K11a203

K11a202

K11a204

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a203 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a203/ThurstonBennequinNumber Hyperbolic Volume 10.2613 A-Polynomial See Data:K11a203/A-polynomial

[edit Notes for K11a203'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 K11a203's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+5 t^3-9 t^2+11 t-11+11 t^{-1} -9 t^{-2} +5 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-3 z^6+z^4+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 63, 6 }
Jones polynomial	[math]\displaystyle{ -q^{12}+3 q^{11}-5 q^{10}+7 q^9-9 q^8+9 q^7-9 q^6+8 q^5-5 q^4+4 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} -12 z^4 a^{-6} +9 z^4 a^{-8} -z^4 a^{-10} +7 z^2 a^{-4} -10 z^2 a^{-6} +10 z^2 a^{-8} -3 z^2 a^{-10} +3 a^{-4} -3 a^{-6} +2 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} +5 z^9 a^{-7} +3 z^9 a^{-9} +z^8 a^{-4} -z^8 a^{-6} +3 z^8 a^{-8} +5 z^8 a^{-10} -11 z^7 a^{-5} -22 z^7 a^{-7} -5 z^7 a^{-9} +6 z^7 a^{-11} -6 z^6 a^{-4} -14 z^6 a^{-6} -25 z^6 a^{-8} -11 z^6 a^{-10} +6 z^6 a^{-12} +18 z^5 a^{-5} +26 z^5 a^{-7} -7 z^5 a^{-9} -10 z^5 a^{-11} +5 z^5 a^{-13} +12 z^4 a^{-4} +32 z^4 a^{-6} +35 z^4 a^{-8} +5 z^4 a^{-10} -7 z^4 a^{-12} +3 z^4 a^{-14} -8 z^3 a^{-5} -7 z^3 a^{-7} +8 z^3 a^{-9} +2 z^3 a^{-11} -4 z^3 a^{-13} +z^3 a^{-15} -10 z^2 a^{-4} -20 z^2 a^{-6} -16 z^2 a^{-8} -4 z^2 a^{-10} +z^2 a^{-12} -z^2 a^{-14} -z a^{-5} +z a^{-13} +3 a^{-4} +3 a^{-6} +2 a^{-8} + a^{-10} }[/math]
The A2 invariant	[math]\displaystyle{ q^{-4} + q^{-8} + q^{-10} +2 q^{-14} - q^{-16} +2 q^{-18} - q^{-20} - q^{-22} -2 q^{-26} + q^{-28} + q^{-32} - q^{-36} }[/math]
The G2 invariant	[math]\displaystyle{ q^{-22} - q^{-24} +4 q^{-26} -5 q^{-28} +6 q^{-30} -4 q^{-32} + q^{-34} +10 q^{-36} -18 q^{-38} +27 q^{-40} -26 q^{-42} +14 q^{-44} +6 q^{-46} -28 q^{-48} +46 q^{-50} -47 q^{-52} +36 q^{-54} -8 q^{-56} -20 q^{-58} +41 q^{-60} -47 q^{-62} +36 q^{-64} -13 q^{-66} -10 q^{-68} +25 q^{-70} -25 q^{-72} +16 q^{-74} +2 q^{-76} -17 q^{-78} +25 q^{-80} -22 q^{-82} +3 q^{-84} +14 q^{-86} -36 q^{-88} +46 q^{-90} -38 q^{-92} +20 q^{-94} +6 q^{-96} -34 q^{-98} +51 q^{-100} -53 q^{-102} +34 q^{-104} -9 q^{-106} -17 q^{-108} +33 q^{-110} -32 q^{-112} +23 q^{-114} -8 q^{-116} -4 q^{-118} +11 q^{-120} -11 q^{-122} +2 q^{-124} +5 q^{-126} -9 q^{-128} +11 q^{-130} -5 q^{-132} - q^{-134} +7 q^{-136} -13 q^{-138} +18 q^{-140} -19 q^{-142} +15 q^{-144} -7 q^{-146} -6 q^{-148} +17 q^{-150} -25 q^{-152} +26 q^{-154} -21 q^{-156} +12 q^{-158} + q^{-160} -16 q^{-162} +22 q^{-164} -24 q^{-166} +20 q^{-168} -11 q^{-170} +2 q^{-172} +6 q^{-174} -12 q^{-176} +14 q^{-178} -11 q^{-180} +8 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

K11a203 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["K11a203"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^4+5 t^3-9 t^2+11 t-11+11 t^{-1} -9 t^{-2} +5 t^{-3} - t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^8-3 z^6+z^4+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]=

{ 63, 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}+7 q^9-9 q^8+9 q^7-9 q^6+8 q^5-5 q^4+4 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} -12 z^4 a^{-6} +9 z^4 a^{-8} -z^4 a^{-10} +7 z^2 a^{-4} -10 z^2 a^{-6} +10 z^2 a^{-8} -3 z^2 a^{-10} +3 a^{-4} -3 a^{-6} +2 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} +5 z^9 a^{-7} +3 z^9 a^{-9} +z^8 a^{-4} -z^8 a^{-6} +3 z^8 a^{-8} +5 z^8 a^{-10} -11 z^7 a^{-5} -22 z^7 a^{-7} -5 z^7 a^{-9} +6 z^7 a^{-11} -6 z^6 a^{-4} -14 z^6 a^{-6} -25 z^6 a^{-8} -11 z^6 a^{-10} +6 z^6 a^{-12} +18 z^5 a^{-5} +26 z^5 a^{-7} -7 z^5 a^{-9} -10 z^5 a^{-11} +5 z^5 a^{-13} +12 z^4 a^{-4} +32 z^4 a^{-6} +35 z^4 a^{-8} +5 z^4 a^{-10} -7 z^4 a^{-12} +3 z^4 a^{-14} -8 z^3 a^{-5} -7 z^3 a^{-7} +8 z^3 a^{-9} +2 z^3 a^{-11} -4 z^3 a^{-13} +z^3 a^{-15} -10 z^2 a^{-4} -20 z^2 a^{-6} -16 z^2 a^{-8} -4 z^2 a^{-10} +z^2 a^{-12} -z^2 a^{-14} -z a^{-5} +z a^{-13} +3 a^{-4} +3 a^{-6} +2 a^{-8} + a^{-10} }[/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["K11a203"];

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

{}

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:

(4, 9)

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{ 16 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{1160}{3} }[/math] [math]\displaystyle{ \frac{160}{3} }[/math] [math]\displaystyle{ 1152 }[/math] [math]\displaystyle{ 2256 }[/math] [math]\displaystyle{ 320 }[/math] [math]\displaystyle{ 328 }[/math] [math]\displaystyle{ \frac{2048}{3} }[/math] [math]\displaystyle{ 2592 }[/math] [math]\displaystyle{ \frac{18560}{3} }[/math] [math]\displaystyle{ \frac{2560}{3} }[/math] [math]\displaystyle{ \frac{203222}{15} }[/math] [math]\displaystyle{ -\frac{1168}{15} }[/math] [math]\displaystyle{ \frac{238448}{45} }[/math] [math]\displaystyle{ \frac{2218}{9} }[/math] [math]\displaystyle{ \frac{10022}{15} }[/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 K11a203. 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                 4 2     2 17               5 3       -2 15             4 4         0 13           5 5           0 11         3 4             -1 9       2 5               3 7     2 3                 -1 5   1 3                   2 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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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.

K11a202 K11a204