10 134

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10 133.gif

10_133

10 135.gif

10_135

10 134.gif Visit 10 134's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

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10 134 Quick Notes


10 134 Further Notes and Views

Knot presentations

Planar diagram presentation X4251 X8493 X9,15,10,14 X5,13,6,12 X13,7,14,6 X11,19,12,18 X15,1,16,20 X19,17,20,16 X17,11,18,10 X2837
Gauss code 1, -10, 2, -1, -4, 5, 10, -2, -3, 9, -6, 4, -5, 3, -7, 8, -9, 6, -8, 7
Dowker-Thistlethwaite code 4 8 -12 2 -14 -18 -6 -20 -10 -16
Conway Notation [221,3,2-]

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [5][-15]
Hyperbolic Volume 8.39292
A-Polynomial See Data:10 134/A-polynomial

[edit Notes for 10 134's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 3 }[/math]
Topological 4 genus [math]\displaystyle{ 3 }[/math]
Concordance genus [math]\displaystyle{ 3 }[/math]
Rasmussen s-Invariant -6

[edit Notes for 10 134's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ 2 t^3-4 t^2+4 t-3+4 t^{-1} -4 t^{-2} +2 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ 2 z^6+8 z^4+6 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 23, 6 }
Jones polynomial [math]\displaystyle{ q^{11}-3 q^{10}+3 q^9-4 q^8+4 q^7-3 q^6+3 q^5-q^4+q^3 }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^6 a^{-6} +z^6 a^{-8} +5 z^4 a^{-6} +4 z^4 a^{-8} -z^4 a^{-10} +7 z^2 a^{-6} +3 z^2 a^{-8} -4 z^2 a^{-10} +3 a^{-6} -3 a^{-10} + a^{-12} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +z^7 a^{-7} +3 z^7 a^{-9} +2 z^7 a^{-11} +z^6 a^{-6} -3 z^6 a^{-8} -3 z^6 a^{-10} +z^6 a^{-12} -3 z^5 a^{-7} -11 z^5 a^{-9} -8 z^5 a^{-11} -5 z^4 a^{-6} +z^4 a^{-8} +5 z^4 a^{-10} -z^4 a^{-12} +11 z^3 a^{-9} +14 z^3 a^{-11} +3 z^3 a^{-13} +7 z^2 a^{-6} -7 z^2 a^{-10} +z^2 a^{-12} +z^2 a^{-14} +2 z a^{-7} -4 z a^{-9} -8 z a^{-11} -2 z a^{-13} -3 a^{-6} +3 a^{-10} + a^{-12} }[/math]
The A2 invariant [math]\displaystyle{ q^{-10} +2 q^{-14} + q^{-16} +2 q^{-18} + q^{-20} + q^{-24} -2 q^{-26} - q^{-28} -2 q^{-30} - q^{-32} + q^{-38} }[/math]
The G2 invariant [math]\displaystyle{ q^{-50} +2 q^{-54} - q^{-56} +2 q^{-58} +5 q^{-64} -5 q^{-66} +8 q^{-68} -4 q^{-70} +6 q^{-74} -7 q^{-76} +10 q^{-78} -5 q^{-80} +2 q^{-82} +5 q^{-84} -6 q^{-86} +6 q^{-88} -5 q^{-92} +9 q^{-94} -6 q^{-96} + q^{-98} +5 q^{-100} -9 q^{-102} +12 q^{-104} -8 q^{-106} +3 q^{-108} + q^{-110} -7 q^{-112} +8 q^{-114} -10 q^{-116} +5 q^{-118} -4 q^{-120} -3 q^{-122} +4 q^{-124} -8 q^{-126} +2 q^{-128} - q^{-130} -6 q^{-132} +5 q^{-134} -6 q^{-136} -2 q^{-138} +6 q^{-140} -9 q^{-142} +9 q^{-144} -4 q^{-146} -2 q^{-148} +7 q^{-150} -7 q^{-152} +6 q^{-154} - q^{-156} +3 q^{-160} - q^{-162} + q^{-164} +2 q^{-168} -2 q^{-174} - q^{-180} + q^{-182} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_134.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{-5} +2 q^{-9} + q^{-13} - q^{-17} -2 q^{-21} + q^{-23} }[/math]
