K11a225

K11a224

K11a226

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a225 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a225/ThurstonBennequinNumber Hyperbolic Volume 9.36224 A-Polynomial See Data:K11a225/A-polynomial

[edit Notes for K11a225's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 3 }[/math] Rasmussen s-Invariant 4

[edit Notes for K11a225's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t^3+8 t^2-11 t+11-11 t^{-1} +8 t^{-2} -2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -2 z^6-4 z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 53, -4 }
Jones polynomial	[math]\displaystyle{ q^2-2 q+3-5 q^{-1} +7 q^{-2} -7 q^{-3} +8 q^{-4} -7 q^{-5} +6 q^{-6} -4 q^{-7} +2 q^{-8} - q^{-9} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^8-2 a^8+2 z^4 a^6+6 z^2 a^6+3 a^6-z^6 a^4-3 z^4 a^4-z^2 a^4-z^6 a^2-4 z^4 a^2-4 z^2 a^2-a^2+z^4+3 z^2+1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^3 a^{11}-z a^{11}+2 z^4 a^{10}-z^2 a^{10}+3 z^5 a^9-2 z^3 a^9+z a^9+4 z^6 a^8-6 z^4 a^8+5 z^2 a^8-2 a^8+4 z^7 a^7-7 z^5 a^7+2 z^3 a^7+z a^7+3 z^8 a^6-4 z^6 a^6-7 z^4 a^6+9 z^2 a^6-3 a^6+2 z^9 a^5-3 z^7 a^5-5 z^5 a^5+4 z^3 a^5+z^{10} a^4-z^8 a^4-5 z^6 a^4+3 z^4 a^4+z^2 a^4+4 z^9 a^3-19 z^7 a^3+28 z^5 a^3-16 z^3 a^3+3 z a^3+z^{10} a^2-3 z^8 a^2-3 z^6 a^2+13 z^4 a^2-8 z^2 a^2+a^2+2 z^9 a-12 z^7 a+23 z^5 a-15 z^3 a+2 z a+z^8-6 z^6+11 z^4-6 z^2+1 }[/math]
The A2 invariant	Data:K11a225/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a225/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a225 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["K11a225"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -2 t^3+8 t^2-11 t+11-11 t^{-1} +8 t^{-2} -2 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -2 z^6-4 z^4+3 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]=

{ 53, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^2 a^8-2 a^8+2 z^4 a^6+6 z^2 a^6+3 a^6-z^6 a^4-3 z^4 a^4-z^2 a^4-z^6 a^2-4 z^4 a^2-4 z^2 a^2-a^2+z^4+3 z^2+1 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^3 a^{11}-z a^{11}+2 z^4 a^{10}-z^2 a^{10}+3 z^5 a^9-2 z^3 a^9+z a^9+4 z^6 a^8-6 z^4 a^8+5 z^2 a^8-2 a^8+4 z^7 a^7-7 z^5 a^7+2 z^3 a^7+z a^7+3 z^8 a^6-4 z^6 a^6-7 z^4 a^6+9 z^2 a^6-3 a^6+2 z^9 a^5-3 z^7 a^5-5 z^5 a^5+4 z^3 a^5+z^{10} a^4-z^8 a^4-5 z^6 a^4+3 z^4 a^4+z^2 a^4+4 z^9 a^3-19 z^7 a^3+28 z^5 a^3-16 z^3 a^3+3 z a^3+z^{10} a^2-3 z^8 a^2-3 z^6 a^2+13 z^4 a^2-8 z^2 a^2+a^2+2 z^9 a-12 z^7 a+23 z^5 a-15 z^3 a+2 z a+z^8-6 z^6+11 z^4-6 z^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["K11a225"];

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+8 t^2-11 t+11-11 t^{-1} +8 t^{-2} -2 t^{-3} }[/math], [math]\displaystyle{ q^2-2 q+3-5 q^{-1} +7 q^{-2} -7 q^{-3} +8 q^{-4} -7 q^{-5} +6 q^{-6} -4 q^{-7} +2 q^{-8} - q^{-9} }[/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:

(3, -8)

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{ 12 }[/math] [math]\displaystyle{ -64 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 318 }[/math] [math]\displaystyle{ 66 }[/math] [math]\displaystyle{ -768 }[/math] [math]\displaystyle{ -\frac{5248}{3} }[/math] [math]\displaystyle{ -\frac{736}{3} }[/math] [math]\displaystyle{ -384 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 2048 }[/math] [math]\displaystyle{ 3816 }[/math] [math]\displaystyle{ 792 }[/math] [math]\displaystyle{ \frac{98991}{10} }[/math] [math]\displaystyle{ -\frac{3822}{5} }[/math] [math]\displaystyle{ \frac{77182}{15} }[/math] [math]\displaystyle{ \frac{1201}{6} }[/math] [math]\displaystyle{ \frac{7631}{10} }[/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]-4 is the signature of K11a225. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 5                       1 1 3                     1   -1 1                   2 1   1 -1                 3 1     -2 -3               4 2       2 -5             4 4         0 -7           4 3           1 -9         3 4             1 -11       3 4               -1 -13     1 3                 2 -15   1 3                   -2 -17   1                     1 -19 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}_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.

K11a224 K11a226