K11n42

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K11n41.gif

K11n41

K11n43.gif

K11n43

K11n42.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n42 at Knotilus!

[edit K11n42 Quick Notes]

K11n42 is the mirror of the "Kinoshita-Terasaka" knot; it is a mutant of the (mirror of the) Conway knot K11n34. See also Heegaard Floer Knot Homology.

Warning: There are two different conventions on whether K11n42 is the "Kinoshita-Terasaka" knot or its mirror. We caution the reader that the invariants below may have been computed using either convention.


[edit K11n42 Further Notes and Views]

K11n42 is not [math]\displaystyle{ k }[/math]-colourable for any [math]\displaystyle{ k }[/math]. See The Determinant and the Signature.

Knot K11n42.
A graph, knot K11n42.
A part of a knot and a part of a graph.

Knot presentations

Planar diagram presentation X4251 X8493 X12,5,13,6 X2837 X9,18,10,19 X11,21,12,20 X6,13,7,14 X15,10,16,11 X17,22,18,1 X19,15,20,14 X21,16,22,17
Gauss code 1, -4, 2, -1, 3, -7, 4, -2, -5, 8, -6, -3, 7, 10, -8, 11, -9, 5, -10, 6, -11, 9
Dowker-Thistlethwaite code 4 8 12 2 -18 -20 6 -10 -22 -14 -16
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation K11n42 ML.gif

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [math]\displaystyle{ 0 }[/math]
Rasmussen s-Invariant 0

[edit Notes for K11n42's four dimensional invariants]

Polynomial invariants

Alexander polynomial 1
Conway polynomial 1
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 1, 0 }
Jones polynomial [math]\displaystyle{ -q^4+2 q^3-2 q^2+2 q+ q^{-2} -2 q^{-3} +2 q^{-4} -2 q^{-5} + q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -a^2 z^6+z^6+a^4 z^4-6 a^2 z^4-z^4 a^{-2} +6 z^4+3 a^4 z^2-11 a^2 z^2-3 z^2 a^{-2} +11 z^2+2 a^4-6 a^2-2 a^{-2} +7 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a z^9+z^9 a^{-1} +a^4 z^8+2 a^2 z^8+2 z^8 a^{-2} +3 z^8+2 a^5 z^7+2 a^3 z^7-5 a z^7-4 z^7 a^{-1} +z^7 a^{-3} +a^6 z^6-4 a^4 z^6-14 a^2 z^6-11 z^6 a^{-2} -20 z^6-9 a^5 z^5-12 a^3 z^5-2 z^5 a^{-1} -5 z^5 a^{-3} -4 a^6 z^4+2 a^4 z^4+26 a^2 z^4+16 z^4 a^{-2} +36 z^4+9 a^5 z^3+16 a^3 z^3+12 a z^3+11 z^3 a^{-1} +6 z^3 a^{-3} +3 a^6 z^2-2 a^4 z^2-20 a^2 z^2-9 z^2 a^{-2} -24 z^2-3 a^5 z-7 a^3 z-7 a z-5 z a^{-1} -2 z a^{-3} +2 a^4+6 a^2+2 a^{-2} +7 }[/math]
The A2 invariant [math]\displaystyle{ q^{18}+q^{14}-q^{12}-q^{10}-q^8-2 q^6+q^4+3+2 q^{-2} + q^{-4} + q^{-6} - q^{-8} - q^{-12} }[/math]
The G2 invariant Data:K11n42/QuantumInvariant/G2/1,0
Further Quantum Invariants
K11n42 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["K11n42"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= 1
In[5]:= Conway[K][z]
Out[5]= 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]= [math]\displaystyle{ \{1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 1, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^4+2 q^3-2 q^2+2 q+ q^{-2} -2 q^{-3} +2 q^{-4} -2 q^{-5} + q^{-6} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -a^2 z^6+z^6+a^4 z^4-6 a^2 z^4-z^4 a^{-2} +6 z^4+3 a^4 z^2-11 a^2 z^2-3 z^2 a^{-2} +11 z^2+2 a^4-6 a^2-2 a^{-2} +7 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ a z^9+z^9 a^{-1} +a^4 z^8+2 a^2 z^8+2 z^8 a^{-2} +3 z^8+2 a^5 z^7+2 a^3 z^7-5 a z^7-4 z^7 a^{-1} +z^7 a^{-3} +a^6 z^6-4 a^4 z^6-14 a^2 z^6-11 z^6 a^{-2} -20 z^6-9 a^5 z^5-12 a^3 z^5-2 z^5 a^{-1} -5 z^5 a^{-3} -4 a^6 z^4+2 a^4 z^4+26 a^2 z^4+16 z^4 a^{-2} +36 z^4+9 a^5 z^3+16 a^3 z^3+12 a z^3+11 z^3 a^{-1} +6 z^3 a^{-3} +3 a^6 z^2-2 a^4 z^2-20 a^2 z^2-9 z^2 a^{-2} -24 z^2-3 a^5 z-7 a^3 z-7 a z-5 z a^{-1} -2 z a^{-3} +2 a^4+6 a^2+2 a^{-2} +7 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {0_1, K11n34,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11n34,}

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["K11n42"];
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]= { 1, [math]\displaystyle{ -q^4+2 q^3-2 q^2+2 q+ q^{-2} -2 q^{-3} +2 q^{-4} -2 q^{-5} + q^{-6} }[/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]= {0_1, K11n34,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {K11n34,}

Vassiliev invariants

V2 and V3: (0, 2)
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
[math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{128}{3} }[/math] [math]\displaystyle{ -\frac{128}{3} }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 312 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 176 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 8 }[/math]

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]0 is the signature of K11n42. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -6-5-4-3-2-1012345χ
9           1-1
7          1 1
5         11 0
3       121  0
1      211   2
-1     132    0
-3    221     1
-5   111      -1
-7  121       0
-9 11         0
-11 1          -1
-131           1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math]
[math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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 Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

Back to the top.

K11n41.gif

K11n41

K11n43.gif

K11n43

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