2 [math]\displaystyle{ q^{-10} +3 q^{-16} + q^{-18} -2 q^{-20} +3 q^{-22} +3 q^{-24} -2 q^{-26} - q^{-28} +2 q^{-30} -2 q^{-32} -3 q^{-34} + q^{-36} -3 q^{-40} +2 q^{-44} - q^{-46} -2 q^{-48} +3 q^{-50} + q^{-52} -3 q^{-54} +2 q^{-56} + q^{-58} - q^{-60} }[/math]
3 [math]\displaystyle{ q^{-15} + q^{-21} +3 q^{-23} + q^{-25} -2 q^{-27} - q^{-29} +5 q^{-31} +5 q^{-33} - q^{-35} -6 q^{-37} +6 q^{-41} +5 q^{-43} -4 q^{-45} -9 q^{-47} -2 q^{-49} +7 q^{-51} +4 q^{-53} -8 q^{-55} -9 q^{-57} +3 q^{-59} +8 q^{-61} -3 q^{-63} -9 q^{-65} + q^{-67} +10 q^{-69} - q^{-71} -6 q^{-73} +7 q^{-77} + q^{-79} -4 q^{-81} -4 q^{-83} +3 q^{-85} +6 q^{-87} +2 q^{-89} -8 q^{-91} -6 q^{-93} +8 q^{-95} +9 q^{-97} -5 q^{-99} -10 q^{-101} +2 q^{-103} +7 q^{-105} +3 q^{-107} -6 q^{-109} -3 q^{-111} + q^{-113} +2 q^{-115} + q^{-117} - q^{-119} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{-10} +2 q^{-14} + q^{-16} +2 q^{-18} + q^{-20} + q^{-24} -2 q^{-26} - q^{-28} -2 q^{-30} - q^{-32} + q^{-38} }[/math]
1,1 [math]\displaystyle{ q^{-20} +4 q^{-24} -2 q^{-26} +12 q^{-28} -8 q^{-30} +24 q^{-32} -16 q^{-34} +26 q^{-36} -18 q^{-38} +14 q^{-40} -6 q^{-42} -16 q^{-44} +14 q^{-46} -38 q^{-48} +32 q^{-50} -45 q^{-52} +36 q^{-54} -32 q^{-56} +28 q^{-58} -12 q^{-60} +6 q^{-62} +10 q^{-64} -16 q^{-66} +23 q^{-68} -24 q^{-70} +20 q^{-72} -16 q^{-74} +10 q^{-76} -4 q^{-78} +2 q^{-84} -2 q^{-90} + q^{-92} }[/math]
2,0 [math]\displaystyle{ q^{-20} +2 q^{-26} +3 q^{-28} + q^{-30} +2 q^{-32} +4 q^{-34} +5 q^{-36} +2 q^{-38} +2 q^{-40} +2 q^{-42} -3 q^{-46} -2 q^{-48} -4 q^{-50} -5 q^{-52} -4 q^{-54} -5 q^{-56} -4 q^{-58} -2 q^{-60} + q^{-62} +2 q^{-64} + q^{-66} +2 q^{-68} +3 q^{-70} + q^{-72} + q^{-78} +2 q^{-80} + q^{-82} - q^{-84} - q^{-88} - q^{-90} - q^{-92} + q^{-96} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{-20} +2 q^{-24} +3 q^{-26} +2 q^{-28} +4 q^{-30} +4 q^{-32} + q^{-34} +4 q^{-36} - q^{-40} -2 q^{-44} -5 q^{-46} -4 q^{-48} -5 q^{-50} -5 q^{-52} -3 q^{-54} - q^{-56} +4 q^{-58} +2 q^{-60} +4 q^{-62} +4 q^{-64} - q^{-66} - q^{-68} + q^{-70} -2 q^{-72} - q^{-74} + q^{-76} }[/math]
1,0,0 [math]\displaystyle{ q^{-15} +2 q^{-19} + q^{-21} +3 q^{-23} + q^{-25} +2 q^{-27} + q^{-29} -2 q^{-35} - q^{-37} -3 q^{-39} - q^{-41} -2 q^{-43} + q^{-49} + q^{-51} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{-30} +2 q^{-34} +3 q^{-36} +4 q^{-38} +4 q^{-40} +7 q^{-42} +5 q^{-44} +6 q^{-46} +5 q^{-48} +2 q^{-50} +2 q^{-52} +3 q^{-54} -3 q^{-58} -2 q^{-60} -4 q^{-62} -10 q^{-64} -14 q^{-66} -10 q^{-68} -11 q^{-70} -11 q^{-72} -2 q^{-74} +4 q^{-76} +4 q^{-78} +10 q^{-80} +12 q^{-82} +7 q^{-84} +3 q^{-86} +3 q^{-88} -5 q^{-92} -4 q^{-94} - q^{-98} -2 q^{-100} + q^{-102} + q^{-104} }[/math]
