T(5,4)

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T(7,3)

T(15,2)

Contents

Image:T(5,4).jpg See other torus knots

Visit T(5,4)'s page at Knotilus!

Visit T(5,4)'s page at the original Knot Atlas!

Edit T(5,4) Quick Notes


Edit T(5,4) Further Notes and Views


[edit] Knot presentations

Planar diagram presentation X17,25,18,24 X10,26,11,25 X3,27,4,26 X11,19,12,18 X4,20,5,19 X27,21,28,20 X5,13,6,12 X28,14,29,13 X21,15,22,14 X29,7,30,6 X22,8,23,7 X15,9,16,8 X23,1,24,30 X16,2,17,1 X9,3,10,2
Gauss code 14, 15, -3, -5, -7, 10, 11, 12, -15, -2, -4, 7, 8, 9, -12, -14, -1, 4, 5, 6, -9, -11, -13, 1, 2, 3, -6, -8, -10, 13
Dowker-Thistlethwaite code 16 -26 -12 22 -2 -18 28 -8 -24 4 -14 -30 10 -20 -6
Braid presentation
Image:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart2.gifImage:BraidPart1.gifImage:BraidPart2.gifImage:BraidPart1.gif
Image:BraidPart0.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 + t2−1 + t−2−t−5 + t−6
Conway polynomial z12 + 11z10 + 44z8 + 77z6 + 56z4 + 15z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 5, 8 }
Jones polynomial −q13−q11 + q10 + q8 + q6
HOMFLY-PT polynomial (db, data sources) z12a−12 + 12z10a−12−z10a−14 + 55z8a−12−11z8a−14 + 121z6a−12−45z6a−14 + z6a−16 + 133z4a−12−84z4a−14 + 7z4a−16 + 70z2a−12−70z2a−14 + 15z2a−16 + 14a−12−21a−14 + 9a−16−a−18
Kauffman polynomial (db, data sources) z12a−12 + z12a−14 + z11a−13 + z11a−15−12z10a−12−12z10a−14−11z9a−13−11z9a−15 + 55z8a−12 + 56z8a−14 + z8a−16 + 45z7a−13 + 46z7a−15 + z7a−17−121z6a−12−129z6a−14−8z6a−16−84z5a−13−91z5a−15−7z5a−17 + 133z4a−12 + 154z4a−14 + 21z4a−16 + 70z3a−13 + 84z3a−15 + 14z3a−17−70z2a−12−91z2a−14−22z2a−16−z2a−18−21za−13−28za−15−8za−17−za−19 + 14a−12 + 21a−14 + 9a−16 + a−18
The A2 invariant Data:T(5,4)/QuantumInvariant/A2/1,0
The G2 invariant Data:T(5,4)/QuantumInvariant/G2/1,0
Further Quantum Invariants
T(5,4) 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(5,4)"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t6−t5 + t2−1 + t−2−t−5 + t−6
In[5]:= Conway[K][z]
Out[5]= z12 + 11z10 + 44z8 + 77z6 + 56z4 + 15z2 + 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]= { 5, 8 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q13−q11 + q10 + 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−12−45z6a−14 + z6a−16 + 133z4a−12−84z4a−14 + 7z4a−16 + 70z2a−12−70z2a−14 + 15z2a−16 + 14a−12−21a−14 + 9a−16−a−18
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 + 56z8a−14 + z8a−16 + 45z7a−13 + 46z7a−15 + z7a−17−121z6a−12−129z6a−14−8z6a−16−84z5a−13−91z5a−15−7z5a−17 + 133z4a−12 + 154z4a−14 + 21z4a−16 + 70z3a−13 + 84z3a−15 + 14z3a−17−70z2a−12−91z2a−14−22z2a−16−z2a−18−21za−13−28za−15−8za−17−za−19 + 14a−12 + 21a−14 + 9a−16 + a−18

[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(5,4)"];
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]= { t6−t5 + t2−1 + t−2−t−5 + t−6, −q13−q11 + q10 + 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: (15, 50)

[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(5,4). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 0123456789χ
27         1-1
25       1  -1
23     1 11 -1
21     11   0
19   11 1   1
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} {\mathbb Z}
r = 7 {\mathbb Z}_2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r = 8 {\mathbb Z}
r = 9 {\mathbb Z}_4 {\mathbb Z}
r = 10 {\mathbb Z}_2 {\mathbb Z}_2

[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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