K11a330

K11a329

K11a331

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a330 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X12,4,13,3 X18,5,19,6 X22,8,1,7 X16,9,17,10 X4,12,5,11 X20,13,21,14 X8,15,9,16 X10,17,11,18 X2,19,3,20 X14,21,15,22
Gauss code	1, -10, 2, -6, 3, -1, 4, -8, 5, -9, 6, -2, 7, -11, 8, -5, 9, -3, 10, -7, 11, -4
Dowker-Thistlethwaite code	6 12 18 22 16 4 20 8 10 2 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a330/ThurstonBennequinNumber Hyperbolic Volume 14.4187 A-Polynomial See Data:K11a330/A-polynomial

[edit Notes for K11a330's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a330's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-5 t^3+12 t^2-17 t+19-17 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+3 z^6+2 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 89, -4 }
Jones polynomial	[math]\displaystyle{ q^2-3 q+6-9 q^{-1} +12 q^{-2} -13 q^{-3} +14 q^{-4} -12 q^{-5} +9 q^{-6} -6 q^{-7} +3 q^{-8} - q^{-9} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^4 z^8-a^6 z^6+6 a^4 z^6-2 a^2 z^6-4 a^6 z^4+14 a^4 z^4-9 a^2 z^4+z^4-5 a^6 z^2+16 a^4 z^2-12 a^2 z^2+3 z^2-3 a^6+7 a^4-5 a^2+2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^3 a^{11}+3 z^4 a^{10}+6 z^5 a^9-4 z^3 a^9+z a^9+9 z^6 a^8-12 z^4 a^8+3 z^2 a^8+11 z^7 a^7-22 z^5 a^7+9 z^3 a^7-2 z a^7+11 z^8 a^6-30 z^6 a^6+23 z^4 a^6-12 z^2 a^6+3 a^6+7 z^9 a^5-16 z^7 a^5-3 z^5 a^5+13 z^3 a^5-4 z a^5+2 z^{10} a^4+7 z^8 a^4-52 z^6 a^4+72 z^4 a^4-35 z^2 a^4+7 a^4+10 z^9 a^3-42 z^7 a^3+49 z^5 a^3-14 z^3 a^3+2 z^{10} a^2-3 z^8 a^2-18 z^6 a^2+43 z^4 a^2-27 z^2 a^2+5 a^2+3 z^9 a-15 z^7 a+24 z^5 a-13 z^3 a+z a+z^8-5 z^6+9 z^4-7 z^2+2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{26}+q^{24}-2 q^{22}-q^{16}+4 q^{14}-q^{12}+3 q^{10}-q^6+q^4-2 q^2+1+ q^{-6} }[/math]
The G2 invariant	Data:K11a330/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a330 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["K11a330"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-5 t^3+12 t^2-17 t+19-17 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+2 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]=

{ 89, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^2-3 q+6-9 q^{-1} +12 q^{-2} -13 q^{-3} +14 q^{-4} -12 q^{-5} +9 q^{-6} -6 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{ a^4 z^8-a^6 z^6+6 a^4 z^6-2 a^2 z^6-4 a^6 z^4+14 a^4 z^4-9 a^2 z^4+z^4-5 a^6 z^2+16 a^4 z^2-12 a^2 z^2+3 z^2-3 a^6+7 a^4-5 a^2+2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^3 a^{11}+3 z^4 a^{10}+6 z^5 a^9-4 z^3 a^9+z a^9+9 z^6 a^8-12 z^4 a^8+3 z^2 a^8+11 z^7 a^7-22 z^5 a^7+9 z^3 a^7-2 z a^7+11 z^8 a^6-30 z^6 a^6+23 z^4 a^6-12 z^2 a^6+3 a^6+7 z^9 a^5-16 z^7 a^5-3 z^5 a^5+13 z^3 a^5-4 z a^5+2 z^{10} a^4+7 z^8 a^4-52 z^6 a^4+72 z^4 a^4-35 z^2 a^4+7 a^4+10 z^9 a^3-42 z^7 a^3+49 z^5 a^3-14 z^3 a^3+2 z^{10} a^2-3 z^8 a^2-18 z^6 a^2+43 z^4 a^2-27 z^2 a^2+5 a^2+3 z^9 a-15 z^7 a+24 z^5 a-13 z^3 a+z a+z^8-5 z^6+9 z^4-7 z^2+2 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a40,}

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["K11a330"];

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^2-17 t+19-17 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ q^2-3 q+6-9 q^{-1} +12 q^{-2} -13 q^{-3} +14 q^{-4} -12 q^{-5} +9 q^{-6} -6 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]=

{K11a40,}

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:

(2, -5)

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{ 8 }[/math] [math]\displaystyle{ -40 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{364}{3} }[/math] [math]\displaystyle{ -\frac{4}{3} }[/math] [math]\displaystyle{ -320 }[/math] [math]\displaystyle{ -\frac{1744}{3} }[/math] [math]\displaystyle{ \frac{128}{3} }[/math] [math]\displaystyle{ -168 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 800 }[/math] [math]\displaystyle{ \frac{2912}{3} }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ \frac{43351}{15} }[/math] [math]\displaystyle{ -\frac{7444}{15} }[/math] [math]\displaystyle{ \frac{64564}{45} }[/math] [math]\displaystyle{ \frac{905}{9} }[/math] [math]\displaystyle{ \frac{2311}{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]-4 is the signature of K11a330. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

K11a329 K11a331