10 109

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10 108.gif

10_108

10 110.gif

10_110

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10 109 Quick Notes


10 109 Further Notes and Views

Knot presentations

Planar diagram presentation X6271 X10,4,11,3 X18,11,19,12 X16,7,17,8 X8,17,9,18 X20,15,1,16 X12,19,13,20 X14,6,15,5 X2,10,3,9 X4,14,5,13
Gauss code 1, -9, 2, -10, 8, -1, 4, -5, 9, -2, 3, -7, 10, -8, 6, -4, 5, -3, 7, -6
Dowker-Thistlethwaite code 6 10 14 16 2 18 4 20 8 12
Conway Notation [2.2.2.2]

Three dimensional invariants

Symmetry type Negative amphicheiral
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-6][-6]
Hyperbolic Volume 14.9002
A-Polynomial See Data:10 109/A-polynomial

[edit Notes for 10 109's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus
Topological 4 genus
Concordance genus
Rasmussen s-Invariant 0

[edit Notes for 10 109's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 85, 0 }
Jones polynomial
HOMFLY-PT polynomial (db, data sources)
Kauffman polynomial (db, data sources)
The A2 invariant
The G2 invariant

Vassiliev invariants

V2 and V3: (3, 0)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where 0 is the signature of 10 109. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
-5-4-3-2-1012345χ
11          1-1
9         2 2
7        51 -4
5       62  4
3      75   -2
1     86    2
-1    68     2
-3   57      -2
-5  26       4
-7 15        -4
-9 2         2
-111          -1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=    
<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=
Crossings[Knot[10, 109]]
Out[2]=  
10
In[3]:=
PD[Knot[10, 109]]
Out[3]=  
PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[18, 11, 19, 12], X[16, 7, 17, 8], 
 X[8, 17, 9, 18], X[20, 15, 1, 16], X[12, 19, 13, 20], 

X[14, 6, 15, 5], X[2, 10, 3, 9], X[4, 14, 5, 13]]
In[4]:=
GaussCode[Knot[10, 109]]
Out[4]=  
GaussCode[1, -9, 2, -10, 8, -1, 4, -5, 9, -2, 3, -7, 10, -8, 6, -4, 5, 
  -3, 7, -6]
In[5]:=
BR[Knot[10, 109]]
Out[5]=  
BR[3, {-1, -1, 2, -1, 2, 2, -1, -1, 2, 2}]
In[6]:=
alex = Alexander[Knot[10, 109]][t]
Out[6]=  
      -4   4    10   17              2      3    4

21 + t - -- + -- - -- - 17 t + 10 t - 4 t + t

           3    2   t
t t
In[7]:=
Conway[Knot[10, 109]][z]
Out[7]=  
       2      4      6    8
1 + 3 z  + 6 z  + 4 z  + z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[10, 109]}
In[9]:=
{KnotDet[Knot[10, 109]], KnotSignature[Knot[10, 109]]}
Out[9]=  
{85, 0}
In[10]:=
J=Jones[Knot[10, 109]][q]
Out[10]=  
      -5   3    7    11   13              2      3      4    5

15 - q + -- - -- + -- - -- - 13 q + 11 q - 7 q + 3 q - q

           4    3    2   q
q q q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[10, 81], Knot[10, 109]}
In[12]:=
A2Invariant[Knot[10, 109]][q]
Out[12]=  
      -14    -12    3     -8    -4   5       2    4    8      10

-1 - q + q - --- + q - q + -- + 5 q - q + q - 3 q +

                   10                2
                  q                 q

  12    14
q - q
In[13]:=
Kauffman[Knot[10, 109]][a, z]
Out[13]=  
                                                                 2
   3       2   z    z    5 z            3      5         2   2 z

7 + -- + 3 a + -- - -- - --- - 5 a z - a z + a z - 18 z + ---- -

    2           5    3    a                                    4
   a           a    a                                         a

    2                          3      3       3
 7 z       2  2      4  2   2 z    4 z    13 z          3      3  3
 ---- - 7 a  z  + 2 a  z  - ---- + ---- + ----- + 13 a z  + 4 a  z  - 
   2                          5      3      a
  a                          a      a

                      4      4                        5      5
    5  3       4   5 z    6 z       2  4      4  4   z    8 z
 2 a  z  + 22 z  - ---- + ---- + 6 a  z  - 5 a  z  + -- - ---- - 
                     4      2                         5     3
                    a      a                         a     a

     5                                          6      6
 16 z          5      3  5    5  5       6   3 z    7 z       2  6
 ----- - 16 a z  - 8 a  z  + a  z  - 20 z  + ---- - ---- - 7 a  z  + 
   a                                           4      2
                                              a      a

              7      7                                 8
    4  6   5 z    6 z         7      3  7       8   5 z       2  8
 3 a  z  + ---- + ---- + 6 a z  + 5 a  z  + 10 z  + ---- + 5 a  z  + 
             3     a                                  2
            a                                        a

    9
 2 z         9
 ---- + 2 a z
a
In[14]:=
{Vassiliev[2][Knot[10, 109]], Vassiliev[3][Knot[10, 109]]}
Out[14]=  
{0, 0}
In[15]:=
Kh[Knot[10, 109]][q, t]
Out[15]=  
8           1        2       1       5       2       6       5

- + 8 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2

         q   t    q  t    q  t    q  t    q  t    q  t    q  t

  7      6               3        3  2      5  2      5  3      7  3
 ---- + --- + 6 q t + 7 q  t + 5 q  t  + 6 q  t  + 2 q  t  + 5 q  t  + 
  3     q t
 q  t

  7  4      9  4    11  5
q t + 2 q t + q t