T(4,3)

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T(7,2)

T(9,2)

Contents

Image:T(4,3).jpg See other torus knots

Visit T(4,3)'s page at Knotilus!

Visit T(4,3)'s page at the original Knot Atlas!

Edit T(4,3) Quick Notes


Edit T(4,3) Further Notes and Views


[edit] Knot presentations

Planar diagram presentation X5,11,6,10 X16,12,1,11 X1726 X12,8,13,7 X13,3,14,2 X8493 X9,15,10,14 X4,16,5,15
Gauss code -3, 5, 6, -8, -1, 3, 4, -6, -7, 1, 2, -4, -5, 7, 8, -2
Dowker-Thistlethwaite code 6 -8 10 -12 14 -16 2 -4
Braid presentation
Image: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.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gif

[edit] Polynomial invariants

Alexander polynomial t3−t2 + 1−t−2 + t−3
Conway polynomial z6 + 5z4 + 5z2 + 1
2nd Alexander ideal (db, data sources) {1}
Determinant and Signature { 3, 6 }
Jones polynomial −q8 + q5 + q3
HOMFLY-PT polynomial (db, data sources) z6a−6 + 6z4a−6−z4a−8 + 10z2a−6−5z2a−8 + 5a−6−5a−8 + a−10
Kauffman polynomial (db, data sources) z6a−6 + z6a−8 + z5a−7 + z5a−9−6z4a−6−6z4a−8−5z3a−7−5z3a−9 + 10z2a−6 + 10z2a−8 + 5za−7 + 5za−9−5a−6−5a−8−a−10
The A2 invariant Data:T(4,3)/QuantumInvariant/A2/1,0
The G2 invariant Data:T(4,3)/QuantumInvariant/G2/1,0
Further Quantum Invariants
T(4,3) 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(4,3)"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= t3−t2 + 1−t−2 + t−3
In[5]:= Conway[K][z]
Out[5]= z6 + 5z4 + 5z2 + 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]= { 3, 6 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= −q8 + q5 + q3
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= z6a−6 + 6z4a−6−z4a−8 + 10z2a−6−5z2a−8 + 5a−6−5a−8 + a−10
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= z6a−6 + z6a−8 + z5a−7 + z5a−9−6z4a−6−6z4a−8−5z3a−7−5z3a−9 + 10z2a−6 + 10z2a−8 + 5za−7 + 5za−9−5a−6−5a−8−a−10

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

Same Alexander/Conway Polynomial: {8_19,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {8_19,}

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(4,3)"];
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]= { t3−t2 + 1−t−2 + t−3, −q8 + q5 + q3 }
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]= {8_19,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {8_19,}

[edit] Vassiliev invariants

V2 and V3: (5, 10)

[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 = 6 is the signature of T(4,3). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 012345χ
17     1-1
15     1-1
13   11 0
11    1 1
9  1   1
71     1
51     1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = 3 i = 5 i = 7
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}

[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(7,2)

T(9,2)

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