K11n132

K11n131

K11n133

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n132 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X5,19,6,18 X7,16,8,17 X9,14,10,15 X2,11,3,12 X13,20,14,21 X15,8,16,9 X17,1,18,22 X19,12,20,13 X21,7,22,6
Gauss code	1, -6, 2, -1, -3, 11, -4, 8, -5, -2, 6, 10, -7, 5, -8, 4, -9, 3, -10, 7, -11, 9
Dowker-Thistlethwaite code	4 10 -18 -16 -14 2 -20 -8 -22 -12 -6

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral 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:K11n132/ThurstonBennequinNumber Hyperbolic Volume 10.2668 A-Polynomial See Data:K11n132/A-polynomial

[edit Notes for K11n132's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n132's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^2-6 t+9-6 t^{-1} +2 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 25, 0 }
Jones polynomial	[math]\displaystyle{ -q+3-3 q^{-1} +4 q^{-2} -4 q^{-3} +4 q^{-4} -3 q^{-5} +2 q^{-6} - q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^6-a^6+z^4 a^4+2 z^2 a^4+a^4+z^4 a^2+2 z^2 a^2+a^2-z^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^5 z^9+a^3 z^9+2 a^6 z^8+3 a^4 z^8+a^2 z^8+a^7 z^7-3 a^5 z^7-4 a^3 z^7-10 a^6 z^6-13 a^4 z^6-3 a^2 z^6-5 a^7 z^5-a^5 z^5+6 a^3 z^5+2 a z^5+14 a^6 z^4+15 a^4 z^4+a^2 z^4+7 a^7 z^3+4 a^5 z^3-8 a^3 z^3-5 a z^3-6 a^6 z^2-7 a^4 z^2+a^2 z^2+2 z^2-2 a^7 z-a^5 z+3 a^3 z+3 a z+z a^{-1} +a^6+a^4-a^2 }[/math]
The A2 invariant	Data:K11n132/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n132/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n132 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["K11n132"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^2-6 t+9-6 t^{-1} +2 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 z^4+2 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]=

{ 25, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q+3-3 q^{-1} +4 q^{-2} -4 q^{-3} +4 q^{-4} -3 q^{-5} +2 q^{-6} - q^{-7} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^2 a^6-a^6+z^4 a^4+2 z^2 a^4+a^4+z^4 a^2+2 z^2 a^2+a^2-z^2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^5 z^9+a^3 z^9+2 a^6 z^8+3 a^4 z^8+a^2 z^8+a^7 z^7-3 a^5 z^7-4 a^3 z^7-10 a^6 z^6-13 a^4 z^6-3 a^2 z^6-5 a^7 z^5-a^5 z^5+6 a^3 z^5+2 a z^5+14 a^6 z^4+15 a^4 z^4+a^2 z^4+7 a^7 z^3+4 a^5 z^3-8 a^3 z^3-5 a z^3-6 a^6 z^2-7 a^4 z^2+a^2 z^2+2 z^2-2 a^7 z-a^5 z+3 a^3 z+3 a z+z a^{-1} +a^6+a^4-a^2 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_8, 10_129, K11n39, K11n45, K11n50,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11n50, K11n133,}

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["K11n132"];

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^2-6 t+9-6 t^{-1} +2 t^{-2} }[/math], [math]\displaystyle{ -q+3-3 q^{-1} +4 q^{-2} -4 q^{-3} +4 q^{-4} -3 q^{-5} +2 q^{-6} - q^{-7} }[/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]=

{8_8, 10_129, K11n39, K11n45, K11n50,}

In[6]:=

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

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

Out[6]=

{K11n50, K11n133,}

Vassiliev invariants

V2 and V3:

(2, -3)

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{ 8 }[/math] [math]\displaystyle{ -24 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{220}{3} }[/math] [math]\displaystyle{ -\frac{4}{3} }[/math] [math]\displaystyle{ -192 }[/math] [math]\displaystyle{ -336 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -88 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ \frac{1760}{3} }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ \frac{20431}{15} }[/math] [math]\displaystyle{ -\frac{1444}{15} }[/math] [math]\displaystyle{ \frac{23524}{45} }[/math] [math]\displaystyle{ \frac{689}{9} }[/math] [math]\displaystyle{ \frac{511}{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]0 is the signature of K11n132. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 χ 3                 1 -1 1               2   2 -1             2 2   0 -3           2 1     1 -5         2 2       0 -7       2 2         0 -9     1 2           1 -11   1 2             -1 -13   1               1 -15 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{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/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.

K11n131 K11n133