K11a22

K11a21

K11a23

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a22 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X12,5,13,6 X2837 X18,10,19,9 X14,11,15,12 X6,13,7,14 X20,16,21,15 X22,18,1,17 X10,20,11,19 X16,22,17,21
Gauss code	1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -3, 7, -6, 8, -11, 9, -5, 10, -8, 11, -9
Dowker-Thistlethwaite code	4 8 12 2 18 14 6 20 22 10 16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a22/ThurstonBennequinNumber Hyperbolic Volume 14.6533 A-Polynomial See Data:K11a22/A-polynomial

[edit Notes for K11a22's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a22's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-5 t^3+13 t^2-20 t+23-20 t^{-1} +13 t^{-2} -5 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+3 z^6+3 z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 101, 4 }
Jones polynomial	[math]\displaystyle{ q^{10}-3 q^9+6 q^8-11 q^7+14 q^6-16 q^5+16 q^4-13 q^3+11 q^2-6 q+3- q^{-1} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-4} -z^6 a^{-2} +6 z^6 a^{-4} -2 z^6 a^{-6} -4 z^4 a^{-2} +15 z^4 a^{-4} -9 z^4 a^{-6} +z^4 a^{-8} -5 z^2 a^{-2} +19 z^2 a^{-4} -14 z^2 a^{-6} +3 z^2 a^{-8} -2 a^{-2} +9 a^{-4} -8 a^{-6} +2 a^{-8} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{10} a^{-4} +z^{10} a^{-6} +3 z^9 a^{-3} +7 z^9 a^{-5} +4 z^9 a^{-7} +3 z^8 a^{-2} +8 z^8 a^{-4} +12 z^8 a^{-6} +7 z^8 a^{-8} +z^7 a^{-1} -6 z^7 a^{-3} -11 z^7 a^{-5} +3 z^7 a^{-7} +7 z^7 a^{-9} -12 z^6 a^{-2} -38 z^6 a^{-4} -41 z^6 a^{-6} -10 z^6 a^{-8} +5 z^6 a^{-10} -4 z^5 a^{-1} -7 z^5 a^{-3} -17 z^5 a^{-5} -26 z^5 a^{-7} -9 z^5 a^{-9} +3 z^5 a^{-11} +16 z^4 a^{-2} +48 z^4 a^{-4} +43 z^4 a^{-6} +6 z^4 a^{-8} -4 z^4 a^{-10} +z^4 a^{-12} +5 z^3 a^{-1} +17 z^3 a^{-3} +32 z^3 a^{-5} +29 z^3 a^{-7} +6 z^3 a^{-9} -3 z^3 a^{-11} -9 z^2 a^{-2} -29 z^2 a^{-4} -25 z^2 a^{-6} -3 z^2 a^{-8} +z^2 a^{-10} -z^2 a^{-12} -2 z a^{-1} -7 z a^{-3} -14 z a^{-5} -11 z a^{-7} -z a^{-9} +z a^{-11} +2 a^{-2} +9 a^{-4} +8 a^{-6} +2 a^{-8} }[/math]
The A2 invariant	[math]\displaystyle{ -q^2+1-2 q^{-2} + q^{-4} +2 q^{-6} +6 q^{-10} - q^{-12} +3 q^{-14} - q^{-16} -3 q^{-18} -4 q^{-22} + q^{-24} + q^{-30} }[/math]
The G2 invariant	[math]\displaystyle{ q^{12}-2 q^{10}+6 q^8-11 q^6+14 q^4-16 q^2+5+17 q^{-2} -49 q^{-4} +82 q^{-6} -97 q^{-8} +72 q^{-10} -7 q^{-12} -91 q^{-14} +185 q^{-16} -233 q^{-18} +204 q^{-20} -95 q^{-22} -72 q^{-24} +227 q^{-26} -309 q^{-28} +285 q^{-30} -147 q^{-32} -33 q^{-34} +193 q^{-36} -259 q^{-38} +215 q^{-40} -72 q^{-42} -87 q^{-44} +211 q^{-46} -220 q^{-48} +128 q^{-50} +45 q^{-52} -215 q^{-54} +321 q^{-56} -304 q^{-58} +173 q^{-60} +35 q^{-62} -247 q^{-64} +384 q^{-66} -395 q^{-68} +272 q^{-70} -61 q^{-72} -166 q^{-74} +309 q^{-76} -336 q^{-78} +227 q^{-80} -51 q^{-82} -122 q^{-84} +211 q^{-86} -194 q^{-88} +76 q^{-90} +69 q^{-92} -183 q^{-94} +208 q^{-96} -141 q^{-98} +7 q^{-100} +127 q^{-102} -216 q^{-104} +231 q^{-106} -168 q^{-108} +63 q^{-110} +49 q^{-112} -134 q^{-114} +170 q^{-116} -158 q^{-118} +115 q^{-120} -46 q^{-122} -13 q^{-124} +59 q^{-126} -83 q^{-128} +81 q^{-130} -64 q^{-132} +40 q^{-134} -11 q^{-136} -11 q^{-138} +25 q^{-140} -30 q^{-142} +26 q^{-144} -18 q^{-146} +10 q^{-148} -2 q^{-150} -4 q^{-152} +5 q^{-154} -6 q^{-156} +4 q^{-158} -2 q^{-160} + q^{-162} }[/math]