1,0,0,0 [math]\displaystyle{ q^{-20} +2 q^{-24} + q^{-26} +3 q^{-28} +2 q^{-30} +2 q^{-32} +2 q^{-34} + q^{-36} + q^{-38} - q^{-40} -2 q^{-44} - q^{-46} -3 q^{-48} -2 q^{-50} -2 q^{-52} -2 q^{-54} + q^{-60} + q^{-62} + q^{-64} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{-20} +2 q^{-24} - q^{-26} +4 q^{-28} -2 q^{-30} +4 q^{-32} - q^{-34} +2 q^{-36} - q^{-40} +2 q^{-42} -4 q^{-44} +5 q^{-46} -6 q^{-48} +5 q^{-50} -5 q^{-52} +3 q^{-54} -3 q^{-56} -2 q^{-62} +2 q^{-64} -3 q^{-66} +3 q^{-68} -3 q^{-70} +2 q^{-72} - q^{-74} + q^{-76} }[/math]
1,0 [math]\displaystyle{ q^{-30} +2 q^{-38} +2 q^{-40} + q^{-42} - q^{-44} +2 q^{-46} +4 q^{-48} +3 q^{-50} -2 q^{-52} +3 q^{-56} +4 q^{-58} -3 q^{-62} - q^{-64} +2 q^{-66} -3 q^{-70} -3 q^{-72} - q^{-74} - q^{-76} -3 q^{-78} -4 q^{-80} -2 q^{-82} + q^{-84} - q^{-86} -3 q^{-88} -2 q^{-90} +3 q^{-92} +3 q^{-94} - q^{-98} +3 q^{-100} +4 q^{-102} + q^{-104} -2 q^{-106} - q^{-108} + q^{-110} +2 q^{-112} - q^{-114} -2 q^{-116} - q^{-118} + q^{-122} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{-30} +2 q^{-34} + q^{-36} +5 q^{-38} + q^{-40} +6 q^{-42} + q^{-44} +6 q^{-46} + q^{-48} +2 q^{-50} + q^{-54} + q^{-56} -2 q^{-58} +2 q^{-60} -5 q^{-62} + q^{-64} -8 q^{-66} - q^{-68} -10 q^{-70} -2 q^{-72} -8 q^{-74} -2 q^{-78} +3 q^{-80} +3 q^{-82} +3 q^{-84} +5 q^{-86} + q^{-88} +4 q^{-90} -2 q^{-92} + q^{-94} -3 q^{-96} +2 q^{-98} -2 q^{-100} - q^{-104} + q^{-106} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{-50} +2 q^{-54} - q^{-56} +2 q^{-58} +5 q^{-64} -5 q^{-66} +8 q^{-68} -4 q^{-70} +6 q^{-74} -7 q^{-76} +10 q^{-78} -5 q^{-80} +2 q^{-82} +5 q^{-84} -6 q^{-86} +6 q^{-88} -5 q^{-92} +9 q^{-94} -6 q^{-96} + q^{-98} +5 q^{-100} -9 q^{-102} +12 q^{-104} -8 q^{-106} +3 q^{-108} + q^{-110} -7 q^{-112} +8 q^{-114} -10 q^{-116} +5 q^{-118} -4 q^{-120} -3 q^{-122} +4 q^{-124} -8 q^{-126} +2 q^{-128} - q^{-130} -6 q^{-132} +5 q^{-134} -6 q^{-136} -2 q^{-138} +6 q^{-140} -9 q^{-142} +9 q^{-144} -4 q^{-146} -2 q^{-148} +7 q^{-150} -7 q^{-152} +6 q^{-154} - q^{-156} +3 q^{-160} - q^{-162} + q^{-164} +2 q^{-168} -2 q^{-174} - q^{-180} + q^{-182} }[/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["10 134"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 2 t^3-4 t^2+4 t-3+4 t^{-1} -4 t^{-2} +2 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 2 z^6+8 z^4+6 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]= { 23, 6 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^{11}-3 q^{10}+3 q^9-4 q^8+4 q^7-3 q^6+3 q^5-q^4+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^{-6} +z^6 a^{-8} +5 z^4 a^{-6} +4 z^4 a^{-8} -z^4 a^{-10} +7 z^2 a^{-6} +3 z^2 a^{-8} -4 z^2 a^{-10} +3 a^{-6} -3 a^{-10} + a^{-12} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +z^7 a^{-7} +3 z^7 a^{-9} +2 z^7 a^{-11} +z^6 a^{-6} -3 z^6 a^{-8} -3 z^6 a^{-10} +z^6 a^{-12} -3 z^5 a^{-7} -11 z^5 a^{-9} -8 z^5 a^{-11} -5 z^4 a^{-6} +z^4 a^{-8} +5 z^4 a^{-10} -z^4 a^{-12} +11 z^3 a^{-9} +14 z^3 a^{-11} +3 z^3 a^{-13} +7 z^2 a^{-6} -7 z^2 a^{-10} +z^2 a^{-12} +z^2 a^{-14} +2 z a^{-7} -4 z a^{-9} -8 z a^{-11} -2 z a^{-13} -3 a^{-6} +3 a^{-10} + a^{-12} }[/math]

