T(11,2)

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T(5,3)

T(13,2)

Contents

Image:T(11,2).jpg See other torus knots

Visit T(11,2)'s page at Knotilus!

Visit T(11,2)'s page at the original Knot Atlas!

Edit T(11,2) Quick Notes


Edit T(11,2) Further Notes and Views


[edit] Knot presentations

Planar diagram presentation X5,17,6,16 X17,7,18,6 X7,19,8,18 X19,9,20,8 X9,21,10,20 X21,11,22,10 X11,1,12,22 X1,13,2,12 X13,3,14,2 X3,15,4,14 X15,5,16,4
Gauss code -8, 9, -10, 11, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 1, -2, 3, -4, 5, -6, 7
Dowker-Thistlethwaite code 12 14 16 18 20 22 2 4 6 8 10
Braid presentation
Image:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart1.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart2.gif

[edit] Polynomial invariants

Alexander polynomial t5−t4 + t3−t2 + t−1 + t−1−t−2 + t−3−t−4 + t−5
Conway polynomial z10 + 9z8 + 28z6 + 35z4 + 15z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 11, 10 }
Jones polynomial −q16 + q15−q14 + q13−q12 + q11−q10 + q9−q8 + q7 + q5
HOMFLY-PT polynomial (db, data sources) z10a−10 + 10z8a−10−z8a−12 + 36z6a−10−8z6a−12 + 56z4a−10−21z4a−12 + 35z2a−10−20z2a−12 + 6a−10−5a−12
Kauffman polynomial (db, data sources) z10a−10 + z10a−12 + z9a−11 + z9a−13−10z8a−10−9z8a−12 + z8a−14−8z7a−11−7z7a−13 + z7a−15 + 36z6a−10 + 29z6a−12−6z6a−14 + z6a−16 + 21z5a−11 + 15z5a−13−5z5a−15 + z5a−17−56z4a−10−41z4a−12 + 10z4a−14−4z4a−16 + z4a−18−20z3a−11−10z3a−13 + 6z3a−15−3z3a−17 + z3a−19 + 35z2a−10 + 25z2a−12−4z2a−14 + 3z2a−16−2z2a−18 + z2a−20 + 5za−11 + za−13−za−15 + za−17−za−19 + za−21−6a−10−5a−12
The A2 invariant Data:T(11,2)/QuantumInvariant/A2/1,0
The G2 invariant Data:T(11,2)/QuantumInvariant/G2/1,0
Further Quantum Invariants
T(11,2) 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(11,2)"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t5−t4 + t3−t2 + t−1 + t−1−t−2 + t−3−t−4 + t−5
In[5]:= Conway[K][z]
Out[5]= z10 + 9z8 + 28z6 + 35z4 + 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]= { 11, 10 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q16 + q15−q14 + q13−q12 + q11−q10 + q9−q8 + q7 + q5
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= z10a−10 + 10z8a−10−z8a−12 + 36z6a−10−8z6a−12 + 56z4a−10−21z4a−12 + 35z2a−10−20z2a−12 + 6a−10−5a−12
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z10a−10 + z10a−12 + z9a−11 + z9a−13−10z8a−10−9z8a−12 + z8a−14−8z7a−11−7z7a−13 + z7a−15 + 36z6a−10 + 29z6a−12−6z6a−14 + z6a−16 + 21z5a−11 + 15z5a−13−5z5a−15 + z5a−17−56z4a−10−41z4a−12 + 10z4a−14−4z4a−16 + z4a−18−20z3a−11−10z3a−13 + 6z3a−15−3z3a−17 + z3a−19 + 35z2a−10 + 25z2a−12−4z2a−14 + 3z2a−16−2z2a−18 + z2a−20 + 5za−11 + za−13−za−15 + za−17−za−19 + za−21−6a−10−5a−12

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

Same Alexander/Conway Polynomial: {K11a367,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a367,}

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(11,2)"];
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]= { t5−t4 + t3−t2 + t−1 + t−1−t−2 + t−3−t−4 + t−5, −q16 + q15−q14 + q13−q12 + q11−q10 + q9−q8 + q7 + q5 }
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]= {K11a367,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {K11a367,}

[edit] Vassiliev invariants

V2 and V3: (15, 55)

[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 = 10 is the signature of T(11,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 01234567891011χ
33           1-1
31            0
29         11 0
27            0
25       11   0
23            0
21     11     0
19            0
17   11       0
15            0
13  1         1
111           1
91           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = 9 i = 11
r = 0 {\mathbb Z} {\mathbb Z}
r = 1
r = 2 {\mathbb Z}
r = 3 {\mathbb Z}_2 {\mathbb Z}
r = 4 {\mathbb Z}
r = 5 {\mathbb Z}_2 {\mathbb Z}
r = 6 {\mathbb Z}
r = 7 {\mathbb Z}_2 {\mathbb Z}
r = 8 {\mathbb Z}
r = 9 {\mathbb Z}_2 {\mathbb Z}
r = 10 {\mathbb Z}
r = 11 {\mathbb Z}_2 {\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(5,3)

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