© | Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table:
10.109
10109 		10.111
10111
    10.110
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   The Alternating Knot 10110   

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Acknowledgement

10.110
KnotPlot

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

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

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

Minimum Braid Representative:


Length is 12, width is 5
Braid index is 5

A Morse Link Presentation:

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

Alexander Polynomial: t-3 - 8t-2 + 20t-1 - 25 + 20t - 8t2 + t3

Conway Polynomial: 1 - 3z2 - 2z4 + z6

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

Determinant and Signature: {83, -2}

Jones Polynomial: q-7 - 3q-6 + 7q-5 - 11q-4 + 13q-3 - 14q-2 + 13q-1 - 10 + 7q - 3q2 + q3

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

A2 (sl(3)) Invariant: q-22 - q-18 + 3q-16 - 2q-14 + q-10 - 3q-8 + 2q-6 - 3q-4 + 2q-2 + 1 - q2 + 3q4 - q6 + q10

HOMFLY-PT Polynomial: a-2 + a-2z2 - 3z2 - 2z4 + a2z2 + 2a2z4 + a2z6 - a4 - 3a4z2 - 2a4z4 + a6 + a6z2

Kauffman Polynomial: - a-2 + 3a-2z2 - 3a-2z4 + a-2z6 - a-1z + 6a-1z3 - 8a-1z5 + 3a-1z7 + 2z2 + z4 - 8z6 + 4z8 - 3az + 13az3 - 19az5 + 4az7 + 2az9 - a2z2 + 8a2z4 - 20a2z6 + 10a2z8 - 6a3z + 21a3z3 - 27a3z5 + 9a3z7 + 2a3z9 - a4 + 6a4z2 - 4a4z4 - 5a4z6 + 6a4z8 - 4a5z + 12a5z3 - 13a5z5 + 8a5z7 - a6 + 5a6z2 - 7a6z4 + 6a6z6 - 2a7z3 + 3a7z5 - a8z2 + a8z4

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

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=-2 is the signature of 10110. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.) 		  
trqj r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4
j = 7          1
j = 5         2 
j = 3        51 
j = 1       52  
j = -1      85   
j = -3     76    
j = -5    67     
j = -7   57      
j = -9  26       
j = -11 15        
j = -13 2         
j = -151          

 n  Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q-20 - 3q-19 + 3q-18 + 5q-17 - 19q-16 + 17q-15 + 21q-14 - 64q-13 + 38q-12 + 63q-11 - 124q-10 + 42q-9 + 114q-8 - 158q-7 + 22q-6 + 144q-5 - 147q-4 - 10q-3 + 139q-2 - 101q-1 - 35 + 101q - 44q2 - 37q3 + 48q4 - 7q5 - 18q6 + 11q7 + q8 - 3q9 + q10
3 q-39 - 3q-38 + 3q-37 + q-36 - 3q-35 - 6q-34 + 14q-33 + 8q-32 - 34q-31 - 18q-30 + 78q-29 + 37q-28 - 137q-27 - 100q-26 + 231q-25 + 195q-24 - 318q-23 - 343q-22 + 382q-21 + 532q-20 - 413q-19 - 719q-18 + 385q-17 + 891q-16 - 318q-15 - 1010q-14 + 210q-13 + 1081q-12 - 91q-11 - 1091q-10 - 39q-9 + 1052q-8 + 168q-7 - 970q-6 - 286q-5 + 844q-4 + 396q-3 - 697q-2 - 457q-1 + 508 + 492q - 330q2 - 460q3 + 156q4 + 393q5 - 35q6 - 281q7 - 50q8 + 184q9 + 66q10 - 90q11 - 61q12 + 37q13 + 37q14 - 10q15 - 18q16 + 3q17 + 5q18 + q19 - 3q20 + q21
4 q-64 - 3q-63 + 3q-62 + q-61 - 7q-60 + 10q-59 - 9q-58 + 11q-57 - 3q-56 - 39q-55 + 49q-54 + 3q-53 + 40q-52 - 52q-51 - 190q-50 + 137q-49 + 165q-48 + 237q-47 - 214q-46 - 770q-45 + 79q-44 + 657q-43 + 1108q-42 - 202q-41 - 2133q-40 - 840q-39 + 1124q-38 + 3111q-37 + 881q-36 - 3771q-35 - 3098q-34 + 479q-33 + 5516q-32 + 3453q-31 - 4343q-30 - 5768q-29 - 1667q-28 + 6806q-27 + 6391q-26 - 3320q-25 - 7298q-24 - 4265q-23 + 6420q-22 + 8244q-21 - 1495q-20 - 7223q-19 - 6150q-18 + 4972q-17 + 8656q-16 + 331q-15 - 6052q-14 - 7124q-13 + 3037q-12 + 8024q-11 + 2018q-10 - 4194q-9 - 7352q-8 + 756q-7 + 6490q-6 + 3449q-5 - 1739q-4 - 6615q-3 - 1499q-2 + 4009q-1 + 3963 + 807q - 4608q2 - 2760q3 + 1157q4 + 2993q5 + 2308q6 - 1950q7 - 2354q8 - 744q9 + 1161q10 + 2090q11 - 71q12 - 990q13 - 1009q14 - 109q15 + 969q16 + 379q17 - 45q18 - 439q19 - 315q20 + 202q21 + 151q22 + 121q23 - 66q24 - 120q25 + 14q26 + 9q27 + 39q28 + 2q29 - 21q30 + 3q31 - 3q32 + 5q33 + q34 - 3q35 + q36

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, 110]]
Out[2]=   
PD[X[1, 6, 2, 7], X[7, 20, 8, 1], X[3, 11, 4, 10], X[5, 16, 6, 17], 
 
