K11a290

K11a289

K11a291

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a290 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a290/ThurstonBennequinNumber Hyperbolic Volume 17.3442 A-Polynomial See Data:K11a290/A-polynomial

[edit Notes for K11a290's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 3 }[/math] Rasmussen s-Invariant 0

[edit Notes for K11a290's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -3 t^3+15 t^2-32 t+41-32 t^{-1} +15 t^{-2} -3 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -3 z^6-3 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 141, 0 }
Jones polynomial	[math]\displaystyle{ q^4-4 q^3+9 q^2-15 q+20-22 q^{-1} +23 q^{-2} -19 q^{-3} +14 q^{-4} -9 q^{-5} +4 q^{-6} - q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^6-a^6+3 z^4 a^4+6 z^2 a^4+2 a^4-2 z^6 a^2-6 z^4 a^2-6 z^2 a^2-a^2-z^6-z^4+z^2+1+z^4 a^{-2} +z^2 a^{-2} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 3 a^4 z^{10}+3 a^2 z^{10}+6 a^5 z^9+16 a^3 z^9+10 a z^9+4 a^6 z^8+5 a^4 z^8+16 a^2 z^8+15 z^8+a^7 z^7-17 a^5 z^7-41 a^3 z^7-9 a z^7+14 z^7 a^{-1} -13 a^6 z^6-39 a^4 z^6-60 a^2 z^6+9 z^6 a^{-2} -25 z^6-3 a^7 z^5+10 a^5 z^5+22 a^3 z^5-14 a z^5-19 z^5 a^{-1} +4 z^5 a^{-3} +13 a^6 z^4+45 a^4 z^4+54 a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} +14 z^4+3 a^7 z^3+3 a^5 z^3+2 a^3 z^3+12 a z^3+9 z^3 a^{-1} -z^3 a^{-3} -5 a^6 z^2-17 a^4 z^2-19 a^2 z^2+2 z^2 a^{-2} -5 z^2-a^7 z-a^5 z-a^3 z-2 a z-z a^{-1} +a^6+2 a^4+a^2+1 }[/math]
The A2 invariant	Data:K11a290/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a290/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a290 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["K11a290"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -3 t^3+15 t^2-32 t+41-32 t^{-1} +15 t^{-2} -3 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 141, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^4-4 q^3+9 q^2-15 q+20-22 q^{-1} +23 q^{-2} -19 q^{-3} +14 q^{-4} -9 q^{-5} +4 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+3 z^4 a^4+6 z^2 a^4+2 a^4-2 z^6 a^2-6 z^4 a^2-6 z^2 a^2-a^2-z^6-z^4+z^2+1+z^4 a^{-2} +z^2 a^{-2} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 3 a^4 z^{10}+3 a^2 z^{10}+6 a^5 z^9+16 a^3 z^9+10 a z^9+4 a^6 z^8+5 a^4 z^8+16 a^2 z^8+15 z^8+a^7 z^7-17 a^5 z^7-41 a^3 z^7-9 a z^7+14 z^7 a^{-1} -13 a^6 z^6-39 a^4 z^6-60 a^2 z^6+9 z^6 a^{-2} -25 z^6-3 a^7 z^5+10 a^5 z^5+22 a^3 z^5-14 a z^5-19 z^5 a^{-1} +4 z^5 a^{-3} +13 a^6 z^4+45 a^4 z^4+54 a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} +14 z^4+3 a^7 z^3+3 a^5 z^3+2 a^3 z^3+12 a z^3+9 z^3 a^{-1} -z^3 a^{-3} -5 a^6 z^2-17 a^4 z^2-19 a^2 z^2+2 z^2 a^{-2} -5 z^2-a^7 z-a^5 z-a^3 z-2 a z-z a^{-1} +a^6+2 a^4+a^2+1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a100,}

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["K11a290"];

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{ -3 t^3+15 t^2-32 t+41-32 t^{-1} +15 t^{-2} -3 t^{-3} }[/math], [math]\displaystyle{ q^4-4 q^3+9 q^2-15 q+20-22 q^{-1} +23 q^{-2} -19 q^{-3} +14 q^{-4} -9 q^{-5} +4 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]=

{K11a100,}

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:

(1, -2)

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{ 4 }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{206}{3} }[/math] [math]\displaystyle{ \frac{82}{3} }[/math] [math]\displaystyle{ -64 }[/math] [math]\displaystyle{ -\frac{640}{3} }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ -112 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{824}{3} }[/math] [math]\displaystyle{ \frac{328}{3} }[/math] [math]\displaystyle{ \frac{22591}{30} }[/math] [math]\displaystyle{ -\frac{4382}{15} }[/math] [math]\displaystyle{ \frac{26462}{45} }[/math] [math]\displaystyle{ \frac{2177}{18} }[/math] [math]\displaystyle{ \frac{3391}{30} }[/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 K11a290. 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 9                       1 1 7                     3   -3 5                   6 1   5 3                 9 3     -6 1               11 6       5 -1             12 10         -2 -3           11 10           1 -5         8 12             4 -7       6 11               -5 -9     3 8                 5 -11   1 6                   -5 -13   3                     3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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.

K11a289 K11a291