K11a223

K11a222

K11a224

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a223 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a223/ThurstonBennequinNumber Hyperbolic Volume 12.6981 A-Polynomial See Data:K11a223/A-polynomial

[edit Notes for K11a223's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 4 }[/math] Rasmussen s-Invariant -6

[edit Notes for K11a223's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+5 t^3-10 t^2+14 t-15+14 t^{-1} -10 t^{-2} +5 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-3 z^6+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 75, 6 }
Jones polynomial	[math]\displaystyle{ -q^{12}+3 q^{11}-5 q^{10}+8 q^9-11 q^8+11 q^7-11 q^6+10 q^5-7 q^4+5 q^3-2 q^2+q }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^8 a^{-6} +z^6 a^{-4} -6 z^6 a^{-6} +2 z^6 a^{-8} +5 z^4 a^{-4} -13 z^4 a^{-6} +9 z^4 a^{-8} -z^4 a^{-10} +8 z^2 a^{-4} -13 z^2 a^{-6} +11 z^2 a^{-8} -3 z^2 a^{-10} +4 a^{-4} -5 a^{-6} +3 a^{-8} - a^{-10} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{10} a^{-6} +z^{10} a^{-8} +2 z^9 a^{-5} +6 z^9 a^{-7} +4 z^9 a^{-9} +z^8 a^{-4} +z^8 a^{-6} +8 z^8 a^{-8} +8 z^8 a^{-10} -10 z^7 a^{-5} -22 z^7 a^{-7} -3 z^7 a^{-9} +9 z^7 a^{-11} -6 z^6 a^{-4} -22 z^6 a^{-6} -44 z^6 a^{-8} -21 z^6 a^{-10} +7 z^6 a^{-12} +15 z^5 a^{-5} +15 z^5 a^{-7} -25 z^5 a^{-9} -20 z^5 a^{-11} +5 z^5 a^{-13} +13 z^4 a^{-4} +43 z^4 a^{-6} +55 z^4 a^{-8} +14 z^4 a^{-10} -8 z^4 a^{-12} +3 z^4 a^{-14} -5 z^3 a^{-5} +9 z^3 a^{-7} +30 z^3 a^{-9} +12 z^3 a^{-11} -3 z^3 a^{-13} +z^3 a^{-15} -12 z^2 a^{-4} -27 z^2 a^{-6} -22 z^2 a^{-8} -6 z^2 a^{-10} -z^2 a^{-14} -3 z a^{-5} -6 z a^{-7} -6 z a^{-9} -3 z a^{-11} +4 a^{-4} +5 a^{-6} +3 a^{-8} + a^{-10} }[/math]
The A2 invariant	[math]\displaystyle{ q^{-4} +2 q^{-8} + q^{-10} +2 q^{-14} -2 q^{-16} +2 q^{-18} -2 q^{-20} - q^{-22} -2 q^{-26} +2 q^{-28} + q^{-32} - q^{-36} }[/math]
The G2 invariant	[math]\displaystyle{ q^{-22} - q^{-24} +5 q^{-26} -7 q^{-28} +10 q^{-30} -9 q^{-32} +3 q^{-34} +15 q^{-36} -31 q^{-38} +47 q^{-40} -46 q^{-42} +27 q^{-44} +13 q^{-46} -57 q^{-48} +91 q^{-50} -90 q^{-52} +63 q^{-54} -4 q^{-56} -56 q^{-58} +93 q^{-60} -96 q^{-62} +63 q^{-64} -7 q^{-66} -48 q^{-68} +69 q^{-70} -58 q^{-72} +19 q^{-74} +30 q^{-76} -66 q^{-78} +71 q^{-80} -48 q^{-82} -4 q^{-84} +57 q^{-86} -105 q^{-88} +112 q^{-90} -80 q^{-92} +26 q^{-94} +44 q^{-96} -100 q^{-98} +121 q^{-100} -100 q^{-102} +47 q^{-104} +16 q^{-106} -71 q^{-108} +85 q^{-110} -57 q^{-112} +14 q^{-114} +31 q^{-116} -52 q^{-118} +47 q^{-120} -14 q^{-122} -29 q^{-124} +54 q^{-126} -61 q^{-128} +47 q^{-130} -10 q^{-132} -25 q^{-134} +48 q^{-136} -53 q^{-138} +47 q^{-140} -31 q^{-142} +8 q^{-144} +9 q^{-146} -27 q^{-148} +33 q^{-150} -32 q^{-152} +26 q^{-154} -13 q^{-156} +4 q^{-158} +7 q^{-160} -19 q^{-162} +21 q^{-164} -21 q^{-166} +15 q^{-168} -7 q^{-170} + q^{-172} +6 q^{-174} -11 q^{-176} +13 q^{-178} -10 q^{-180} +7 q^{-182} -2 q^{-184} - q^{-186} +2 q^{-188} -4 q^{-190} +3 q^{-192} -2 q^{-194} + q^{-196} }[/math]

