K11a132

K11a131

K11a133

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a132 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X16,6,17,5 X12,8,13,7 X18,9,19,10 X2,11,3,12 X20,14,21,13 X6,16,7,15 X22,18,1,17 X8,19,9,20 X14,22,15,21
Gauss code	1, -6, 2, -1, 3, -8, 4, -10, 5, -2, 6, -4, 7, -11, 8, -3, 9, -5, 10, -7, 11, -9
Dowker-Thistlethwaite code	4 10 16 12 18 2 20 6 22 8 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number [math]\displaystyle{ \{1,2\} }[/math] 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a132/ThurstonBennequinNumber Hyperbolic Volume 16.7788 A-Polynomial See Data:K11a132/A-polynomial

[edit Notes for K11a132's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a132's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^3-13 t^2+32 t-41+32 t^{-1} -13 t^{-2} +2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^6-z^4-2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \left\{2,t^2+t+1\right\} }[/math]
Determinant and Signature	{ 135, 2 }
Jones polynomial	[math]\displaystyle{ -q^8+4 q^7-9 q^6+15 q^5-19 q^4+22 q^3-22 q^2+18 q-13+8 q^{-1} -3 q^{-2} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-2} +z^6 a^{-4} +2 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2-4 z^2 a^{-2} +4 z^2 a^{-4} -z^2 a^{-6} -2 z^2+a^2-4 a^{-2} +4 a^{-4} - a^{-6} +1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10} a^{-4} +5 z^9 a^{-1} +12 z^9 a^{-3} +7 z^9 a^{-5} +9 z^8 a^{-2} +14 z^8 a^{-4} +10 z^8 a^{-6} +5 z^8+3 a z^7-7 z^7 a^{-1} -23 z^7 a^{-3} -5 z^7 a^{-5} +8 z^7 a^{-7} +a^2 z^6-30 z^6 a^{-2} -40 z^6 a^{-4} -17 z^6 a^{-6} +4 z^6 a^{-8} -10 z^6-7 a z^5+2 z^5 a^{-1} +17 z^5 a^{-3} -5 z^5 a^{-5} -12 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+33 z^4 a^{-2} +42 z^4 a^{-4} +12 z^4 a^{-6} -5 z^4 a^{-8} +5 z^4+4 a z^3-4 z^3 a^{-1} -8 z^3 a^{-3} +6 z^3 a^{-5} +5 z^3 a^{-7} -z^3 a^{-9} +3 a^2 z^2-20 z^2 a^{-2} -21 z^2 a^{-4} -5 z^2 a^{-6} +z^2 a^{-8} -2 z^2+3 z a^{-1} +3 z a^{-3} -z a^{-5} -z a^{-7} -a^2+4 a^{-2} +4 a^{-4} + a^{-6} +1 }[/math]
The A2 invariant	Data:K11a132/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a132/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a132 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["K11a132"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^3-13 t^2+32 t-41+32 t^{-1} -13 t^{-2} +2 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 z^6-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{ \left\{2,t^2+t+1\right\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 135, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^8+4 q^7-9 q^6+15 q^5-19 q^4+22 q^3-22 q^2+18 q-13+8 q^{-1} -3 q^{-2} + q^{-3} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^6 a^{-2} +z^6 a^{-4} +2 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2-4 z^2 a^{-2} +4 z^2 a^{-4} -z^2 a^{-6} -2 z^2+a^2-4 a^{-2} +4 a^{-4} - a^{-6} +1 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10} a^{-4} +5 z^9 a^{-1} +12 z^9 a^{-3} +7 z^9 a^{-5} +9 z^8 a^{-2} +14 z^8 a^{-4} +10 z^8 a^{-6} +5 z^8+3 a z^7-7 z^7 a^{-1} -23 z^7 a^{-3} -5 z^7 a^{-5} +8 z^7 a^{-7} +a^2 z^6-30 z^6 a^{-2} -40 z^6 a^{-4} -17 z^6 a^{-6} +4 z^6 a^{-8} -10 z^6-7 a z^5+2 z^5 a^{-1} +17 z^5 a^{-3} -5 z^5 a^{-5} -12 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+33 z^4 a^{-2} +42 z^4 a^{-4} +12 z^4 a^{-6} -5 z^4 a^{-8} +5 z^4+4 a z^3-4 z^3 a^{-1} -8 z^3 a^{-3} +6 z^3 a^{-5} +5 z^3 a^{-7} -z^3 a^{-9} +3 a^2 z^2-20 z^2 a^{-2} -21 z^2 a^{-4} -5 z^2 a^{-6} +z^2 a^{-8} -2 z^2+3 z a^{-1} +3 z a^{-3} -z a^{-5} -z a^{-7} -a^2+4 a^{-2} +4 a^{-4} + a^{-6} +1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a6, K11a352,}

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

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["K11a132"];

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^3-13 t^2+32 t-41+32 t^{-1} -13 t^{-2} +2 t^{-3} }[/math], [math]\displaystyle{ -q^8+4 q^7-9 q^6+15 q^5-19 q^4+22 q^3-22 q^2+18 q-13+8 q^{-1} -3 q^{-2} + q^{-3} }[/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]=

{K11a6, K11a352,}

In[6]:=

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

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

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(-2, 0)

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{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{212}{3} }[/math] [math]\displaystyle{ \frac{76}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -\frac{256}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{1696}{3} }[/math] [math]\displaystyle{ -\frac{608}{3} }[/math] [math]\displaystyle{ -\frac{9511}{15} }[/math] [math]\displaystyle{ -\frac{1796}{15} }[/math] [math]\displaystyle{ -\frac{14524}{45} }[/math] [math]\displaystyle{ \frac{391}{9} }[/math] [math]\displaystyle{ -\frac{871}{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 K11a132. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     3   3 13                   6 1   -5 11                 9 3     6 9               10 6       -4 7             12 9         3 5           10 10           0 3         8 12             -4 1       6 11               5 -1     2 7                 -5 -3   1 6                   5 -5   2                     -2 -7 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=3 }[/math] [math]\displaystyle{ r=-4 }[/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}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=7 }[/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.

K11a131 K11a133