T(7,4)

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T(10,3).jpg

T(10,3)

T(21,2).jpg

T(21,2)

T(7,4).jpg See other torus knots

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Knot presentations

Planar diagram presentation X9,41,10,40 X20,42,21,41 X31,1,32,42 X21,11,22,10 X32,12,33,11 X1,13,2,12 X33,23,34,22 X2,24,3,23 X13,25,14,24 X3,35,4,34 X14,36,15,35 X25,37,26,36 X15,5,16,4 X26,6,27,5 X37,7,38,6 X27,17,28,16 X38,18,39,17 X7,19,8,18 X39,29,40,28 X8,30,9,29 X19,31,20,30
Gauss code -6, -8, -10, 13, 14, 15, -18, -20, -1, 4, 5, 6, -9, -11, -13, 16, 17, 18, -21, -2, -4, 7, 8, 9, -12, -14, -16, 19, 20, 21, -3, -5, -7, 10, 11, 12, -15, -17, -19, 1, 2, 3
Dowker-Thistlethwaite code 12 34 -26 18 40 -32 24 4 -38 30 10 -2 36 16 -8 42 22 -14 6 28 -20
Braid presentation
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Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^9-t^8+t^5-t^4+t^2-1+ t^{-2} - t^{-4} + t^{-5} - t^{-8} + t^{-9} }[/math]
Conway polynomial [math]\displaystyle{ z^{18}+17 z^{16}+119 z^{14}+442 z^{12}+936 z^{10}+1131 z^8+741 z^6+235 z^4+30 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 7, 14 }
Jones polynomial [math]\displaystyle{ -q^{18}-q^{16}+q^{15}-q^{14}+q^{13}+q^{11}+q^9 }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^{18} a^{-18} +18 z^{16} a^{-18} -z^{16} a^{-20} +136 z^{14} a^{-18} -17 z^{14} a^{-20} +561 z^{12} a^{-18} -120 z^{12} a^{-20} +z^{12} a^{-22} +1378 z^{10} a^{-18} -455 z^{10} a^{-20} +13 z^{10} a^{-22} +2067 z^8 a^{-18} -1002 z^8 a^{-20} +66 z^8 a^{-22} +1873 z^6 a^{-18} -1296 z^6 a^{-20} +165 z^6 a^{-22} -z^6 a^{-24} +981 z^4 a^{-18} -951 z^4 a^{-20} +211 z^4 a^{-22} -6 z^4 a^{-24} +270 z^2 a^{-18} -360 z^2 a^{-20} +130 z^2 a^{-22} -10 z^2 a^{-24} +30 a^{-18} -54 a^{-20} +30 a^{-22} -5 a^{-24} }[/math]
Kauffman polynomial (db, data sources) Data:T(7,4)/Kauffman Polynomial
The A2 invariant Data:T(7,4)/QuantumInvariant/A2/1,0
The G2 invariant Data:T(7,4)/QuantumInvariant/G2/1,0
Further Quantum Invariants
T(7,4) 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(7,4)"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^9-t^8+t^5-t^4+t^2-1+ t^{-2} - t^{-4} + t^{-5} - t^{-8} + t^{-9} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^{18}+17 z^{16}+119 z^{14}+442 z^{12}+936 z^{10}+1131 z^8+741 z^6+235 z^4+30 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]= { 7, 14 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^{18}-q^{16}+q^{15}-q^{14}+q^{13}+q^{11}+q^9 }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^{18} a^{-18} +18 z^{16} a^{-18} -z^{16} a^{-20} +136 z^{14} a^{-18} -17 z^{14} a^{-20} +561 z^{12} a^{-18} -120 z^{12} a^{-20} +z^{12} a^{-22} +1378 z^{10} a^{-18} -455 z^{10} a^{-20} +13 z^{10} a^{-22} +2067 z^8 a^{-18} -1002 z^8 a^{-20} +66 z^8 a^{-22} +1873 z^6 a^{-18} -1296 z^6 a^{-20} +165 z^6 a^{-22} -z^6 a^{-24} +981 z^4 a^{-18} -951 z^4 a^{-20} +211 z^4 a^{-22} -6 z^4 a^{-24} +270 z^2 a^{-18} -360 z^2 a^{-20} +130 z^2 a^{-22} -10 z^2 a^{-24} +30 a^{-18} -54 a^{-20} +30 a^{-22} -5 a^{-24} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= Data:T(7,4)/Kauffman Polynomial

"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(7,4)"];
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^9-t^8+t^5-t^4+t^2-1+ t^{-2} - t^{-4} + t^{-5} - t^{-8} + t^{-9} }[/math], [math]\displaystyle{ -q^{18}-q^{16}+q^{15}-q^{14}+q^{13}+q^{11}+q^9 }[/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: (30, 140)
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(7,4)/V 2,1 Data:T(7,4)/V 3,1 Data:T(7,4)/V 4,1 Data:T(7,4)/V 4,2 Data:T(7,4)/V 4,3 Data:T(7,4)/V 5,1 Data:T(7,4)/V 5,2 Data:T(7,4)/V 5,3 Data:T(7,4)/V 5,4 Data:T(7,4)/V 6,1 Data:T(7,4)/V 6,2 Data:T(7,4)/V 6,3 Data:T(7,4)/V 6,4 Data:T(7,4)/V 6,5 Data:T(7,4)/V 6,6 Data:T(7,4)/V 6,7 Data:T(7,4)/V 6,8 Data:T(7,4)/V 6,9

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 [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]14 is the signature of T(7,4). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 012345678910111213χ
39            110
37           1  -1
35           21 -1
33         21   -1
31       1  1   0
29     1 12     0
27     11       0
25   11 1       1
23    1         1
21  1           1
191             1
171             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{ i=15 }[/math] [math]\displaystyle{ i=17 }[/math] [math]\displaystyle{ i=19 }[/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{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=9 }[/math] [math]\displaystyle{ {\mathbb Z}_2\oplus{\mathbb Z}_4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math]
[math]\displaystyle{ r=11 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=12 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=13 }[/math] [math]\displaystyle{ {\mathbb Z}_4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=14 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/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).

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T(10,3).jpg

T(10,3)

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T(21,2)

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