K11a218

K11a217

K11a219

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a218 at Knotilus!

Knot presentations

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

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:K11a218/ThurstonBennequinNumber Hyperbolic Volume 16.744 A-Polynomial See Data:K11a218/A-polynomial

[edit Notes for K11a218's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 3 }[/math] Rasmussen s-Invariant 2

[edit Notes for K11a218's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-10 t^2+32 t-45+32 t^{-1} -10 t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6-4 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 131, -2 }
Jones polynomial	[math]\displaystyle{ q^3-4 q^2+9 q-14+19 q^{-1} -21 q^{-2} +21 q^{-3} -18 q^{-4} +13 q^{-5} -7 q^{-6} +3 q^{-7} - q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^8+3 z^2 a^6+2 a^6-3 z^4 a^4-3 z^2 a^4-a^4+z^6 a^2+z^4 a^2+z^2 a^2-2 z^4-z^2+1+z^2 a^{-2} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 a^4 z^{10}+2 a^2 z^{10}+5 a^5 z^9+11 a^3 z^9+6 a z^9+6 a^6 z^8+9 a^4 z^8+10 a^2 z^8+7 z^8+5 a^7 z^7-a^5 z^7-18 a^3 z^7-8 a z^7+4 z^7 a^{-1} +3 a^8 z^6-5 a^6 z^6-20 a^4 z^6-29 a^2 z^6+z^6 a^{-2} -16 z^6+a^9 z^5-6 a^7 z^5-6 a^5 z^5+8 a^3 z^5-2 a z^5-9 z^5 a^{-1} -5 a^8 z^4-2 a^6 z^4+11 a^4 z^4+20 a^2 z^4-2 z^4 a^{-2} +10 z^4-2 a^9 z^3+2 a^7 z^3+6 a^5 z^3-3 a^3 z^3+5 z^3 a^{-1} +3 a^8 z^2+6 a^6 z^2+a^4 z^2-6 a^2 z^2+z^2 a^{-2} -3 z^2+a^9 z-a^5 z+a^3 z+a z-a^8-2 a^6-a^4+1 }[/math]
The A2 invariant	Data:K11a218/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a218/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a218 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["K11a218"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^3-10 t^2+32 t-45+32 t^{-1} -10 t^{-2} + t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 131, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^3-4 q^2+9 q-14+19 q^{-1} -21 q^{-2} +21 q^{-3} -18 q^{-4} +13 q^{-5} -7 q^{-6} +3 q^{-7} - q^{-8} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 a^4 z^{10}+2 a^2 z^{10}+5 a^5 z^9+11 a^3 z^9+6 a z^9+6 a^6 z^8+9 a^4 z^8+10 a^2 z^8+7 z^8+5 a^7 z^7-a^5 z^7-18 a^3 z^7-8 a z^7+4 z^7 a^{-1} +3 a^8 z^6-5 a^6 z^6-20 a^4 z^6-29 a^2 z^6+z^6 a^{-2} -16 z^6+a^9 z^5-6 a^7 z^5-6 a^5 z^5+8 a^3 z^5-2 a z^5-9 z^5 a^{-1} -5 a^8 z^4-2 a^6 z^4+11 a^4 z^4+20 a^2 z^4-2 z^4 a^{-2} +10 z^4-2 a^9 z^3+2 a^7 z^3+6 a^5 z^3-3 a^3 z^3+5 z^3 a^{-1} +3 a^8 z^2+6 a^6 z^2+a^4 z^2-6 a^2 z^2+z^2 a^{-2} -3 z^2+a^9 z-a^5 z+a^3 z+a z-a^8-2 a^6-a^4+1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a131,}

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["K11a218"];

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{ t^3-10 t^2+32 t-45+32 t^{-1} -10 t^{-2} + t^{-3} }[/math], [math]\displaystyle{ q^3-4 q^2+9 q-14+19 q^{-1} -21 q^{-2} +21 q^{-3} -18 q^{-4} +13 q^{-5} -7 q^{-6} +3 q^{-7} - q^{-8} }[/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]=

{K11a131,}

Vassiliev invariants

V2 and V3:

(1, -3)

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{ -24 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{350}{3} }[/math] [math]\displaystyle{ \frac{106}{3} }[/math] [math]\displaystyle{ -96 }[/math] [math]\displaystyle{ -464 }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ -184 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ \frac{1400}{3} }[/math] [math]\displaystyle{ \frac{424}{3} }[/math] [math]\displaystyle{ \frac{55471}{30} }[/math] [math]\displaystyle{ -\frac{9622}{15} }[/math] [math]\displaystyle{ \frac{68822}{45} }[/math] [math]\displaystyle{ \frac{2129}{18} }[/math] [math]\displaystyle{ \frac{7951}{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]-2 is the signature of K11a218. 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 7                       1 1 5                     3   -3 3                   6 1   5 1                 8 3     -5 -1               11 6       5 -3             11 9         -2 -5           10 10           0 -7         8 11             3 -9       5 10               -5 -11     2 8                 6 -13   1 5                   -4 -15   2                     2 -17 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=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\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^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/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.

K11a217 K11a219