Vassiliev invariants

V2 and V3: (6, 13)
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{ 24 }[/math] [math]\displaystyle{ 104 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 588 }[/math] [math]\displaystyle{ 76 }[/math] [math]\displaystyle{ 2496 }[/math] [math]\displaystyle{ \frac{11216}{3} }[/math] [math]\displaystyle{ \frac{1856}{3} }[/math] [math]\displaystyle{ 392 }[/math] [math]\displaystyle{ 2304 }[/math] [math]\displaystyle{ 5408 }[/math] [math]\displaystyle{ 14112 }[/math] [math]\displaystyle{ 1824 }[/math] [math]\displaystyle{ \frac{123271}{5} }[/math] [math]\displaystyle{ \frac{24548}{15} }[/math] [math]\displaystyle{ \frac{38348}{5} }[/math] [math]\displaystyle{ 131 }[/math] [math]\displaystyle{ \frac{4551}{5} }[/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]6 is the signature of 10 134. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 012345678χ
23        11
21       2 -2
19      11 0
17     32  -1
15    11   0
13   23    1
11  11     0
9  2      2
711       0
51        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, 134]]
Out[2]=  
10
In[3]:=
PD[Knot[10, 134]]
Out[3]=  
PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[9, 15, 10, 14], X[5, 13, 6, 12], 

 X[13, 7, 14, 6], X[11, 19, 12, 18], X[15, 1, 16, 20], 



  X[19, 17, 20, 16], X[17, 11, 18, 10], X[2, 8, 3, 7]]
In[4]:=
GaussCode[Knot[10, 134]]
Out[4]=  
GaussCode[1, -10, 2, -1, -4, 5, 10, -2, -3, 9, -6, 4, -5, 3, -7, 8, -9, 
  6, -8, 7]
In[5]:=
BR[Knot[10, 134]]
Out[5]=  
BR[4, {1, 1, 1, 2, 1, 1, 2, 3, -2, 3, 3}]
In[6]:=
alex = Alexander[Knot[10, 134]][t]
Out[6]=  
     2    4    4            2      3

-3 + -- - -- + - + 4 t - 4 t  + 2 t



     3    2   t


     t    t
In[7]:=
Conway[Knot[10, 134]][z]
Out[7]=  
       2      4      6
1 + 6 z  + 8 z  + 2 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[10, 134]}
In[9]:=
{KnotDet[Knot[10, 134]], KnotSignature[Knot[10, 134]]}
Out[9]=  
{23, 6}
In[10]:=
J=Jones[Knot[10, 134]][q]
Out[10]=  
 3    4      5      6      7      8      9      10    11
q  - q  + 3 q  - 3 q  + 4 q  - 4 q  + 3 q  - 3 q   + q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[10, 134]}
In[12]:=
A2Invariant[Knot[10, 134]][q]
Out[12]=  
 10      14    16      18    20    24      26    28      30    32    38
q   + 2 q   + q   + 2 q   + q   + q   - 2 q   - q   - 2 q   - q   + q
In[13]:=
Kauffman[Knot[10, 134]][a, z]
Out[13]=  
                                           2     2       2      2

-12    3    3    2 z   8 z   4 z   2 z   z     z     7 z    7 z



a    + --- - -- - --- - --- - --- + --- + --- + --- - ---- + ---- + 



       10    6    13    11    9     7     14    12    10      6
      a     a    a     a     a     a     a     a     a       a

    3       3       3    4       4    4      4      5       5      5
 3 z    14 z    11 z    z     5 z    z    5 z    8 z    11 z    3 z
 ---- + ----- + ----- - --- + ---- + -- - ---- - ---- - ----- - ---- + 
  13      11      9      12    10     8     6     11      9       7
 a       a       a      a     a      a     a     a       a       a

  6       6      6    6      7      7    7    8     8
 z     3 z    3 z    z    2 z    3 z    z    z     z
 --- - ---- - ---- + -- + ---- + ---- + -- + --- + --
  12    10      8     6    11      9     7    10    8


  a     a       a     a    a       a     a    a     a
In[14]:=
{Vassiliev[2][Knot[10, 134]], Vassiliev[3][Knot[10, 134]]}
Out[14]=  
{0, 13}
In[15]:=
Kh[Knot[10, 134]][q, t]
Out[15]=  
 5    7    7        9  2    11  2    11  3      13  3      13  4

q  + q  + q  t + 2 q  t  + q   t  + q   t  + 2 q   t  + 3 q   t  + 



  15  4    15  5      17  5      17  6    19  6    19  7      21  7
 q   t  + q   t  + 3 q   t  + 2 q   t  + q   t  + q   t  + 2 q   t  + 

  23  8


  q   t
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