K11a321

K11a320

K11a322

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a321 at Knotilus!

Knot presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a321/ThurstonBennequinNumber Hyperbolic Volume 16.2583 A-Polynomial See Data:K11a321/A-polynomial

[edit Notes for K11a321'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 K11a321's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -3 t^3+15 t^2-27 t+31-27 t^{-1} +15 t^{-2} -3 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -3 z^6-3 z^4+6 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{11,t+1\} }[/math]
Determinant and Signature	{ 121, -4 }
Jones polynomial	[math]\displaystyle{ 1-3 q^{-1} +7 q^{-2} -12 q^{-3} +17 q^{-4} -19 q^{-5} +20 q^{-6} -17 q^{-7} +13 q^{-8} -8 q^{-9} +3 q^{-10} - q^{-11} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^{10}-2 a^{10}+3 z^4 a^8+7 z^2 a^8+3 a^8-2 z^6 a^6-6 z^4 a^6-5 z^2 a^6-2 a^6-z^6 a^4-z^4 a^4+3 z^2 a^4+2 a^4+z^4 a^2+2 z^2 a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^5 a^{13}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-4 z^4 a^{12}+z^2 a^{12}+6 z^7 a^{11}-10 z^5 a^{11}+8 z^3 a^{11}-4 z a^{11}+7 z^8 a^{10}-10 z^6 a^{10}+7 z^4 a^{10}-5 z^2 a^{10}+2 a^{10}+5 z^9 a^9-z^7 a^9-10 z^5 a^9+12 z^3 a^9-4 z a^9+2 z^{10} a^8+7 z^8 a^8-18 z^6 a^8+16 z^4 a^8-8 z^2 a^8+3 a^8+10 z^9 a^7-21 z^7 a^7+20 z^5 a^7-11 z^3 a^7+2 z a^7+2 z^{10} a^6+5 z^8 a^6-19 z^6 a^6+20 z^4 a^6-11 z^2 a^6+2 a^6+5 z^9 a^5-11 z^7 a^5+11 z^5 a^5-8 z^3 a^5+z a^5+5 z^8 a^4-13 z^6 a^4+12 z^4 a^4-7 z^2 a^4+2 a^4+3 z^7 a^3-8 z^5 a^3+5 z^3 a^3+z^6 a^2-3 z^4 a^2+2 z^2 a^2 }[/math]
The A2 invariant	Data:K11a321/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a321/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a321 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["K11a321"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -3 t^3+15 t^2-27 t+31-27 t^{-1} +15 t^{-2} -3 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -3 z^6-3 z^4+6 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{ \{11,t+1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 121, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ 1-3 q^{-1} +7 q^{-2} -12 q^{-3} +17 q^{-4} -19 q^{-5} +20 q^{-6} -17 q^{-7} +13 q^{-8} -8 q^{-9} +3 q^{-10} - q^{-11} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^2 a^{10}-2 a^{10}+3 z^4 a^8+7 z^2 a^8+3 a^8-2 z^6 a^6-6 z^4 a^6-5 z^2 a^6-2 a^6-z^6 a^4-z^4 a^4+3 z^2 a^4+2 a^4+z^4 a^2+2 z^2 a^2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^5 a^{13}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-4 z^4 a^{12}+z^2 a^{12}+6 z^7 a^{11}-10 z^5 a^{11}+8 z^3 a^{11}-4 z a^{11}+7 z^8 a^{10}-10 z^6 a^{10}+7 z^4 a^{10}-5 z^2 a^{10}+2 a^{10}+5 z^9 a^9-z^7 a^9-10 z^5 a^9+12 z^3 a^9-4 z a^9+2 z^{10} a^8+7 z^8 a^8-18 z^6 a^8+16 z^4 a^8-8 z^2 a^8+3 a^8+10 z^9 a^7-21 z^7 a^7+20 z^5 a^7-11 z^3 a^7+2 z a^7+2 z^{10} a^6+5 z^8 a^6-19 z^6 a^6+20 z^4 a^6-11 z^2 a^6+2 a^6+5 z^9 a^5-11 z^7 a^5+11 z^5 a^5-8 z^3 a^5+z a^5+5 z^8 a^4-13 z^6 a^4+12 z^4 a^4-7 z^2 a^4+2 a^4+3 z^7 a^3-8 z^5 a^3+5 z^3 a^3+z^6 a^2-3 z^4 a^2+2 z^2 a^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["K11a321"];

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^2-27 t+31-27 t^{-1} +15 t^{-2} -3 t^{-3} }[/math], [math]\displaystyle{ 1-3 q^{-1} +7 q^{-2} -12 q^{-3} +17 q^{-4} -19 q^{-5} +20 q^{-6} -17 q^{-7} +13 q^{-8} -8 q^{-9} +3 q^{-10} - q^{-11} }[/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:

(6, -16)

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{ 24 }[/math] [math]\displaystyle{ -128 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 860 }[/math] [math]\displaystyle{ 164 }[/math] [math]\displaystyle{ -3072 }[/math] [math]\displaystyle{ -\frac{18656}{3} }[/math] [math]\displaystyle{ -\frac{3200}{3} }[/math] [math]\displaystyle{ -1120 }[/math] [math]\displaystyle{ 2304 }[/math] [math]\displaystyle{ 8192 }[/math] [math]\displaystyle{ 20640 }[/math] [math]\displaystyle{ 3936 }[/math] [math]\displaystyle{ \frac{225791}{5} }[/math] [math]\displaystyle{ -\frac{28612}{15} }[/math] [math]\displaystyle{ \frac{331804}{15} }[/math] [math]\displaystyle{ \frac{1649}{3} }[/math] [math]\displaystyle{ \frac{16111}{5} }[/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 K11a321. 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 1                       1 1 -1                     2   -2 -3                   5 1   4 -5                 8 3     -5 -7               9 4       5 -9             10 8         -2 -11           10 9           1 -13         7 10             3 -15       6 10               -4 -17     2 7                 5 -19   1 6                   -5 -21   2                     2 -23 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=-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}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/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.

K11a320 K11a322