10 1

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9 49.gif

9_49

10 2.gif

10_2

10 1.gif Visit 10 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

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10 1 Quick Notes


10 1 Further Notes and Views

Knot presentations

Planar diagram presentation X1425 X11,14,12,15 X3,13,4,12 X13,3,14,2 X5,20,6,1 X7,18,8,19 X9,16,10,17 X15,10,16,11 X17,8,18,9 X19,6,20,7
Gauss code -1, 4, -3, 1, -5, 10, -6, 9, -7, 8, -2, 3, -4, 2, -8, 7, -9, 6, -10, 5
Dowker-Thistlethwaite code 4 12 20 18 16 14 2 10 8 6
Conway Notation [82]

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 1
Bridge index 2
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-9][-3]
Hyperbolic Volume 3.5262
A-Polynomial See Data:10 1/A-polynomial

[edit Notes for 10 1'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{ 1 }[/math]
Rasmussen s-Invariant 0

[edit Notes for 10 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -4 t+9-4 t^{-1} }[/math]
Conway polynomial [math]\displaystyle{ 1-4 z^2 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 17, 0 }
Jones polynomial [math]\displaystyle{ q^2-q+2-2 q^{-1} +2 q^{-2} -2 q^{-3} +2 q^{-4} -2 q^{-5} + q^{-6} - q^{-7} + q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ a^8-z^2 a^6-a^6-z^2 a^4-z^2 a^2-z^2+ a^{-2} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a^7 z^9+a^5 z^9+a^8 z^8+2 a^6 z^8+a^4 z^8-7 a^7 z^7-6 a^5 z^7+a^3 z^7-7 a^8 z^6-12 a^6 z^6-4 a^4 z^6+a^2 z^6+16 a^7 z^5+12 a^5 z^5-3 a^3 z^5+a z^5+15 a^8 z^4+21 a^6 z^4+3 a^4 z^4-2 a^2 z^4+z^4-14 a^7 z^3-11 a^5 z^3+a^3 z^3-a z^3+z^3 a^{-1} -10 a^8 z^2-11 a^6 z^2+z^2 a^{-2} +4 a^7 z+4 a^5 z+a^8+a^6- a^{-2} }[/math]
The A2 invariant [math]\displaystyle{ q^{26}+q^{24}-q^{18}-q^{16}+ q^{-2} + q^{-6} + q^{-8} }[/math]
The G2 invariant Data:10 1/QuantumInvariant/G2/1,0
Further Quantum Invariants
Further quantum knot invariants for 10_1.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{17}-q^{11}+ q^{-1} + q^{-5} }[/math]
2 [math]\displaystyle{ q^{50}-q^{46}-q^{40}+q^{36}+q^{16}+q^{14}-q^4-q^2+ q^{-2} + q^{-8} + q^{-14} }[/math]
3 [math]\displaystyle{ q^{99}-q^{95}-q^{93}+q^{89}-q^{85}+q^{81}+q^{79}-q^{75}+q^{49}+q^{47}-q^{43}-q^{37}-q^{35}-q^{29}+q^{25}+q^{23}+q^{15}+q^{13}+q^{11}-q^9+q^5+q^3-q- q^{-1} + q^{-3} + q^{-5} - q^{-7} - q^{-9} + q^{-19} + q^{-27} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{26}+q^{24}-q^{18}-q^{16}+ q^{-2} + q^{-6} + q^{-8} }[/math]
2,0 [math]\displaystyle{ q^{68}+q^{66}+q^{64}-q^{62}-q^{60}-q^{58}-q^{56}-q^{54}-q^{52}+q^{50}+q^{48}+q^{46}+q^{24}+2 q^{22}+q^{20}+q^{18}+q^{16}-q^{10}-2 q^8-2 q^6-q^4+ q^{-4} + q^{-10} + q^{-12} + q^{-16} + q^{-18} + q^{-20} }[/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 1"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -4 t+9-4 t^{-1} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 1-4 z^2 }[/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]= { 17, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^2-q+2-2 q^{-1} +2 q^{-2} -2 q^{-3} +2 q^{-4} -2 q^{-5} + q^{-6} - q^{-7} + q^{-8} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ a^8-z^2 a^6-a^6-z^2 a^4-z^2 a^2-z^2+ a^{-2} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ a^7 z^9+a^5 z^9+a^8 z^8+2 a^6 z^8+a^4 z^8-7 a^7 z^7-6 a^5 z^7+a^3 z^7-7 a^8 z^6-12 a^6 z^6-4 a^4 z^6+a^2 z^6+16 a^7 z^5+12 a^5 z^5-3 a^3 z^5+a z^5+15 a^8 z^4+21 a^6 z^4+3 a^4 z^4-2 a^2 z^4+z^4-14 a^7 z^3-11 a^5 z^3+a^3 z^3-a z^3+z^3 a^{-1} -10 a^8 z^2-11 a^6 z^2+z^2 a^{-2} +4 a^7 z+4 a^5 z+a^8+a^6- a^{-2} }[/math]

