K11a296

K11a295

K11a297

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a296 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X10,3,11,4 X16,5,17,6 X22,8,1,7 X18,9,19,10 X2,11,3,12 X20,13,21,14 X8,15,9,16 X4,17,5,18 X14,19,15,20 X12,21,13,22
Gauss code	1, -6, 2, -9, 3, -1, 4, -8, 5, -2, 6, -11, 7, -10, 8, -3, 9, -5, 10, -7, 11, -4
Dowker-Thistlethwaite code	6 10 16 22 18 2 20 8 4 14 12

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:K11a296/ThurstonBennequinNumber Hyperbolic Volume 15.7452 A-Polynomial See Data:K11a296/A-polynomial

[edit Notes for K11a296'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 K11a296's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -6 t^2+28 t-43+28 t^{-1} -6 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -6 z^4+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 111, -2 }
Jones polynomial	[math]\displaystyle{ q-3+7 q^{-1} -12 q^{-2} +16 q^{-3} -17 q^{-4} +18 q^{-5} -15 q^{-6} +11 q^{-7} -7 q^{-8} +3 q^{-9} - q^{-10} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^{10}+3 z^2 a^8+a^8-2 z^4 a^6-3 z^4 a^4-2 z^2 a^4-z^4 a^2+2 z^2 a^2+a^2+z^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^7 a^{11}-4 z^5 a^{11}+5 z^3 a^{11}-2 z a^{11}+3 z^8 a^{10}-11 z^6 a^{10}+12 z^4 a^{10}-4 z^2 a^{10}+a^{10}+4 z^9 a^9-12 z^7 a^9+8 z^5 a^9+z a^9+2 z^{10} a^8+2 z^8 a^8-21 z^6 a^8+23 z^4 a^8-9 z^2 a^8+a^8+10 z^9 a^7-27 z^7 a^7+21 z^5 a^7-11 z^3 a^7+3 z a^7+2 z^{10} a^6+8 z^8 a^6-30 z^6 a^6+25 z^4 a^6-10 z^2 a^6+6 z^9 a^5-5 z^7 a^5-8 z^5 a^5+7 z^3 a^5-z a^5+9 z^8 a^4-14 z^6 a^4+7 z^4 a^4+9 z^7 a^3-14 z^5 a^3+11 z^3 a^3-z a^3+6 z^6 a^2-6 z^4 a^2+4 z^2 a^2-a^2+3 z^5 a-2 z^3 a+z^4-z^2 }[/math]
The A2 invariant	Data:K11a296/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a296/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a296 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["K11a296"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -6 t^2+28 t-43+28 t^{-1} -6 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -6 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]=

{ 111, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q-3+7 q^{-1} -12 q^{-2} +16 q^{-3} -17 q^{-4} +18 q^{-5} -15 q^{-6} +11 q^{-7} -7 q^{-8} +3 q^{-9} - q^{-10} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -a^{10}+3 z^2 a^8+a^8-2 z^4 a^6-3 z^4 a^4-2 z^2 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{ z^7 a^{11}-4 z^5 a^{11}+5 z^3 a^{11}-2 z a^{11}+3 z^8 a^{10}-11 z^6 a^{10}+12 z^4 a^{10}-4 z^2 a^{10}+a^{10}+4 z^9 a^9-12 z^7 a^9+8 z^5 a^9+z a^9+2 z^{10} a^8+2 z^8 a^8-21 z^6 a^8+23 z^4 a^8-9 z^2 a^8+a^8+10 z^9 a^7-27 z^7 a^7+21 z^5 a^7-11 z^3 a^7+3 z a^7+2 z^{10} a^6+8 z^8 a^6-30 z^6 a^6+25 z^4 a^6-10 z^2 a^6+6 z^9 a^5-5 z^7 a^5-8 z^5 a^5+7 z^3 a^5-z a^5+9 z^8 a^4-14 z^6 a^4+7 z^4 a^4+9 z^7 a^3-14 z^5 a^3+11 z^3 a^3-z a^3+6 z^6 a^2-6 z^4 a^2+4 z^2 a^2-a^2+3 z^5 a-2 z^3 a+z^4-z^2 }[/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["K11a296"];

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{ -6 t^2+28 t-43+28 t^{-1} -6 t^{-2} }[/math], [math]\displaystyle{ q-3+7 q^{-1} -12 q^{-2} +16 q^{-3} -17 q^{-4} +18 q^{-5} -15 q^{-6} +11 q^{-7} -7 q^{-8} +3 q^{-9} - q^{-10} }[/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:

(4, -10)

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{ -80 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{1448}{3} }[/math] [math]\displaystyle{ \frac{328}{3} }[/math] [math]\displaystyle{ -1280 }[/math] [math]\displaystyle{ -\frac{9056}{3} }[/math] [math]\displaystyle{ -\frac{1472}{3} }[/math] [math]\displaystyle{ -688 }[/math] [math]\displaystyle{ \frac{2048}{3} }[/math] [math]\displaystyle{ 3200 }[/math] [math]\displaystyle{ \frac{23168}{3} }[/math] [math]\displaystyle{ \frac{5248}{3} }[/math] [math]\displaystyle{ \frac{280982}{15} }[/math] [math]\displaystyle{ -\frac{10696}{5} }[/math] [math]\displaystyle{ \frac{495368}{45} }[/math] [math]\displaystyle{ \frac{3130}{9} }[/math] [math]\displaystyle{ \frac{25862}{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 K11a296. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                       1 1 1                     2   -2 -1                   5 1   4 -3                 8 3     -5 -5               8 4       4 -7             9 8         -1 -9           9 8           1 -11         6 9             3 -13       5 9               -4 -15     2 6                 4 -17   1 5                   -4 -19   2                     2 -21 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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=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.

K11a295 K11a297