K11a1: Difference between revisions

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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[Knot[11, Alternating, 1]]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[Knot[11, Alternating, 1]]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[6]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>alex = Alexander[Knot[11, Alternating, 1]][t]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[6]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>alex = Alexander[Knot[11, Alternating, 1]][t]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 12 30
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 12 30 2 3
-39 + ------------ - ------------ + ----------- + 30 Alternating -
-39 + -- - -- + -- + 30 t - 12 t + 2 t
3 2 Alternating
3 2 t
Alternating Alternating
t t</nowiki></pre></td></tr>
2 3
12 Alternating + 2 Alternating</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[7]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Conway[Knot[11, Alternating, 1]][z]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[7]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Conway[Knot[11, Alternating, 1]][z]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 6
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 6
Line 142: Line 139:
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{0, 2}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{0, 2}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[15]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kh[Knot[11, Alternating, 1]][q, t]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[15]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kh[Knot[11, Alternating, 1]][q, t]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 1 2 1 5
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 3 1 2 1 5 2 7 5 q
--------------- + --------------- + --------------- + --------------- +
10 q + 8 q + ----- + ----- + ----- + ----- + ---- + --- + --- +
4 7 3 5 3 3 2 3
7 4 5 3 3 3 3 2 2 q t t
Alternating q Alternating q Alternating q Alternating q
q t q t q t q t q t
2 7 5 q 3
-------------- + ------------- + 10 q + ----------- + 8 q +
2 Alternating q Alternating
Alternating q
3 5 2 5
11 Alternating q + 9 Alternating q + 10 Alternating q +
2 7 3 7 3 9
11 Alternating q + 8 Alternating q + 10 Alternating q +
4 9 4 11 5 11
6 Alternating q + 8 Alternating q + 3 Alternating q +
5 13 6 13 6 15
3 5 5 2 7 2 7 3 9 3
6 Alternating q + Alternating q + 3 Alternating q +
11 q t + 9 q t + 10 q t + 11 q t + 8 q t + 10 q t +
7 17
9 4 11 4 11 5 13 5 13 6 15 6 17 7
Alternating q</nowiki></pre></td></tr>
6 q t + 8 q t + 3 q t + 6 q t + q t + 3 q t + q t</nowiki></pre></td></tr>
</table>
</table>



Revision as of 20:53, 28 August 2005

10 165.gif

10_165

K11a2.gif

K11a2

K11a1.gif Visit K11a1's page at Knotilus!

Visit K11a1's page at the original Knot Atlas!

K11a1 Quick Notes


K11a1.
A graph, Knot K11a1.

Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X14,7,15,8 X2,9,3,10 X16,12,17,11 X20,14,21,13 X6,15,7,16 X22,18,1,17 X12,20,13,19 X18,22,19,21
Gauss code 1, -5, 2, -1, 3, -8, 4, -2, 5, -3, 6, -10, 7, -4, 8, -6, 9, -11, 10, -7, 11, -9
Dowker-Thistlethwaite code 4 8 10 14 2 16 20 6 22 12 18
Conway Notation [221,211,2]

Three dimensional invariants

Symmetry type Reversible
Unknotting number
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a1/ThurstonBennequinNumber
Hyperbolic Volume 15.6439
A-Polynomial See Data:K11a1/A-polynomial

[edit Notes for K11a1's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus
Rasmussen s-Invariant -2

[edit Notes for K11a1's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 127, 2 }
Jones polynomial
HOMFLY-PT polynomial (db, data sources)
Kauffman polynomial (db, data sources)
The A2 invariant Data:K11a1/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a1/QuantumInvariant/G2/1,0

Vassiliev invariants

V2 and V3: (0, 2)
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 2 is the signature of K11a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
17           1-1
15          3 3
13         61 -5
11        83  5
9       106   -4
7      118    3
5     910     1
3    811      -3
1   510       5
-1  27        -5
-3 15         4
-5 2          -2
-71           1
Integral Khovanov Homology

(db, data source)

  

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[11, Alternating, 1]]
Out[2]=  
11
In[3]:=
PD[Knot[11, Alternating, 1]]
Out[3]=  
PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[14, 7, 15, 8], 
 X[2, 9, 3, 10], X[16, 12, 17, 11], X[20, 14, 21, 13], 