Further Quantum Invariants

K11a22 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["K11a22"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-5 t^3+13 t^2-20 t+23-20 t^{-1} +13 t^{-2} -5 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+3 z^6+3 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]=

{ 101, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^{10}-3 q^9+6 q^8-11 q^7+14 q^6-16 q^5+16 q^4-13 q^3+11 q^2-6 q+3- q^{-1} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^8 a^{-4} -z^6 a^{-2} +6 z^6 a^{-4} -2 z^6 a^{-6} -4 z^4 a^{-2} +15 z^4 a^{-4} -9 z^4 a^{-6} +z^4 a^{-8} -5 z^2 a^{-2} +19 z^2 a^{-4} -14 z^2 a^{-6} +3 z^2 a^{-8} -2 a^{-2} +9 a^{-4} -8 a^{-6} +2 a^{-8} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^{10} a^{-4} +z^{10} a^{-6} +3 z^9 a^{-3} +7 z^9 a^{-5} +4 z^9 a^{-7} +3 z^8 a^{-2} +8 z^8 a^{-4} +12 z^8 a^{-6} +7 z^8 a^{-8} +z^7 a^{-1} -6 z^7 a^{-3} -11 z^7 a^{-5} +3 z^7 a^{-7} +7 z^7 a^{-9} -12 z^6 a^{-2} -38 z^6 a^{-4} -41 z^6 a^{-6} -10 z^6 a^{-8} +5 z^6 a^{-10} -4 z^5 a^{-1} -7 z^5 a^{-3} -17 z^5 a^{-5} -26 z^5 a^{-7} -9 z^5 a^{-9} +3 z^5 a^{-11} +16 z^4 a^{-2} +48 z^4 a^{-4} +43 z^4 a^{-6} +6 z^4 a^{-8} -4 z^4 a^{-10} +z^4 a^{-12} +5 z^3 a^{-1} +17 z^3 a^{-3} +32 z^3 a^{-5} +29 z^3 a^{-7} +6 z^3 a^{-9} -3 z^3 a^{-11} -9 z^2 a^{-2} -29 z^2 a^{-4} -25 z^2 a^{-6} -3 z^2 a^{-8} +z^2 a^{-10} -z^2 a^{-12} -2 z a^{-1} -7 z a^{-3} -14 z a^{-5} -11 z a^{-7} -z a^{-9} +z a^{-11} +2 a^{-2} +9 a^{-4} +8 a^{-6} +2 a^{-8} }[/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["K11a22"];

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^4-5 t^3+13 t^2-20 t+23-20 t^{-1} +13 t^{-2} -5 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ q^{10}-3 q^9+6 q^8-11 q^7+14 q^6-16 q^5+16 q^4-13 q^3+11 q^2-6 q+3- q^{-1} }[/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, 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{ 12 }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 78 }[/math] [math]\displaystyle{ 10 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 304 }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ 56 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 936 }[/math] [math]\displaystyle{ 120 }[/math] [math]\displaystyle{ \frac{11871}{10} }[/math] [math]\displaystyle{ -\frac{1186}{15} }[/math] [math]\displaystyle{ \frac{10502}{15} }[/math] [math]\displaystyle{ \frac{65}{6} }[/math] [math]\displaystyle{ \frac{991}{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 K11a22. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 21                       1 1 19                     2   -2 17                   4 1   3 15                 7 2     -5 13               7 4       3 11             9 7         -2 9           7 7           0 7         6 9             3 5       5 7               -2 3     2 7                 5 1   1 4                   -3 -1   2                     2 -3 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=5 }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=8 }[/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.

K11a21 K11a23