q  - 3 q  + q   - q
In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=   
{Knot[9, 10]}
In[16]:=
A2Invariant[Knot[9, 10]][q]
Out[16]=   
 6    8    10      16      20    22    24    26      28    30    32    34
q  - q  + q   + 2 q   + 2 q   + q   + q   + q   - 2 q   - q   - q   - q
In[17]:=
HOMFLYPT[Knot[9, 10]][a, z]
Out[17]=   
                  2       2      2      2    4      4    4
-2     -8   2    z     2 z    5 z    2 z    z    2 z    z
--- + a   + -- - --- + ---- + ---- + ---- + -- + ---- + --
 10          6    10     8      6      4     8     6     4
a           a    a      a      a      a     a     a     a
In[18]:=
Kauffman[Knot[9, 10]][a, z]
Out[18]=   
                                 2      2      2      2      3    3       3
 2     -8   2    4 z   4 z   11 z    2 z    7 z    2 z    4 z    z     9 z
--- + a   - -- + --- - --- - ----- - ---- + ---- - ---- - ---- - --- + ---- + 
 10          6    13    9      10      8      6      4     13     11     9
a           a    a     a      a       a      a      a     a      a      a
 
       3      3      4      4      4      4    4    5     5       5      5
    3 z    3 z    2 z    9 z    3 z    7 z    z    z     z     7 z    3 z
>   ---- - ---- - ---- + ---- + ---- - ---- + -- + --- - --- - ---- - ---- + 
      7      5     12     10      8      6     4    13    11     9      7
     a      a     a      a       a      a     a    a     a      a      a
 
       5    6       6    6      6    7       7      7    8     8
    2 z    z     3 z    z    3 z    z     3 z    2 z    z     z
>   ---- + --- - ---- - -- + ---- + --- + ---- + ---- + --- + --
      5     12    10     8     6     11     9      7     10    8
     a     a     a      a     a     a      a      a     a     a
In[19]:=
{Vassiliev[2][Knot[9, 10]], Vassiliev[3][Knot[9, 10]]}
Out[19]=   
{8, 22}
In[20]:=
Kh[Knot[9, 10]][q, t]
Out[20]=   
 3    5      5        7  2      9  2      9  3      11  3      11  4
q  + q  + 2 q  t + 2 q  t  + 2 q  t  + 3 q  t  + 2 q   t  + 3 q   t  + 
 
       13  4      13  5      15  5      15  6      17  6      19  7    19  8
>   3 q   t  + 2 q   t  + 3 q   t  + 3 q   t  + 2 q   t  + 3 q   t  + q   t  + 
 
     23  9
>   q   t
In[21]:=
ColouredJones[Knot[9, 10], 2][q]
Out[21]=   
 4      5    6      7      8       10       11      12       13       14
q  - 2 q  + q  + 5 q  - 8 q  + 15 q   - 16 q   - 5 q   + 28 q   - 22 q   - 
 
        15       16       17       18       19       20       21       22
>   11 q   + 35 q   - 22 q   - 13 q   + 32 q   - 15 q   - 14 q   + 22 q   - 
 
       23       24       25      26      27      28    30    31
>   7 q   - 10 q   + 10 q   - 2 q   - 4 q   + 3 q   - q   + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 910
9.9
99 		9.11
911