8 15

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8 14.gif

8_14

8 16.gif

8_16

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Two trefoil knots along a closed loop, mutually interlinked. (See also 10 120.)



Symmetrical depiction.

Knot presentations

Planar diagram presentation X1425 X3849 X5,12,6,13 X13,16,14,1 X9,14,10,15 X15,10,16,11 X11,6,12,7 X7283
Gauss code -1, 8, -2, 1, -3, 7, -8, 2, -5, 6, -7, 3, -4, 5, -6, 4
Dowker-Thistlethwaite code 4 8 12 2 14 6 16 10
Conway Notation [21,21,2]

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 3
Super bridge index
Nakanishi index 1
Maximal Thurston-Bennequin number [-13][3]
Hyperbolic Volume 9.93065
A-Polynomial See Data:8 15/A-polynomial

[edit Notes for 8 15's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 8 15's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 33, -4 }
Jones polynomial
HOMFLY-PT polynomial (db, data sources)
Kauffman polynomial (db, data sources)
The A2 invariant
The G2 invariant

Vassiliev invariants

V2 and V3: (4, -7)
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 -4 is the signature of 8 15. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
-8-7-6-5-4-3-2-10χ
-3        11
-5       21-1
-7      3  3
-9     22  0
-11    43   1
-13   22    0
-15  24     -2
-17 12      1
-19 2       -2
-211        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[8, 15]]
Out[2]=  
8
In[3]:=
PD[Knot[8, 15]]
Out[3]=  
PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 12, 6, 13], X[13, 16, 14, 1], 
  X[9, 14, 10, 15], X[15, 10, 16, 11], X[11, 6, 12, 7], X[7, 2, 8, 3]]
In[4]:=
GaussCode[Knot[8, 15]]
Out[4]=  
GaussCode[-1, 8, -2, 1, -3, 7, -8, 2, -5, 6, -7, 3, -4, 5, -6, 4]
In[5]:=
BR[Knot[8, 15]]
Out[5]=  
BR[4, {-1, -1, 2, -1, -3, -2, -2, -2, -3}]
In[6]:=
alex = Alexander[Knot[8, 15]][t]
Out[6]=  
     3    8            2

11 + -- - - - 8 t + 3 t

     2   t
t
In[7]:=
Conway[Knot[8, 15]][z]
Out[7]=  
       2      4
1 + 4 z  + 3 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[8, 15], Knot[11, NonAlternating, 65]}
In[9]:=
{KnotDet[Knot[8, 15]], KnotSignature[Knot[8, 15]]}
Out[9]=  
{33, -4}
In[10]:=
J=Jones[Knot[8, 15]][q]
Out[10]=  
 -10   3    4    6    6    5    5    2     -2

q - -- + -- - -- + -- - -- + -- - -- + q

       9    8    7    6    5    4    3
q q q q q q q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[8, 15]}
In[12]:=
A2Invariant[Knot[8, 15]][q]
Out[12]=  
 -32    -30    2     -26    2     2     -20    3     -14    -12    2

q + q - --- - q - --- - --- + q + --- + q + q + --- -

              28           24    22           16                  10
             q            q     q            q                   q

  -8    -6
q + q
In[13]:=
Kauffman[Knot[8, 15]][a, z]
Out[13]=  
 4      6      8    10      7        9        11        4  2

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

    6  2      8  2    12  2      5  3       7  3       9  3
 5 a  z  + 8 a  z  - a   z  - 2 a  z  - 11 a  z  - 14 a  z  - 

    11  3    4  4      6  4       8  4      10  4    12  4      5  5
 5 a   z  + a  z  - 5 a  z  - 10 a  z  - 3 a   z  + a   z  + 2 a  z  + 

    7  5      9  5      11  5      6  6      8  6      10  6    7  7
 5 a  z  + 6 a  z  + 3 a   z  + 3 a  z  + 6 a  z  + 3 a   z  + a  z  + 

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

q + q + ------ + ------ + ------ + ------ + ------ + ------ +

            21  8    19  7    17  7    17  6    15  6    15  5
           q   t    q   t    q   t    q   t    q   t    q   t

   2        2        4        3        2       2       3      2
 ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----
  13  5    13  4    11  4    11  3    9  3    9  2    7  2    5
q t q t q t q t q t q t q t q t