10 138

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10 137.gif

10_137

10 139.gif

10_139

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10 138 Quick Notes


10 138 Further Notes and Views

Knot presentations

Planar diagram presentation X4251 X10,6,11,5 X8394 X2,9,3,10 X16,12,17,11 X7,15,8,14 X15,7,16,6 X20,18,1,17 X18,13,19,14 X12,19,13,20
Gauss code 1, -4, 3, -1, 2, 7, -6, -3, 4, -2, 5, -10, 9, 6, -7, -5, 8, -9, 10, -8
Dowker-Thistlethwaite code 4 8 10 -14 2 16 18 -6 20 12
Conway Notation [211,211,2-]

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-3][-7]
Hyperbolic Volume 10.4672
A-Polynomial See Data:10 138/A-polynomial

[edit Notes for 10 138's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 1 }[/math]
Topological 4 genus [math]\displaystyle{ 1 }[/math]
Concordance genus [math]\displaystyle{ 3 }[/math]
Rasmussen s-Invariant -2

[edit Notes for 10 138's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^3-5 t^2+8 t-7+8 t^{-1} -5 t^{-2} + t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ z^6+z^4-3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 35, 2 }
Jones polynomial [math]\displaystyle{ 2 q^5-4 q^4+5 q^3-6 q^2+6 q-5+4 q^{-1} -2 q^{-2} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^6 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -2 z^4+a^2 z^2+5 z^2 a^{-2} -3 z^2 a^{-4} -6 z^2+2 a^2+3 a^{-2} -2 a^{-4} + a^{-6} -3 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^8 a^{-2} +z^8+2 a z^7+5 z^7 a^{-1} +3 z^7 a^{-3} +a^2 z^6+3 z^6 a^{-2} +3 z^6 a^{-4} +z^6-7 a z^5-14 z^5 a^{-1} -6 z^5 a^{-3} +z^5 a^{-5} -4 a^2 z^4-13 z^4 a^{-2} -5 z^4 a^{-4} -12 z^4+6 a z^3+8 z^3 a^{-1} +5 z^3 a^{-3} +3 z^3 a^{-5} +5 a^2 z^2+10 z^2 a^{-2} +6 z^2 a^{-4} +3 z^2 a^{-6} +12 z^2-a z-z a^{-1} -2 z a^{-3} -2 z a^{-5} -2 a^2-3 a^{-2} -2 a^{-4} - a^{-6} -3 }[/math]
The A2 invariant [math]\displaystyle{ q^{10}+q^8+q^4-q^2- q^{-4} +2 q^{-6} - q^{-8} + q^{-10} - q^{-12} - q^{-14} + q^{-16} + q^{-20} }[/math]
The G2 invariant [math]\displaystyle{ q^{46}-q^{44}+4 q^{42}-5 q^{40}+5 q^{38}-2 q^{36}-4 q^{34}+14 q^{32}-18 q^{30}+20 q^{28}-11 q^{26}-4 q^{24}+19 q^{22}-27 q^{20}+29 q^{18}-16 q^{16}+17 q^{12}-25 q^{10}+19 q^8-5 q^6-13 q^4+20 q^2-21+7 q^{-2} +8 q^{-4} -25 q^{-6} +33 q^{-8} -29 q^{-10} +14 q^{-12} +5 q^{-14} -25 q^{-16} +35 q^{-18} -34 q^{-20} +24 q^{-22} -4 q^{-24} -12 q^{-26} +28 q^{-28} -27 q^{-30} +17 q^{-32} + q^{-34} -15 q^{-36} +22 q^{-38} -17 q^{-40} +16 q^{-44} -24 q^{-46} +25 q^{-48} -15 q^{-50} -5 q^{-52} +18 q^{-54} -26 q^{-56} +22 q^{-58} -13 q^{-60} + q^{-62} +9 q^{-64} -12 q^{-66} +11 q^{-68} -8 q^{-70} +5 q^{-72} + q^{-74} -3 q^{-76} + q^{-78} -2 q^{-80} +2 q^{-82} - q^{-84} +2 q^{-86} + q^{-88} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_138.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^7-q^5+2 q^3-q+ q^{-1} - q^{-5} + q^{-7} -2 q^{-9} +2 q^{-11} }[/math]
2 [math]\displaystyle{ q^{22}-q^{20}-2 q^{18}+4 q^{16}+q^{14}-6 q^{12}+4 q^{10}+5 q^8-7 q^6-q^4+7 q^2-3-4 q^{-2} +5 q^{-4} + q^{-6} -5 q^{-8} + q^{-10} +6 q^{-12} -2 q^{-14} -5 q^{-16} +7 q^{-18} + q^{-20} -8 q^{-22} +4 q^{-24} +2 q^{-26} -4 q^{-28} + q^{-30} + q^{-32} }[/math]
