K11a1

10_165

K11a2

	Visit K11a1's page at Knotilus! Visit K11a1's page at the original Knot Atlas!
	K11a1 Quick Notes

K11a1.

A graph, Knot K11a1.

Knot presentations

Planar diagram presentation	X4251 X8394 X10,6,11,5 X14,7,15,8 X2,9,3,10 X16,12,17,11 X20,14,21,13 X6,15,7,16 X22,18,1,17 X12,20,13,19 X18,22,19,21
Gauss code	1, -5, 2, -1, 3, -8, 4, -2, 5, -3, 6, -10, 7, -4, 8, -6, 9, -11, 10, -7, 11, -9
Dowker-Thistlethwaite code	4 8 10 14 2 16 20 6 22 12 18
Conway Notation	[221,211,2]

Three dimensional invariants

Symmetry type Reversible Unknotting number [math]\displaystyle{ \{1,2\} }[/math] 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a1/ThurstonBennequinNumber Hyperbolic Volume 15.6439 A-Polynomial See Data:K11a1/A-polynomial

[edit Notes for K11a1'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 K11a1'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^8+4 q^7-9 q^6+14 q^5-18 q^4+21 q^3-20 q^2+17 q-12+7 q^{-1} -3 q^{-2} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-2} +z^6 a^{-4} +z^4 a^{-2} +2 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2+3 z^2 a^{-4} -z^2 a^{-6} -3 z^2+a^2+2 a^{-4} - a^{-6} -1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{10} a^{-2} +z^{10} a^{-4} +3 z^9 a^{-1} +8 z^9 a^{-3} +5 z^9 a^{-5} +11 z^8 a^{-2} +16 z^8 a^{-4} +9 z^8 a^{-6} +4 z^8+3 a z^7+3 z^7 a^{-1} -4 z^7 a^{-3} +4 z^7 a^{-5} +8 z^7 a^{-7} +a^2 z^6-25 z^6 a^{-2} -36 z^6 a^{-4} -14 z^6 a^{-6} +4 z^6 a^{-8} -6 z^6-8 a z^5-18 z^5 a^{-1} -16 z^5 a^{-3} -20 z^5 a^{-5} -13 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+15 z^4 a^{-2} +28 z^4 a^{-4} +9 z^4 a^{-6} -5 z^4 a^{-8} -2 z^4+7 a z^3+15 z^3 a^{-1} +16 z^3 a^{-3} +16 z^3 a^{-5} +7 z^3 a^{-7} -z^3 a^{-9} +3 a^2 z^2-3 z^2 a^{-2} -10 z^2 a^{-4} -4 z^2 a^{-6} +z^2 a^{-8} +5 z^2-2 a z-4 z a^{-1} -4 z a^{-3} -4 z a^{-5} -2 z a^{-7} -a^2+2 a^{-4} + a^{-6} -1 }[/math]
The A2 invariant	Data:K11a1/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a1/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a1 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["K11a1"];

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^8+4 q^7-9 q^6+14 q^5-18 q^4+21 q^3-20 q^2+17 q-12+7 q^{-1} -3 q^{-2} + q^{-3} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^6 a^{-2} +z^6 a^{-4} +z^4 a^{-2} +2 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2+3 z^2 a^{-4} -z^2 a^{-6} -3 z^2+a^2+2 a^{-4} - a^{-6} -1 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^{10} a^{-2} +z^{10} a^{-4} +3 z^9 a^{-1} +8 z^9 a^{-3} +5 z^9 a^{-5} +11 z^8 a^{-2} +16 z^8 a^{-4} +9 z^8 a^{-6} +4 z^8+3 a z^7+3 z^7 a^{-1} -4 z^7 a^{-3} +4 z^7 a^{-5} +8 z^7 a^{-7} +a^2 z^6-25 z^6 a^{-2} -36 z^6 a^{-4} -14 z^6 a^{-6} +4 z^6 a^{-8} -6 z^6-8 a z^5-18 z^5 a^{-1} -16 z^5 a^{-3} -20 z^5 a^{-5} -13 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+15 z^4 a^{-2} +28 z^4 a^{-4} +9 z^4 a^{-6} -5 z^4 a^{-8} -2 z^4+7 a z^3+15 z^3 a^{-1} +16 z^3 a^{-3} +16 z^3 a^{-5} +7 z^3 a^{-7} -z^3 a^{-9} +3 a^2 z^2-3 z^2 a^{-2} -10 z^2 a^{-4} -4 z^2 a^{-6} +z^2 a^{-8} +5 z^2-2 a z-4 z a^{-1} -4 z a^{-3} -4 z a^{-5} -2 z a^{-7} -a^2+2 a^{-4} + a^{-6} -1 }[/math]

