T(13,2)

T(11,2)

T(7,3)

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Edit T(13,2) Further Notes and Views

Knot presentations

Planar diagram presentation	X11,25,12,24 X25,13,26,12 X13,1,14,26 X1,15,2,14 X15,3,16,2 X3,17,4,16 X17,5,18,4 X5,19,6,18 X19,7,20,6 X7,21,8,20 X21,9,22,8 X9,23,10,22 X23,11,24,10
Gauss code	-4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 1, -2, 3
Dowker-Thistlethwaite code	14 16 18 20 22 24 26 2 4 6 8 10 12

Braid presentation

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^6-t^5+t^4-t^3+t^2-t+1- t^{-1} + t^{-2} - t^{-3} + t^{-4} - t^{-5} + t^{-6} }[/math]
Conway polynomial	[math]\displaystyle{ z^{12}+11 z^{10}+45 z^8+84 z^6+70 z^4+21 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 13, 12 }
Jones polynomial	[math]\displaystyle{ -q^{19}+q^{18}-q^{17}+q^{16}-q^{15}+q^{14}-q^{13}+q^{12}-q^{11}+q^{10}-q^9+q^8+q^6 }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^{12} a^{-12} +12 z^{10} a^{-12} -z^{10} a^{-14} +55 z^8 a^{-12} -10 z^8 a^{-14} +120 z^6 a^{-12} -36 z^6 a^{-14} +126 z^4 a^{-12} -56 z^4 a^{-14} +56 z^2 a^{-12} -35 z^2 a^{-14} +7 a^{-12} -6 a^{-14} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{12} a^{-12} +z^{12} a^{-14} +z^{11} a^{-13} +z^{11} a^{-15} -12 z^{10} a^{-12} -11 z^{10} a^{-14} +z^{10} a^{-16} -10 z^9 a^{-13} -9 z^9 a^{-15} +z^9 a^{-17} +55 z^8 a^{-12} +46 z^8 a^{-14} -8 z^8 a^{-16} +z^8 a^{-18} +36 z^7 a^{-13} +28 z^7 a^{-15} -7 z^7 a^{-17} +z^7 a^{-19} -120 z^6 a^{-12} -92 z^6 a^{-14} +21 z^6 a^{-16} -6 z^6 a^{-18} +z^6 a^{-20} -56 z^5 a^{-13} -35 z^5 a^{-15} +15 z^5 a^{-17} -5 z^5 a^{-19} +z^5 a^{-21} +126 z^4 a^{-12} +91 z^4 a^{-14} -20 z^4 a^{-16} +10 z^4 a^{-18} -4 z^4 a^{-20} +z^4 a^{-22} +35 z^3 a^{-13} +15 z^3 a^{-15} -10 z^3 a^{-17} +6 z^3 a^{-19} -3 z^3 a^{-21} +z^3 a^{-23} -56 z^2 a^{-12} -41 z^2 a^{-14} +5 z^2 a^{-16} -4 z^2 a^{-18} +3 z^2 a^{-20} -2 z^2 a^{-22} +z^2 a^{-24} -6 z a^{-13} -z a^{-15} +z a^{-17} -z a^{-19} +z a^{-21} -z a^{-23} +z a^{-25} +7 a^{-12} +6 a^{-14} }[/math]
The A2 invariant	Data:T(13,2)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(13,2)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(13,2) Quantum Invariants.