3 [math]\displaystyle{ q^{45}-q^{43}-2 q^{41}+5 q^{37}+3 q^{35}-7 q^{33}-8 q^{31}+6 q^{29}+14 q^{27}-19 q^{23}-8 q^{21}+17 q^{19}+19 q^{17}-10 q^{15}-26 q^{13}+q^{11}+28 q^9+10 q^7-27 q^5-19 q^3+22 q+24 q^{-1} -15 q^{-3} -25 q^{-5} +11 q^{-7} +28 q^{-9} -6 q^{-11} -24 q^{-13} - q^{-15} +22 q^{-17} +7 q^{-19} -18 q^{-21} -17 q^{-23} +10 q^{-25} +24 q^{-27} + q^{-29} -28 q^{-31} -13 q^{-33} +28 q^{-35} +21 q^{-37} -21 q^{-39} -24 q^{-41} +14 q^{-43} +21 q^{-45} -3 q^{-47} -17 q^{-49} +8 q^{-53} -2 q^{-57} -2 q^{-59} +2 q^{-61} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{10}+q^8+q^4-q^2- q^{-4} +2 q^{-6} - q^{-8} + q^{-10} - q^{-12} - q^{-14} + q^{-16} + q^{-20} }[/math]
2,0 [math]\displaystyle{ q^{28}+q^{26}-2 q^{22}+3 q^{18}+q^{16}-3 q^{14}-q^{12}+3 q^{10}+2 q^8-3 q^6+3 q^2-2 q^{-2} - q^{-6} -2 q^{-8} + q^{-10} +2 q^{-16} +7 q^{-18} +2 q^{-20} -4 q^{-22} + q^{-26} -3 q^{-28} -4 q^{-30} +2 q^{-34} +2 q^{-36} - q^{-48} + q^{-52} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{20}-q^{18}+2 q^{16}+2 q^{14}-2 q^{12}+4 q^{10}+q^8-4 q^6+4 q^4-2 q^2-5+3 q^{-2} -3 q^{-6} +2 q^{-8} +2 q^{-10} + q^{-12} - q^{-14} +4 q^{-18} -4 q^{-20} +5 q^{-24} -5 q^{-26} - q^{-28} +4 q^{-30} -3 q^{-32} - q^{-34} +3 q^{-36} }[/math]
1,0,0 [math]\displaystyle{ q^{13}+q^{11}+2 q^9+q^5-2 q^3-2 q^{-1} + q^{-7} +2 q^{-9} + q^{-13} -2 q^{-15} -2 q^{-19} + q^{-21} + q^{-25} + q^{-27} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{20}-q^{18}+4 q^{16}-4 q^{14}+6 q^{12}-6 q^{10}+7 q^8-6 q^6+4 q^4-2 q^2-3+5 q^{-2} -8 q^{-4} +11 q^{-6} -12 q^{-8} +14 q^{-10} -11 q^{-12} +9 q^{-14} -6 q^{-16} +2 q^{-18} -4 q^{-22} +5 q^{-24} -7 q^{-26} +7 q^{-28} -6 q^{-30} +5 q^{-32} -3 q^{-34} +3 q^{-36} }[/math]
1,0 [math]\displaystyle{ q^{34}-q^{30}-q^{28}+3 q^{26}+3 q^{24}-2 q^{22}-4 q^{20}+q^{18}+6 q^{16}+3 q^{14}-6 q^{12}-5 q^{10}+4 q^8+7 q^6-q^4-8 q^2-2+5 q^{-2} +3 q^{-4} -3 q^{-6} -4 q^{-8} +2 q^{-10} +4 q^{-12} - q^{-14} -5 q^{-16} + q^{-18} +6 q^{-20} + q^{-22} -5 q^{-24} -3 q^{-26} +5 q^{-28} +5 q^{-30} -3 q^{-32} -6 q^{-34} +2 q^{-36} +7 q^{-38} + q^{-40} -6 q^{-42} -4 q^{-44} +3 q^{-46} +5 q^{-48} - q^{-50} -4 q^{-52} -2 q^{-54} + q^{-56} +3 q^{-58} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{46}-q^{44}+4 q^{42}-5 q^{40}+5 q^{38}-2 q^{36}-4 q^{34}+14 q^{32}-18 q^{30}+20 q^{28}-11 q^{26}-4 q^{24}+19 q^{22}-27 q^{20}+29 q^{18}-16 q^{16}+17 q^{12}-25 q^{10}+19 q^8-5 q^6-13 q^4+20 q^2-21+7 q^{-2} +8 q^{-4} -25 q^{-6} +33 q^{-8} -29 q^{-10} +14 q^{-12} +5 q^{-14} -25 q^{-16} +35 q^{-18} -34 q^{-20} +24 q^{-22} -4 q^{-24} -12 q^{-26} +28 q^{-28} -27 q^{-30} +17 q^{-32} + q^{-34} -15 q^{-36} +22 q^{-38} -17 q^{-40} +16 q^{-44} -24 q^{-46} +25 q^{-48} -15 q^{-50} -5 q^{-52} +18 q^{-54} -26 q^{-56} +22 q^{-58} -13 q^{-60} + q^{-62} +9 q^{-64} -12 q^{-66} +11 q^{-68} -8 q^{-70} +5 q^{-72} + q^{-74} -3 q^{-76} + q^{-78} -2 q^{-80} +2 q^{-82} - q^{-84} +2 q^{-86} + q^{-88} }[/math]

