K11a122

K11a121

K11a123

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a122 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a122/ThurstonBennequinNumber Hyperbolic Volume 16.4594 A-Polynomial See Data:K11a122/A-polynomial

[edit Notes for K11a122's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 3 }[/math] Rasmussen s-Invariant 2

[edit Notes for K11a122's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^3-12 t^2+30 t-39+30 t^{-1} -12 t^{-2} +2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^6+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 127, -2 }
Jones polynomial	[math]\displaystyle{ -q^2+4 q-8+14 q^{-1} -18 q^{-2} +21 q^{-3} -20 q^{-4} +17 q^{-5} -13 q^{-6} +7 q^{-7} -3 q^{-8} + q^{-9} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^2 a^8+a^8-2 z^4 a^6-3 z^2 a^6-2 a^6+z^6 a^4+z^4 a^4+z^6 a^2+2 z^4 a^2+3 z^2 a^2+2 a^2-z^4-z^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{10}-3 z^4 a^{10}+2 z^2 a^{10}+3 z^7 a^9-8 z^5 a^9+6 z^3 a^9-2 z a^9+5 z^8 a^8-12 z^6 a^8+9 z^4 a^8-4 z^2 a^8+a^8+5 z^9 a^7-9 z^7 a^7+4 z^5 a^7-2 z a^7+2 z^{10} a^6+7 z^8 a^6-26 z^6 a^6+28 z^4 a^6-13 z^2 a^6+2 a^6+11 z^9 a^5-23 z^7 a^5+20 z^5 a^5-9 z^3 a^5+2 z a^5+2 z^{10} a^4+10 z^8 a^4-27 z^6 a^4+23 z^4 a^4-6 z^2 a^4+6 z^9 a^3-4 z^7 a^3-3 z^5 a^3+z^3 a^3+2 z a^3+8 z^8 a^2-10 z^6 a^2+z^4 a^2+3 z^2 a^2-2 a^2+7 z^7 a-10 z^5 a+3 z^3 a+4 z^6-6 z^4+2 z^2+z^5 a^{-1} -z^3 a^{-1} }[/math]
The A2 invariant	Data:K11a122/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a122/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a122 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["K11a122"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^3-12 t^2+30 t-39+30 t^{-1} -12 t^{-2} +2 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 127, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^2+4 q-8+14 q^{-1} -18 q^{-2} +21 q^{-3} -20 q^{-4} +17 q^{-5} -13 q^{-6} +7 q^{-7} -3 q^{-8} + q^{-9} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^2 a^8+a^8-2 z^4 a^6-3 z^2 a^6-2 a^6+z^6 a^4+z^4 a^4+z^6 a^2+2 z^4 a^2+3 z^2 a^2+2 a^2-z^4-z^2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^6 a^{10}-3 z^4 a^{10}+2 z^2 a^{10}+3 z^7 a^9-8 z^5 a^9+6 z^3 a^9-2 z a^9+5 z^8 a^8-12 z^6 a^8+9 z^4 a^8-4 z^2 a^8+a^8+5 z^9 a^7-9 z^7 a^7+4 z^5 a^7-2 z a^7+2 z^{10} a^6+7 z^8 a^6-26 z^6 a^6+28 z^4 a^6-13 z^2 a^6+2 a^6+11 z^9 a^5-23 z^7 a^5+20 z^5 a^5-9 z^3 a^5+2 z a^5+2 z^{10} a^4+10 z^8 a^4-27 z^6 a^4+23 z^4 a^4-6 z^2 a^4+6 z^9 a^3-4 z^7 a^3-3 z^5 a^3+z^3 a^3+2 z a^3+8 z^8 a^2-10 z^6 a^2+z^4 a^2+3 z^2 a^2-2 a^2+7 z^7 a-10 z^5 a+3 z^3 a+4 z^6-6 z^4+2 z^2+z^5 a^{-1} -z^3 a^{-1} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a1, K11a149,}

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["K11a122"];

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{ 2 t^3-12 t^2+30 t-39+30 t^{-1} -12 t^{-2} +2 t^{-3} }[/math], [math]\displaystyle{ -q^2+4 q-8+14 q^{-1} -18 q^{-2} +21 q^{-3} -20 q^{-4} +17 q^{-5} -13 q^{-6} +7 q^{-7} -3 q^{-8} + q^{-9} }[/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]=

{K11a1, K11a149,}

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:

(0, 2)

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{ 0 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -64 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{352}{3} }[/math] [math]\displaystyle{ -\frac{128}{3} }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 160 }[/math] [math]\displaystyle{ \frac{128}{3} }[/math] [math]\displaystyle{ 256 }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ 32 }[/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]-2 is the signature of K11a122. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                       1 -1 3                     3   3 1                   5 1   -4 -1                 9 3     6 -3               10 6       -4 -5             11 8         3 -7           9 10           1 -9         8 11             -3 -11       5 9               4 -13     2 8                 -6 -15   1 5                   4 -17   2                     -2 -19 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=-1 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/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.

K11a121 K11a123