K11a206

K11a205

K11a207

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a206 at Knotilus!

Knot presentations

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

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:K11a206/ThurstonBennequinNumber Hyperbolic Volume 8.62876 A-Polynomial See Data:K11a206/A-polynomial

[edit Notes for K11a206'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 K11a206's four dimensional invariants]

Polynomial invariants

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

Further Quantum Invariants

K11a206 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["K11a206"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^4+5 t^3-7 t^2+7 t-7+7 t^{-1} -7 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+8 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]=

{ 47, -6 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^4 a^{10}-4 z^2 a^{10}-3 a^{10}+2 z^6 a^8+10 z^4 a^8+13 z^2 a^8+4 a^8-z^8 a^6-6 z^6 a^6-11 z^4 a^6-7 z^2 a^6-a^6+z^6 a^4+5 z^4 a^4+6 z^2 a^4+a^4 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^3 a^{15}-z a^{15}+2 z^4 a^{14}-z^2 a^{14}+3 z^5 a^{13}-3 z^3 a^{13}+2 z a^{13}+3 z^6 a^{12}-3 z^4 a^{12}+z^2 a^{12}+3 z^7 a^{11}-5 z^5 a^{11}+2 z^3 a^{11}-z a^{11}+3 z^8 a^{10}-9 z^6 a^{10}+11 z^4 a^{10}-11 z^2 a^{10}+3 a^{10}+2 z^9 a^9-5 z^7 a^9-z^5 a^9+5 z^3 a^9-3 z a^9+z^{10} a^8-z^8 a^8-10 z^6 a^8+19 z^4 a^8-13 z^2 a^8+4 a^8+4 z^9 a^7-20 z^7 a^7+29 z^5 a^7-14 z^3 a^7+2 z a^7+z^{10} a^6-3 z^8 a^6-4 z^6 a^6+14 z^4 a^6-7 z^2 a^6+a^6+2 z^9 a^5-12 z^7 a^5+22 z^5 a^5-13 z^3 a^5+z a^5+z^8 a^4-6 z^6 a^4+11 z^4 a^4-7 z^2 a^4+a^4 }[/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["K11a206"];

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

(8, -23)

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{ 32 }[/math] [math]\displaystyle{ -184 }[/math] [math]\displaystyle{ 512 }[/math] [math]\displaystyle{ \frac{4048}{3} }[/math] [math]\displaystyle{ \frac{680}{3} }[/math] [math]\displaystyle{ -5888 }[/math] [math]\displaystyle{ -\frac{32656}{3} }[/math] [math]\displaystyle{ -\frac{5728}{3} }[/math] [math]\displaystyle{ -1624 }[/math] [math]\displaystyle{ \frac{16384}{3} }[/math] [math]\displaystyle{ 16928 }[/math] [math]\displaystyle{ \frac{129536}{3} }[/math] [math]\displaystyle{ \frac{21760}{3} }[/math] [math]\displaystyle{ \frac{1334044}{15} }[/math] [math]\displaystyle{ \frac{6184}{15} }[/math] [math]\displaystyle{ \frac{1694776}{45} }[/math] [math]\displaystyle{ \frac{6308}{9} }[/math] [math]\displaystyle{ \frac{77164}{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 K11a206. 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 -3                     1   -1 -5                   2 1   1 -7                 3 2     -1 -9               3 1       2 -11             3 3         0 -13           4 3           1 -15         2 3             1 -17       3 4               -1 -19     1 2                 1 -21   1 3                   -2 -23   1                     1 -25 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-7 }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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.

K11a205 K11a207