.

Computer Talk
The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["10 138"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^3-5 t^2+8 t-7+8 t^{-1} -5 t^{-2} + t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^6+z^4-3 z^2+1 }[/math]
In[6]:= Alexander[K, 2][t]
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
Out[6]= [math]\displaystyle{ \{1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 35, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ 2 q^5-4 q^4+5 q^3-6 q^2+6 q-5+4 q^{-1} -2 q^{-2} + q^{-3} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^6 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -2 z^4+a^2 z^2+5 z^2 a^{-2} -3 z^2 a^{-4} -6 z^2+2 a^2+3 a^{-2} -2 a^{-4} + a^{-6} -3 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^8 a^{-2} +z^8+2 a z^7+5 z^7 a^{-1} +3 z^7 a^{-3} +a^2 z^6+3 z^6 a^{-2} +3 z^6 a^{-4} +z^6-7 a z^5-14 z^5 a^{-1} -6 z^5 a^{-3} +z^5 a^{-5} -4 a^2 z^4-13 z^4 a^{-2} -5 z^4 a^{-4} -12 z^4+6 a z^3+8 z^3 a^{-1} +5 z^3 a^{-3} +3 z^3 a^{-5} +5 a^2 z^2+10 z^2 a^{-2} +6 z^2 a^{-4} +3 z^2 a^{-6} +12 z^2-a z-z a^{-1} -2 z a^{-3} -2 z a^{-5} -2 a^2-3 a^{-2} -2 a^{-4} - a^{-6} -3 }[/math]

