K11n138

K11n137

K11n139

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n138 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X12,3,13,4 X5,16,6,17 X7,14,8,15 X9,19,10,18 X11,21,12,20 X2,13,3,14 X15,6,16,7 X17,22,18,1 X19,11,20,10 X21,9,22,8
Gauss code	1, -7, 2, -1, -3, 8, -4, 11, -5, 10, -6, -2, 7, 4, -8, 3, -9, 5, -10, 6, -11, 9
Dowker-Thistlethwaite code	4 12 -16 -14 -18 -20 2 -6 -22 -10 -8

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number [math]\displaystyle{ \{1,2\} }[/math] 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n138/ThurstonBennequinNumber Hyperbolic Volume 7.77671 A-Polynomial See Data:K11n138/A-polynomial

[edit Notes for K11n138's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 2 }[/math] Rasmussen s-Invariant 0

[edit Notes for K11n138's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t^2+4 t-3+4 t^{-1} -2 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -2 z^4-4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 15, 2 }
Jones polynomial	[math]\displaystyle{ q^3-q^2+2 q-2+2 q^{-1} -3 q^{-2} +2 q^{-3} - q^{-4} + q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^2 a^4+2 a^4-z^4 a^2-3 z^2 a^2-2 a^2-z^4-3 z^2-1+z^2 a^{-2} +2 a^{-2} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^3 z^9+a z^9+a^4 z^8+3 a^2 z^8+2 z^8-6 a^3 z^7-4 a z^7+2 z^7 a^{-1} -7 a^4 z^6-18 a^2 z^6+z^6 a^{-2} -10 z^6+11 a^3 z^5+a z^5-10 z^5 a^{-1} +16 a^4 z^4+32 a^2 z^4-5 z^4 a^{-2} +11 z^4-8 a^3 z^3+3 a z^3+11 z^3 a^{-1} -13 a^4 z^2-19 a^2 z^2+5 z^2 a^{-2} -z^2+3 a^3 z+a z-3 z a^{-1} -z a^{-3} +2 a^4+2 a^2-2 a^{-2} -1 }[/math]
The A2 invariant	Data:K11n138/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n138/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n138 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["K11n138"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ -2 t^2+4 t-3+4 t^{-1} -2 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -2 z^4-4 z^2+1 }[/math]

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]=

{ 15, 2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ q^3-q^2+2 q-2+2 q^{-1} -3 q^{-2} +2 q^{-3} - q^{-4} + q^{-5} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ z^2 a^4+2 a^4-z^4 a^2-3 z^2 a^2-2 a^2-z^4-3 z^2-1+z^2 a^{-2} +2 a^{-2} }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ a^3 z^9+a z^9+a^4 z^8+3 a^2 z^8+2 z^8-6 a^3 z^7-4 a z^7+2 z^7 a^{-1} -7 a^4 z^6-18 a^2 z^6+z^6 a^{-2} -10 z^6+11 a^3 z^5+a z^5-10 z^5 a^{-1} +16 a^4 z^4+32 a^2 z^4-5 z^4 a^{-2} +11 z^4-8 a^3 z^3+3 a z^3+11 z^3 a^{-1} -13 a^4 z^2-19 a^2 z^2+5 z^2 a^{-2} -z^2+3 a^3 z+a z-3 z a^{-1} -z a^{-3} +2 a^4+2 a^2-2 a^{-2} -1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n79,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11n79,}

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["K11n138"];

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]=

{ [math]\displaystyle{ -2 t^2+4 t-3+4 t^{-1} -2 t^{-2} }[/math], [math]\displaystyle{ q^3-q^2+2 q-2+2 q^{-1} -3 q^{-2} +2 q^{-3} - q^{-4} + q^{-5} }[/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]=

{K11n79,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11n79,}

Vassiliev invariants

V2 and V3:

(-4, 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{ -16 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{568}{3} }[/math] [math]\displaystyle{ \frac{248}{3} }[/math] [math]\displaystyle{ -256 }[/math] [math]\displaystyle{ -\frac{1376}{3} }[/math] [math]\displaystyle{ -\frac{224}{3} }[/math] [math]\displaystyle{ -144 }[/math] [math]\displaystyle{ -\frac{2048}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ -\frac{9088}{3} }[/math] [math]\displaystyle{ -\frac{3968}{3} }[/math] [math]\displaystyle{ -\frac{37382}{15} }[/math] [math]\displaystyle{ \frac{2936}{5} }[/math] [math]\displaystyle{ -\frac{115208}{45} }[/math] [math]\displaystyle{ \frac{3494}{9} }[/math] [math]\displaystyle{ -\frac{9062}{15} }[/math]

V_2,1 through V_6,9 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]2 is the signature of K11n138. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 7                 1 1 5                   0 3             2 1   1 1           1 1     0 -1         1 2 1     0 -3       2 1         -1 -5       1           -1 -7   1 2             1 -9                   0 -11 1                 1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/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} }[/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} }[/math] [math]\displaystyle{ r=2 }[/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.

K11n137 K11n139