T(7,3)

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T(13,2)

T(5,4)

Contents

Image:T(7,3).jpg See other torus knots

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Visit T(7,3)'s page at the original Knot Atlas!

Edit T(7,3) Quick Notes


Edit T(7,3) Further Notes and Views


[edit] Knot presentations

Planar diagram presentation X1,11,2,10 X20,12,21,11 X21,3,22,2 X12,4,13,3 X13,23,14,22 X4,24,5,23 X5,15,6,14 X24,16,25,15 X25,7,26,6 X16,8,17,7 X17,27,18,26 X8,28,9,27 X9,19,10,18 X28,20,1,19
Gauss code -1, 3, 4, -6, -7, 9, 10, -12, -13, 1, 2, -4, -5, 7, 8, -10, -11, 13, 14, -2, -3, 5, 6, -8, -9, 11, 12, -14
Dowker-Thistlethwaite code 10 -12 14 -16 18 -20 22 -24 26 -28 2 -4 6 -8
Braid presentation
Image:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart2.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gif

[edit] Polynomial invariants

Alexander polynomial t6−t5 + t3−t2 + 1−t−2 + t−3−t−5 + t−6
Conway polynomial z12 + 11z10 + 44z8 + 78z6 + 60z4 + 16z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 1, 8 }
Jones polynomial −q14 + q8 + q6
HOMFLY-PT polynomial (db, data sources) z12a−12 + 12z10a−12−z10a−14 + 55z8a−12−11z8a−14 + 121z6a−12−44z6a−14 + z6a−16 + 132z4a−12−78z4a−14 + 6z4a−16 + 66z2a−12−60z2a−14 + 10z2a−16 + 12a−12−16a−14 + 5a−16
Kauffman polynomial (db, data sources) z12a−12 + z12a−14 + z11a−13 + z11a−15−12z10a−12−12z10a−14−11z9a−13−11z9a−15 + 55z8a−12 + 55z8a−14 + 44z7a−13 + 44z7a−15−121z6a−12−122z6a−14−z6a−16−78z5a−13−78z5a−15 + 132z4a−12 + 138z4a−14 + 6z4a−16 + 60z3a−13 + 60z3a−15−66z2a−12−76z2a−14−10z2a−16−16za−13−16za−15 + 12a−12 + 16a−14 + 5a−16
The A2 invariant Data:T(7,3)/QuantumInvariant/A2/1,0
The G2 invariant Data:T(7,3)/QuantumInvariant/G2/1,0
Further Quantum Invariants
T(7,3) 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(7,3)"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t6−t5 + t3−t2 + 1−t−2 + t−3−t−5 + t−6
In[5]:= Conway[K][z]
Out[5]= z12 + 11z10 + 44z8 + 78z6 + 60z4 + 16z2 + 1
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]= {1}
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 1, 8 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q14 + q8 + q6
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= z12a−12 + 12z10a−12−z10a−14 + 55z8a−12−11z8a−14 + 121z6a−12−44z6a−14 + z6a−16 + 132z4a−12−78z4a−14 + 6z4a−16 + 66z2a−12−60z2a−14 + 10z2a−16 + 12a−12−16a−14 + 5a−16
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z12a−12 + z12a−14 + z11a−13 + z11a−15−12z10a−12−12z10a−14−11z9a−13−11z9a−15 + 55z8a−12 + 55z8a−14 + 44z7a−13 + 44z7a−15−121z6a−12−122z6a−14−z6a−16−78z5a−13−78z5a−15 + 132z4a−12 + 138z4a−14 + 6z4a−16 + 60z3a−13 + 60z3a−15−66z2a−12−76z2a−14−10z2a−16−16za−13−16za−15 + 12a−12 + 16a−14 + 5a−16

[edit] "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {}

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(7,3)"];
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]= { t6−t5 + t3−t2 + 1−t−2 + t−3−t−5 + t−6, −q14 + q8 + q6 }
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]= {}

[edit] Vassiliev invariants

V2 and V3: (16, 56)

[edit] Khovanov Homology

The coefficients of the monomials trqj are shown, along with their alternating sums χ (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j−2r = s + 1 or j−2r = s−1, where s = 8 is the signature of T(7,3). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 0123456789χ
29         1-1
27         1-1
25       11 0
23     1  1 0
21     11   0
19   11     0
17    1     1
15  1       1
131         1
111         1
Integral Khovanov Homology

(db, data source)

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

[edit] Computer Talk

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


[edit] 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(13,2)

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