Vassiliev invariants

V2 and V3: (-3, -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
[math]\displaystyle{ -12 }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 66 }[/math] [math]\displaystyle{ 30 }[/math] [math]\displaystyle{ 192 }[/math] [math]\displaystyle{ \frac{704}{3} }[/math] [math]\displaystyle{ \frac{128}{3} }[/math] [math]\displaystyle{ 48 }[/math] [math]\displaystyle{ -288 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ -792 }[/math] [math]\displaystyle{ -360 }[/math] [math]\displaystyle{ -\frac{3311}{10} }[/math] [math]\displaystyle{ \frac{382}{5} }[/math] [math]\displaystyle{ -\frac{6142}{15} }[/math] [math]\displaystyle{ \frac{431}{6} }[/math] [math]\displaystyle{ -\frac{1071}{10} }[/math]

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s+1 }[/math], where [math]\displaystyle{ s= }[/math]2 is the signature of 10 138. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 -4-3-2-101234χ
11        22
9       2 -2
7      32 1
5     32  -1
3    33   0
1   34    1
-1  12     -1
-3 13      2
-5 1       -1
-71        1

Computer Talk

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

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math] 

In[1]:=    
 
<< KnotTheory`
 
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=
Crossings[Knot[10, 138]]
Out[2]=  
10
In[3]:=
PD[Knot[10, 138]]
Out[3]=  
PD[X[4, 2, 5, 1], X[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10], 

 X[16, 12, 17, 11], X[7, 15, 8, 14], X[15, 7, 16, 6], 



  X[20, 18, 1, 17], X[18, 13, 19, 14], X[12, 19, 13, 20]]
In[4]:=
GaussCode[Knot[10, 138]]
Out[4]=  
GaussCode[1, -4, 3, -1, 2, 7, -6, -3, 4, -2, 5, -10, 9, 6, -7, -5, 8, 
  -9, 10, -8]
In[5]:=
BR[Knot[10, 138]]
Out[5]=  
BR[5, {-1, 2, -1, 2, 3, 2, 2, -4, 3, -4}]
In[6]:=
alex = Alexander[Knot[10, 138]][t]
Out[6]=  
      -3   5    8            2    3

-7 + t   - -- + - + 8 t - 5 t  + t



           2   t


           t
In[7]:=
Conway[Knot[10, 138]][z]
Out[7]=  
       2    4    6
1 - 3 z  + z  + z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[10, 138]}
In[9]:=
{KnotDet[Knot[10, 138]], KnotSignature[Knot[10, 138]]}
Out[9]=  
{35, 2}
In[10]:=
J=Jones[Knot[10, 138]][q]
Out[10]=  
      -3   2    4            2      3      4      5

-5 + q   - -- + - + 6 q - 6 q  + 5 q  - 4 q  + 2 q



           2   q


           q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[10, 138], Knot[11, NonAlternating, 117]}
In[12]:=
A2Invariant[Knot[10, 138]][q]
Out[12]=  
 -10    -8    -4    -2    4      6    8    10    12    14    16    20
q    + q   + q   - q   - q  + 2 q  - q  + q   - q   - q   + q   + q
In[13]:=
Kauffman[Knot[10, 138]][a, z]
Out[13]=  
                                                             2      2

     -6   2    3       2   2 z   2 z   z             2   3 z    6 z



-3 - a   - -- - -- - 2 a  - --- - --- - - - a z + 12 z  + ---- + ---- + 



           4    2           5     3    a                   6      4
          a    a           a     a                        a      a

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

     4              5      5       5                    6      6
 13 z       2  4   z    6 z    14 z         5    6   3 z    3 z
 ----- - 4 a  z  + -- - ---- - ----- - 7 a z  + z  + ---- + ---- + 
   2                5     3      a                     4      2
  a                a     a                            a      a

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


           a                          a
In[14]:=
{Vassiliev[2][Knot[10, 138]], Vassiliev[3][Knot[10, 138]]}
Out[14]=  
{0, -2}
In[15]:=
Kh[Knot[10, 138]][q, t]
Out[15]=  
         3     1       1       1       3      1      2    3 q

4 q + 3 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  4


  3 q  t + 3 q  t + 2 q  t  + 3 q  t  + 2 q  t  + 2 q  t  + 2 q   t
Retrieved from "https://katlas.org/index.php?title=10_138&oldid=26340"