 X[6, 15, 7, 16], X[22, 18, 1, 17], X[12, 20, 13, 19], 

X[18, 22, 19, 21]]
In[4]:=
GaussCode[Knot[11, Alternating, 1]]
Out[4]=  
GaussCode[1, -5, 2, -1, 3, -8, 4, -2, 5, -3, 6, -10, 7, -4, 8, -6, 9, 
  -11, 10, -7, 11, -9]
In[5]:=
BR[Knot[11, Alternating, 1]]
Out[5]=  
BR[Knot[11, Alternating, 1]]
In[6]:=
alex = Alexander[Knot[11, Alternating, 1]][t]
Out[6]=  
      2    12   30              2      3

-39 + -- - -- + -- + 30 t - 12 t + 2 t

      3    2   t
t t
In[7]:=
Conway[Knot[11, Alternating, 1]][z]
Out[7]=  
       6
1 + 2 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[11, Alternating, 1], Knot[11, Alternating, 122], 
  Knot[11, Alternating, 149]}
In[9]:=
{KnotDet[Knot[11, Alternating, 1]], KnotSignature[Knot[11, Alternating, 1]]}
Out[9]=  
{127, 2}
In[10]:=
J=Jones[Knot[11, Alternating, 1]][q]
Out[10]=  
       -3   3    7              2       3       4       5      6

-12 + q - -- + - + 17 q - 20 q + 21 q - 18 q + 14 q - 9 q +

            2   q
           q

    7    8
4 q - q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[11, Alternating, 1], Knot[11, Alternating, 149]}
In[12]:=
A2Invariant[Knot[11, Alternating, 1]][q]
Out[12]=  
      -10    -6   3    2       2      4      6    8    10      12

-1 + q - q + -- - -- + 3 q - 4 q + 3 q - q + q + 3 q -

                  4    2
                 q    q

    14      16      18      20      22    24
3 q + 4 q - 2 q - 2 q + 2 q - q
In[13]:=
Kauffman[Knot[11, Alternating, 1]][a, z]
Out[13]=  
                                                             2      2
     -6   2     2   2 z   4 z   4 z   4 z              2   z    4 z

-1 + a + -- - a - --- - --- - --- - --- - 2 a z + 5 z + -- - ---- -

           4         7     5     3     a                    8     6
          a         a     a     a                          a     a

     2      2              3      3       3       3       3
 10 z    3 z       2  2   z    7 z    16 z    16 z    15 z         3
 ----- - ---- + 3 a  z  - -- + ---- + ----- + ----- + ----- + 7 a z  - 
   4       2               9     7      5       3       a
  a       a               a     a      a       a

           4      4       4       4              5       5       5
    4   5 z    9 z    28 z    15 z       2  4   z    13 z    20 z
 2 z  - ---- + ---- + ----- + ----- - 3 a  z  + -- - ----- - ----- - 
          8      6      4       2                9     7       5
         a      a      a       a                a     a       a

     5       5                      6       6       6       6
 16 z    18 z         5      6   4 z    14 z    36 z    25 z
 ----- - ----- - 8 a z  - 6 z  + ---- - ----- - ----- - ----- + 
   3       a                       8      6       4       2
  a                               a      a       a       a

            7      7      7      7                      8       8
  2  6   8 z    4 z    4 z    3 z         7      8   9 z    16 z
 a  z  + ---- + ---- - ---- + ---- + 3 a z  + 4 z  + ---- + ----- + 
           7      5      3     a                       6      4
          a      a      a                             a      a

     8      9      9      9    10    10
 11 z    5 z    8 z    3 z    z     z
 ----- + ---- + ---- + ---- + --- + ---
   2       5      3     a      4     2
a a a a a
In[14]:=
{Vassiliev[2][Knot[11, Alternating, 1]], Vassiliev[3][Knot[11, Alternating, 1]]}
Out[14]=  
{0, 2}
In[15]:=
Kh[Knot[11, Alternating, 1]][q, t]
Out[15]=  
          3     1       2       1       5      2      7    5 q

10 q + 8 q + ----- + ----- + ----- + ----- + ---- + --- + --- +

              7  4    5  3    3  3    3  2      2   q t    t
             q  t    q  t    q  t    q  t    q t

     3        5         5  2       7  2      7  3       9  3
 11 q  t + 9 q  t + 10 q  t  + 11 q  t  + 8 q  t  + 10 q  t  + 

    9  4      11  4      11  5      13  5    13  6      15  6    17  7
6 q t + 8 q t + 3 q t + 6 q t + q t + 3 q t + q t