Vassiliev invariants

V2 and V3: (-4, 6)
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{ -16 }[/math] [math]\displaystyle{ 48 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{232}{3} }[/math] [math]\displaystyle{ \frac{200}{3} }[/math] [math]\displaystyle{ -768 }[/math] [math]\displaystyle{ -1088 }[/math] [math]\displaystyle{ -256 }[/math] [math]\displaystyle{ -272 }[/math] [math]\displaystyle{ -\frac{2048}{3} }[/math] [math]\displaystyle{ 1152 }[/math] [math]\displaystyle{ -\frac{3712}{3} }[/math] [math]\displaystyle{ -\frac{3200}{3} }[/math] [math]\displaystyle{ \frac{30898}{15} }[/math] [math]\displaystyle{ \frac{5768}{15} }[/math] [math]\displaystyle{ -\frac{6248}{45} }[/math] [math]\displaystyle{ \frac{4046}{9} }[/math] [math]\displaystyle{ -\frac{4142}{15} }[/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]0 is the signature of 10 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 -8-7-6-5-4-3-2-1012χ
5          11
3           0
1        21 1
-1       11  0
-3      11   0
-5     11    0
-7    11     0
-9   11      0
-11   1       -1
-13 11        0
-15           0
-171          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, 1]]
Out[2]=  
10
In[3]:=
PD[Knot[10, 1]]
Out[3]=  
PD[X[1, 4, 2, 5], X[11, 14, 12, 15], X[3, 13, 4, 12], X[13, 3, 14, 2], 

 X[5, 20, 6, 1], X[7, 18, 8, 19], X[9, 16, 10, 17], X[15, 10, 16, 11], 



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

9 - - - 4 t


    t
In[7]:=
Conway[Knot[10, 1]][z]
Out[7]=  
       2
1 - 4 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[8, 3], Knot[10, 1]}
In[9]:=
{KnotDet[Knot[10, 1]], KnotSignature[Knot[10, 1]]}
Out[9]=  
{17, 0}
In[10]:=
J=Jones[Knot[10, 1]][q]
Out[10]=  
     -8    -7    -6   2    2    2    2    2        2

2 + q   - q   + q   - -- + -- - -- + -- - - - q + q



                      5    4    3    2   q


                      q    q    q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[10, 1]}
In[12]:=
A2Invariant[Knot[10, 1]][q]
Out[12]=  
 -26    -24    -18    -16    2    6    8
q    + q    - q    - q    + q  + q  + q
In[13]:=
Kauffman[Knot[10, 1]][a, z]
Out[13]=  
                                    2                          3

 -2    6    8      5        7     z        6  2       8  2   z



-a   + a  + a  + 4 a  z + 4 a  z + -- - 11 a  z  - 10 a  z  + -- - 



                                   2                         a
                                  a

    3    3  3       5  3       7  3    4      2  4      4  4
 a z  + a  z  - 11 a  z  - 14 a  z  + z  - 2 a  z  + 3 a  z  + 

     6  4       8  4      5      3  5       5  5       7  5    2  6
 21 a  z  + 15 a  z  + a z  - 3 a  z  + 12 a  z  + 16 a  z  + a  z  - 

    4  6       6  6      8  6    3  7      5  7      7  7    4  8
 4 a  z  - 12 a  z  - 7 a  z  + a  z  - 6 a  z  - 7 a  z  + a  z  + 

    6  8    8  8    5  9    7  9


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

- + 2 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 
q          17  8    13  7    13  6    11  5    9  5    9  4    7  4



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

   1       1       1       1      1      1           5  2
 ----- + ----- + ----- + ----- + ---- + --- + q t + q  t
  7  3    5  3    5  2    3  2    3     q t


  q  t    q  t    q  t    q  t    q  t
Retrieved from "https://katlas.org/index.php?title=10_1&oldid=26096"