K11a106

K11a105

K11a107

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a106 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a106/ThurstonBennequinNumber Hyperbolic Volume 13.4001 A-Polynomial See Data:K11a106/A-polynomial

[edit Notes for K11a106's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 4 }[/math] Rasmussen s-Invariant 0

[edit Notes for K11a106's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-5 t^3+12 t^2-18 t+21-18 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+3 z^6+2 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 93, 0 }
Jones polynomial	[math]\displaystyle{ q^6-3 q^5+5 q^4-9 q^3+13 q^2-14 q+15-13 q^{-1} +10 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8-a^2 z^6-2 z^6 a^{-2} +6 z^6-4 a^2 z^4-9 z^4 a^{-2} +z^4 a^{-4} +14 z^4-5 a^2 z^2-12 z^2 a^{-2} +3 z^2 a^{-4} +15 z^2-2 a^2-4 a^{-2} + a^{-4} +6 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{10} a^{-2} +z^{10}+3 a z^9+6 z^9 a^{-1} +3 z^9 a^{-3} +4 a^2 z^8+6 z^8 a^{-2} +4 z^8 a^{-4} +6 z^8+4 a^3 z^7-2 a z^7-13 z^7 a^{-1} -4 z^7 a^{-3} +3 z^7 a^{-5} +3 a^4 z^6-4 a^2 z^6-27 z^6 a^{-2} -12 z^6 a^{-4} +z^6 a^{-6} -21 z^6+a^5 z^5-5 a^3 z^5+7 z^5 a^{-1} -9 z^5 a^{-3} -10 z^5 a^{-5} -6 a^4 z^4+a^2 z^4+39 z^4 a^{-2} +10 z^4 a^{-4} -3 z^4 a^{-6} +33 z^4-2 a^5 z^3-a^3 z^3+a z^3+9 z^3 a^{-1} +17 z^3 a^{-3} +8 z^3 a^{-5} +3 a^4 z^2-4 a^2 z^2-22 z^2 a^{-2} -5 z^2 a^{-4} +z^2 a^{-6} -23 z^2+a^5 z+a^3 z-2 a z-6 z a^{-1} -6 z a^{-3} -2 z a^{-5} +2 a^2+4 a^{-2} + a^{-4} +6 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{14}+q^{12}-2 q^{10}+q^8+q^6-q^4+4 q^2-2+3 q^{-2} +2 q^{-8} -3 q^{-10} - q^{-14} + q^{-18} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+8 q^{72}-6 q^{70}-2 q^{68}+16 q^{66}-29 q^{64}+41 q^{62}-42 q^{60}+27 q^{58}-2 q^{56}-37 q^{54}+74 q^{52}-101 q^{50}+102 q^{48}-79 q^{46}+25 q^{44}+47 q^{42}-117 q^{40}+170 q^{38}-174 q^{36}+125 q^{34}-38 q^{32}-72 q^{30}+154 q^{28}-185 q^{26}+155 q^{24}-59 q^{22}-46 q^{20}+126 q^{18}-135 q^{16}+69 q^{14}+37 q^{12}-137 q^{10}+180 q^8-143 q^6+34 q^4+117 q^2-236+292 q^{-2} -242 q^{-4} +106 q^{-6} +61 q^{-8} -211 q^{-10} +291 q^{-12} -270 q^{-14} +173 q^{-16} -22 q^{-18} -114 q^{-20} +200 q^{-22} -191 q^{-24} +99 q^{-26} +17 q^{-28} -118 q^{-30} +151 q^{-32} -107 q^{-34} + q^{-36} +114 q^{-38} -186 q^{-40} +195 q^{-42} -128 q^{-44} -5 q^{-46} +121 q^{-48} -198 q^{-50} +207 q^{-52} -152 q^{-54} +59 q^{-56} +37 q^{-58} -105 q^{-60} +135 q^{-62} -120 q^{-64} +77 q^{-66} -23 q^{-68} -18 q^{-70} +42 q^{-72} -48 q^{-74} +41 q^{-76} -25 q^{-78} +11 q^{-80} +2 q^{-82} -8 q^{-84} +7 q^{-86} -7 q^{-88} +4 q^{-90} -2 q^{-92} + q^{-94} }[/math]

Further Quantum Invariants

K11a106 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["K11a106"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-5 t^3+12 t^2-18 t+21-18 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+3 z^6+2 z^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]=

{ 93, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^6-3 q^5+5 q^4-9 q^3+13 q^2-14 q+15-13 q^{-1} +10 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^8-a^2 z^6-2 z^6 a^{-2} +6 z^6-4 a^2 z^4-9 z^4 a^{-2} +z^4 a^{-4} +14 z^4-5 a^2 z^2-12 z^2 a^{-2} +3 z^2 a^{-4} +15 z^2-2 a^2-4 a^{-2} + a^{-4} +6 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^{10} a^{-2} +z^{10}+3 a z^9+6 z^9 a^{-1} +3 z^9 a^{-3} +4 a^2 z^8+6 z^8 a^{-2} +4 z^8 a^{-4} +6 z^8+4 a^3 z^7-2 a z^7-13 z^7 a^{-1} -4 z^7 a^{-3} +3 z^7 a^{-5} +3 a^4 z^6-4 a^2 z^6-27 z^6 a^{-2} -12 z^6 a^{-4} +z^6 a^{-6} -21 z^6+a^5 z^5-5 a^3 z^5+7 z^5 a^{-1} -9 z^5 a^{-3} -10 z^5 a^{-5} -6 a^4 z^4+a^2 z^4+39 z^4 a^{-2} +10 z^4 a^{-4} -3 z^4 a^{-6} +33 z^4-2 a^5 z^3-a^3 z^3+a z^3+9 z^3 a^{-1} +17 z^3 a^{-3} +8 z^3 a^{-5} +3 a^4 z^2-4 a^2 z^2-22 z^2 a^{-2} -5 z^2 a^{-4} +z^2 a^{-6} -23 z^2+a^5 z+a^3 z-2 a z-6 z a^{-1} -6 z a^{-3} -2 z a^{-5} +2 a^2+4 a^{-2} + a^{-4} +6 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a194, K11a346,}

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["K11a106"];

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+12 t^2-18 t+21-18 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ q^6-3 q^5+5 q^4-9 q^3+13 q^2-14 q+15-13 q^{-1} +10 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5} }[/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]=

{K11a194, K11a346,}

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:

(1, -1)

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{ 4 }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ -\frac{34}{3} }[/math] [math]\displaystyle{ -\frac{38}{3} }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ \frac{16}{3} }[/math] [math]\displaystyle{ \frac{64}{3} }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -\frac{136}{3} }[/math] [math]\displaystyle{ -\frac{152}{3} }[/math] [math]\displaystyle{ -\frac{209}{30} }[/math] [math]\displaystyle{ -\frac{662}{15} }[/math] [math]\displaystyle{ \frac{1862}{45} }[/math] [math]\displaystyle{ -\frac{463}{18} }[/math] [math]\displaystyle{ \frac{751}{30} }[/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]0 is the signature of K11a106. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     2   -2 9                   3 1   2 7                 6 2     -4 5               7 3       4 3             7 6         -1 1           8 7           1 -1         6 8             2 -3       4 7               -3 -5     2 6                 4 -7   1 4                   -3 -9   2                     2 -11 1                       -1

Integral Khovanov Homology (db, data source)

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

K11a105 K11a107