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{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{64}{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{ 208 }[/math] [math]\displaystyle{ -\frac{544}{3} }[/math] [math]\displaystyle{ 224 }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ 48 }[/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 K11a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math]

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[11, Alternating, 1]]
Out[2]=	11
In[3]:=	PD[Knot[11, Alternating, 1]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[14, 7, 15, 8], X[2, 9, 3, 10], X[16, 12, 17, 11], X[20, 14, 21, 13], X[6, 15, 7, 16], X[22, 18, 1, 17], X[12, 20, 13, 19], X[18, 22, 19, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 1]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -8, 4, -2, 5, -3, 6, -10, 7, -4, 8, -6, 9, -11, 10, -7, 11, -9]
In[5]:=	BR[Knot[11, Alternating, 1]]
Out[5]=	BR[Knot[11, Alternating, 1]]
In[6]:=	alex = Alexander[Knot[11, Alternating, 1]][t]
Out[6]=	2 12 30 -39 + ------------ - ------------ + ----------- + 30 Alternating - 3 2 Alternating Alternating Alternating 2 3 12 Alternating + 2 Alternating
In[7]:=	Conway[Knot[11, Alternating, 1]][z]
Out[7]=	6 1 + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 1], Knot[11, Alternating, 122], Knot[11, Alternating, 149]}
In[9]:=	{KnotDet[Knot[11, Alternating, 1]], KnotSignature[Knot[11, Alternating, 1]]}
Out[9]=	{127, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 1]][q]
Out[10]=	-3 3 7 2 3 4 5 6 -12 + q - -- + - + 17 q - 20 q + 21 q - 18 q + 14 q - 9 q + 2 q q 7 8 4 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 1], Knot[11, Alternating, 149]}
In[12]:=	A2Invariant[Knot[11, Alternating, 1]][q]
Out[12]=	-10 -6 3 2 2 4 6 8 10 12 -1 + q - q + -- - -- + 3 q - 4 q + 3 q - q + q + 3 q - 4 2 q q 14 16 18 20 22 24 3 q + 4 q - 2 q - 2 q + 2 q - q
In[13]:=	Kauffman[Knot[11, Alternating, 1]][a, z]
Out[13]=	2 2 -6 2 2 2 z 4 z 4 z 4 z 2 z 4 z -1 + a + -- - a - --- - --- - --- - --- - 2 a z + 5 z + -- - ---- - 4 7 5 3 a 8 6 a a a a a a 2 2 3 3 3 3 3 10 z 3 z 2 2 z 7 z 16 z 16 z 15 z 3 ----- - ---- + 3 a z - -- + ---- + ----- + ----- + ----- + 7 a z - 4 2 9 7 5 3 a a a a a a a 4 4 4 4 5 5 5 4 5 z 9 z 28 z 15 z 2 4 z 13 z 20 z 2 z - ---- + ---- + ----- + ----- - 3 a z + -- - ----- - ----- - 8 6 4 2 9 7 5 a a a a a a a 5 5 6 6 6 6 16 z 18 z 5 6 4 z 14 z 36 z 25 z ----- - ----- - 8 a z - 6 z + ---- - ----- - ----- - ----- + 3 a 8 6 4 2 a a a a a 7 7 7 7 8 8 2 6 8 z 4 z 4 z 3 z 7 8 9 z 16 z a z + ---- + ---- - ---- + ---- + 3 a z + 4 z + ---- + ----- + 7 5 3 a 6 4 a a a a a 8 9 9 9 10 10 11 z 5 z 8 z 3 z z z ----- + ---- + ---- + ---- + --- + --- 2 5 3 a 4 2 a a a a a
In[14]:=	{Vassiliev[2][Knot[11, Alternating, 1]], Vassiliev[3][Knot[11, Alternating, 1]]}
Out[14]=	{0, 2}
In[15]:=	Kh[Knot[11, Alternating, 1]][q, t]
Out[15]=	1 2 1 5 --------------- + --------------- + --------------- + --------------- + 4 7 3 5 3 3 2 3 Alternating q Alternating q Alternating q Alternating q 2 7 5 q 3 -------------- + ------------- + 10 q + ----------- + 8 q + 2 Alternating q Alternating Alternating q 3 5 2 5 11 Alternating q + 9 Alternating q + 10 Alternating q + 2 7 3 7 3 9 11 Alternating q + 8 Alternating q + 10 Alternating q + 4 9 4 11 5 11 6 Alternating q + 8 Alternating q + 3 Alternating q + 5 13 6 13 6 15 6 Alternating q + Alternating q + 3 Alternating q + 7 17 Alternating q