Further Quantum Invariants

K11a223 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["K11a223"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^4+5 t^3-10 t^2+14 t-15+14 t^{-1} -10 t^{-2} +5 t^{-3} - t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 75, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^8 a^{-6} +z^6 a^{-4} -6 z^6 a^{-6} +2 z^6 a^{-8} +5 z^4 a^{-4} -13 z^4 a^{-6} +9 z^4 a^{-8} -z^4 a^{-10} +8 z^2 a^{-4} -13 z^2 a^{-6} +11 z^2 a^{-8} -3 z^2 a^{-10} +4 a^{-4} -5 a^{-6} +3 a^{-8} - a^{-10} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^{10} a^{-6} +z^{10} a^{-8} +2 z^9 a^{-5} +6 z^9 a^{-7} +4 z^9 a^{-9} +z^8 a^{-4} +z^8 a^{-6} +8 z^8 a^{-8} +8 z^8 a^{-10} -10 z^7 a^{-5} -22 z^7 a^{-7} -3 z^7 a^{-9} +9 z^7 a^{-11} -6 z^6 a^{-4} -22 z^6 a^{-6} -44 z^6 a^{-8} -21 z^6 a^{-10} +7 z^6 a^{-12} +15 z^5 a^{-5} +15 z^5 a^{-7} -25 z^5 a^{-9} -20 z^5 a^{-11} +5 z^5 a^{-13} +13 z^4 a^{-4} +43 z^4 a^{-6} +55 z^4 a^{-8} +14 z^4 a^{-10} -8 z^4 a^{-12} +3 z^4 a^{-14} -5 z^3 a^{-5} +9 z^3 a^{-7} +30 z^3 a^{-9} +12 z^3 a^{-11} -3 z^3 a^{-13} +z^3 a^{-15} -12 z^2 a^{-4} -27 z^2 a^{-6} -22 z^2 a^{-8} -6 z^2 a^{-10} -z^2 a^{-14} -3 z a^{-5} -6 z a^{-7} -6 z a^{-9} -3 z a^{-11} +4 a^{-4} +5 a^{-6} +3 a^{-8} + a^{-10} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n148,}

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["K11a223"];

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^3-10 t^2+14 t-15+14 t^{-1} -10 t^{-2} +5 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ -q^{12}+3 q^{11}-5 q^{10}+8 q^9-11 q^8+11 q^7-11 q^6+10 q^5-7 q^4+5 q^3-2 q^2+q }[/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]=

{K11n148,}

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, 6)

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{ 48 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 238 }[/math] [math]\displaystyle{ 34 }[/math] [math]\displaystyle{ 576 }[/math] [math]\displaystyle{ 1248 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ 240 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 1152 }[/math] [math]\displaystyle{ 2856 }[/math] [math]\displaystyle{ 408 }[/math] [math]\displaystyle{ \frac{66591}{10} }[/math] [math]\displaystyle{ -\frac{10826}{15} }[/math] [math]\displaystyle{ \frac{48302}{15} }[/math] [math]\displaystyle{ \frac{1505}{6} }[/math] [math]\displaystyle{ \frac{4191}{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]6 is the signature of K11a223. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 25                       1 -1 23                     2   2 21                   3 1   -2 19                 5 2     3 17               6 3       -3 15             5 5         0 13           6 6           0 11         4 5             -1 9       3 6               3 7     2 4                 -2 5   1 4                   3 3   1                     -1 1 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=7 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/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}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=9 }[/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.

K11a222 K11a224