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["T(13,2)"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ t^6-t^5+t^4-t^3+t^2-t+1- t^{-1} + t^{-2} - t^{-3} + t^{-4} - t^{-5} + t^{-6} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^{12}+11 z^{10}+45 z^8+84 z^6+70 z^4+21 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]=

{ 13, 12 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ -q^{19}+q^{18}-q^{17}+q^{16}-q^{15}+q^{14}-q^{13}+q^{12}-q^{11}+q^{10}-q^9+q^8+q^6 }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ z^{12} a^{-12} +12 z^{10} a^{-12} -z^{10} a^{-14} +55 z^8 a^{-12} -10 z^8 a^{-14} +120 z^6 a^{-12} -36 z^6 a^{-14} +126 z^4 a^{-12} -56 z^4 a^{-14} +56 z^2 a^{-12} -35 z^2 a^{-14} +7 a^{-12} -6 a^{-14} }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ z^{12} a^{-12} +z^{12} a^{-14} +z^{11} a^{-13} +z^{11} a^{-15} -12 z^{10} a^{-12} -11 z^{10} a^{-14} +z^{10} a^{-16} -10 z^9 a^{-13} -9 z^9 a^{-15} +z^9 a^{-17} +55 z^8 a^{-12} +46 z^8 a^{-14} -8 z^8 a^{-16} +z^8 a^{-18} +36 z^7 a^{-13} +28 z^7 a^{-15} -7 z^7 a^{-17} +z^7 a^{-19} -120 z^6 a^{-12} -92 z^6 a^{-14} +21 z^6 a^{-16} -6 z^6 a^{-18} +z^6 a^{-20} -56 z^5 a^{-13} -35 z^5 a^{-15} +15 z^5 a^{-17} -5 z^5 a^{-19} +z^5 a^{-21} +126 z^4 a^{-12} +91 z^4 a^{-14} -20 z^4 a^{-16} +10 z^4 a^{-18} -4 z^4 a^{-20} +z^4 a^{-22} +35 z^3 a^{-13} +15 z^3 a^{-15} -10 z^3 a^{-17} +6 z^3 a^{-19} -3 z^3 a^{-21} +z^3 a^{-23} -56 z^2 a^{-12} -41 z^2 a^{-14} +5 z^2 a^{-16} -4 z^2 a^{-18} +3 z^2 a^{-20} -2 z^2 a^{-22} +z^2 a^{-24} -6 z a^{-13} -z a^{-15} +z a^{-17} -z a^{-19} +z a^{-21} -z a^{-23} +z a^{-25} +7 a^{-12} +6 a^{-14} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["T(13,2)"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ [math]\displaystyle{ t^6-t^5+t^4-t^3+t^2-t+1- t^{-1} + t^{-2} - t^{-3} + t^{-4} - t^{-5} + t^{-6} }[/math], [math]\displaystyle{ -q^{19}+q^{18}-q^{17}+q^{16}-q^{15}+q^{14}-q^{13}+q^{12}-q^{11}+q^{10}-q^9+q^8+q^6 }[/math] }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(21, 91)

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 Data:T(13,2)/V 2,1 Data:T(13,2)/V 3,1 Data:T(13,2)/V 4,1 Data:T(13,2)/V 4,2 Data:T(13,2)/V 4,3 Data:T(13,2)/V 5,1 Data:T(13,2)/V 5,2 Data:T(13,2)/V 5,3 Data:T(13,2)/V 5,4 Data:T(13,2)/V 6,1 Data:T(13,2)/V 6,2 Data:T(13,2)/V 6,3 Data:T(13,2)/V 6,4 Data:T(13,2)/V 6,5 Data:T(13,2)/V 6,6 Data:T(13,2)/V 6,7 Data:T(13,2)/V 6,8 Data:T(13,2)/V 6,9

V_2,1 through V_6,9 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]12 is the signature of T(13,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 χ 39                           1 -1 37                             0 35                       1 1   0 33                             0 31                   1 1       0 29                             0 27               1 1           0 25                             0 23           1 1               0 21                             0 19       1 1                   0 17                             0 15     1                       1 13 1                           1 11 1                           1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=11 }[/math] [math]\displaystyle{ i=13 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=9 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=11 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=12 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=13 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

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

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Torus Knot Page master template (intermediate). See/edit the Torus Knot_Splice_Base (expert). Back to the top.

T(11,2) T(7,3)