K11n141

K11n140

K11n142

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n141 at Knotilus!
	K11n141 is also known as the pretzel knot P(5,-3,-3).

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number [math]\displaystyle{ \{1,2,3\} }[/math] 3-genus 1 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n141/ThurstonBennequinNumber Hyperbolic Volume 7.80349 A-Polynomial See Data:K11n141/A-polynomial

[edit Notes for K11n141's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 1 }[/math] Rasmussen s-Invariant 0

[edit Notes for K11n141's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -5 t+11-5 t^{-1} }[/math]
Conway polynomial	[math]\displaystyle{ 1-5 z^2 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 21, 0 }
Jones polynomial	[math]\displaystyle{ 2 q^2-2 q+3-4 q^{-1} +3 q^{-2} -3 q^{-3} +2 q^{-4} - q^{-5} + q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^6-z^2 a^4-2 z^2 a^2-a^2-2 z^2-1+2 a^{-2} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^3 z^9+a z^9+a^4 z^8+2 a^2 z^8+z^8+a^5 z^7-6 a^3 z^7-7 a z^7+a^6 z^6-4 a^4 z^6-11 a^2 z^6-6 z^6-4 a^5 z^5+15 a^3 z^5+20 a z^5+z^5 a^{-1} -5 a^6 z^4+4 a^4 z^4+22 a^2 z^4+13 z^4+3 a^5 z^3-19 a^3 z^3-24 a z^3-2 z^3 a^{-1} +6 a^6 z^2-2 a^4 z^2-16 a^2 z^2+2 z^2 a^{-2} -6 z^2+8 a^3 z+11 a z+3 z a^{-1} -a^6+a^2-2 a^{-2} -1 }[/math]
The A2 invariant	Data:K11n141/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n141/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n141 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["K11n141"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -5 t+11-5 t^{-1} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 1-5 z^2 }[/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]=

{ 21, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ 2 q^2-2 q+3-4 q^{-1} +3 q^{-2} -3 q^{-3} +2 q^{-4} - 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^6-z^2 a^4-2 z^2 a^2-a^2-2 z^2-1+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+2 a^2 z^8+z^8+a^5 z^7-6 a^3 z^7-7 a z^7+a^6 z^6-4 a^4 z^6-11 a^2 z^6-6 z^6-4 a^5 z^5+15 a^3 z^5+20 a z^5+z^5 a^{-1} -5 a^6 z^4+4 a^4 z^4+22 a^2 z^4+13 z^4+3 a^5 z^3-19 a^3 z^3-24 a z^3-2 z^3 a^{-1} +6 a^6 z^2-2 a^4 z^2-16 a^2 z^2+2 z^2 a^{-2} -6 z^2+8 a^3 z+11 a z+3 z a^{-1} -a^6+a^2-2 a^{-2} -1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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["K11n141"];

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{ -5 t+11-5 t^{-1} }[/math], [math]\displaystyle{ 2 q^2-2 q+3-4 q^{-1} +3 q^{-2} -3 q^{-3} +2 q^{-4} - 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]=

{}

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:

(-5, 4)

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{ -20 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 200 }[/math] [math]\displaystyle{ \frac{746}{3} }[/math] [math]\displaystyle{ \frac{310}{3} }[/math] [math]\displaystyle{ -640 }[/math] [math]\displaystyle{ -\frac{3040}{3} }[/math] [math]\displaystyle{ -\frac{640}{3} }[/math] [math]\displaystyle{ -224 }[/math] [math]\displaystyle{ -\frac{4000}{3} }[/math] [math]\displaystyle{ 512 }[/math] [math]\displaystyle{ -\frac{14920}{3} }[/math] [math]\displaystyle{ -\frac{6200}{3} }[/math] [math]\displaystyle{ -\frac{19279}{6} }[/math] [math]\displaystyle{ 778 }[/math] [math]\displaystyle{ -\frac{30782}{9} }[/math] [math]\displaystyle{ \frac{9995}{18} }[/math] [math]\displaystyle{ -\frac{5071}{6} }[/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 K11n141. 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 5                 2 2 3                   0 1             3 2   1 -1           2 1     -1 -3         1 2       -1 -5       2 2         0 -7       1           -1 -9   1 2             1 -11                   0 -13 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=-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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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.

K11n140 K11n142