>   X[17, 8, 18, 9], X[9, 14, 10, 15], X[11, 3, 12, 2], X[15, 4, 16, 5], 
 
>   X[13, 19, 14, 18], X[19, 13, 20, 12]]
In[3]:=
GaussCode[Knot[10, 110]]
Out[3]=   
GaussCode[-1, 7, -3, 8, -4, 1, -2, 5, -6, 3, -7, 10, -9, 6, -8, 4, -5, 9, -10, 
 
>   2]
In[4]:=
DTCode[Knot[10, 110]]
Out[4]=   
DTCode[6, 10, 16, 20, 14, 2, 18, 4, 8, 12]
In[5]:=
br = BR[Knot[10, 110]]
Out[5]=   
BR[5, {-1, 2, -1, -3, -2, -2, -2, 4, 3, -2, 3, 4}]
In[6]:=
{First[br], Crossings[br]}
Out[6]=   
{5, 12}
In[7]:=
BraidIndex[Knot[10, 110]]
Out[7]=   
5
In[8]:=
Show[DrawMorseLink[Knot[10, 110]]]
Out[8]=   
 -Graphics- 
In[9]:=
#[Knot[10, 110]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=   
{Chiral, 2, 3, 3, NotAvailable, 1}
In[10]:=
alex = Alexander[Knot[10, 110]][t]
Out[10]=   
       -3   8    20             2    3
-25 + t   - -- + -- + 20 t - 8 t  + t
             2   t
            t
In[11]:=
Conway[Knot[10, 110]][z]
Out[11]=   
       2      4    6
1 - 3 z  - 2 z  + z
In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=   
{Knot[10, 110]}
In[13]:=
{KnotDet[Knot[10, 110]], KnotSignature[Knot[10, 110]]}
Out[13]=   
{83, -2}
In[14]:=
Jones[Knot[10, 110]][q]
Out[14]=   
       -7   3    7    11   13   14   13            2    3
-10 + q   - -- + -- - -- + -- - -- + -- + 7 q - 3 q  + q
             6    5    4    3    2   q
            q    q    q    q    q
In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=   
{Knot[10, 110]}
In[16]:=
A2Invariant[Knot[10, 110]][q]
Out[16]=   
     -22    -18    3     2     -10   3    2    3    2     2      4    6    10
1 + q    - q    + --- - --- + q    - -- + -- - -- + -- - q  + 3 q  - q  + q
                   16    14           8    6    4    2
                  q     q            q    q    q    q
In[17]:=
HOMFLYPT[Knot[10, 110]][a, z]
Out[17]=   
                        2
 -2    4    6      2   z     2  2      4  2    6  2      4      2  4
a   - a  + a  - 3 z  + -- + a  z  - 3 a  z  + a  z  - 2 z  + 2 a  z  - 
                        2
                       a
 
       4  4    2  6
>   2 a  z  + a  z
In[18]:=
Kauffman[Knot[10, 110]][a, z]
Out[18]=   
                                                         2
  -2    4    6   z              3        5        2   3 z     2  2      4  2
-a   - a  - a  - - - 3 a z - 6 a  z - 4 a  z + 2 z  + ---- - a  z  + 6 a  z  + 
                 a                                      2
                                                       a
 
                         3
       6  2    8  2   6 z          3       3  3       5  3      7  3    4
>   5 a  z  - a  z  + ---- + 13 a z  + 21 a  z  + 12 a  z  - 2 a  z  + z  - 
                       a
 
       4                                            5
    3 z       2  4      4  4      6  4    8  4   8 z          5       3  5
>   ---- + 8 a  z  - 4 a  z  - 7 a  z  + a  z  - ---- - 19 a z  - 27 a  z  - 
      2                                           a
     a
 
                                 6                                     7
        5  5      7  5      6   z        2  6      4  6      6  6   3 z
>   13 a  z  + 3 a  z  - 8 z  + -- - 20 a  z  - 5 a  z  + 6 a  z  + ---- + 
                                 2                                   a
                                a
 
         7      3  7      5  7      8       2  8      4  8        9      3  9
>   4 a z  + 9 a  z  + 8 a  z  + 4 z  + 10 a  z  + 6 a  z  + 2 a z  + 2 a  z
In[19]:=
{Vassiliev[2][Knot[10, 110]], Vassiliev[3][Knot[10, 110]]}
Out[19]=   
{-3, 3}
In[20]:=
Kh[Knot[10, 110]][q, t]
Out[20]=   
6    8     1        2        1        5        2       6       5       7
-- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 
 3   q    15  6    13  5    11  5    11  4    9  4    9  3    7  3    7  2
q        q   t    q   t    q   t    q   t    q  t    q  t    q  t    q  t
 
      6      7      7     5 t                2      3  2    3  3      5  3
>   ----- + ---- + ---- + --- + 5 q t + 2 q t  + 5 q  t  + q  t  + 2 q  t  + 
     5  2    5      3      q
    q  t    q  t   q  t
 
     7  4
>   q  t
In[21]:=
ColouredJones[Knot[10, 110], 2][q]
Out[21]=   
       -20    3     3     5    19    17    21    64    38    63    124   42
-35 + q    - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + -- + 
              19    18    17    16    15    14    13    12    11    10    9
             q     q     q     q     q     q     q     q     q     q     q
 
    114   158   22   144   147   10   139   101               2       3
>   --- - --- + -- + --- - --- - -- + --- - --- + 101 q - 44 q  - 37 q  + 
     8     7     6    5     4     3    2     q
    q     q     q    q     q     q    q
 
        4      5       6       7    8      9    10
>   48 q  - 7 q  - 18 q  + 11 q  + q  - 3 q  + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10110
10.109
10109 		10.111
10111