K11a327

K11a326

K11a328

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a327 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

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:K11a327/ThurstonBennequinNumber Hyperbolic Volume 19.2356 A-Polynomial See Data:K11a327/A-polynomial

[edit Notes for K11a327'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 K11a327's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 3 t^3-17 t^2+44 t-59+44 t^{-1} -17 t^{-2} +3 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ 3 z^6+z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 187, 2 }
Jones polynomial	[math]\displaystyle{ q^9-5 q^8+11 q^7-19 q^6+26 q^5-30 q^4+31 q^3-26 q^2+20 q-12+5 q^{-1} - q^{-2} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-2} +2 z^6 a^{-4} +5 z^4 a^{-4} -3 z^4 a^{-6} -z^4-z^2 a^{-2} +7 z^2 a^{-4} -4 z^2 a^{-6} +z^2 a^{-8} +3 a^{-4} -2 a^{-6} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 4 z^{10} a^{-4} +4 z^{10} a^{-6} +13 z^9 a^{-3} +23 z^9 a^{-5} +10 z^9 a^{-7} +17 z^8 a^{-2} +26 z^8 a^{-4} +19 z^8 a^{-6} +10 z^8 a^{-8} +12 z^7 a^{-1} -8 z^7 a^{-3} -36 z^7 a^{-5} -11 z^7 a^{-7} +5 z^7 a^{-9} -26 z^6 a^{-2} -71 z^6 a^{-4} -61 z^6 a^{-6} -20 z^6 a^{-8} +z^6 a^{-10} +5 z^6+a z^5-14 z^5 a^{-1} -16 z^5 a^{-3} -3 z^5 a^{-5} -11 z^5 a^{-7} -9 z^5 a^{-9} +14 z^4 a^{-2} +54 z^4 a^{-4} +50 z^4 a^{-6} +12 z^4 a^{-8} -z^4 a^{-10} -3 z^4+4 z^3 a^{-1} +15 z^3 a^{-3} +23 z^3 a^{-5} +16 z^3 a^{-7} +4 z^3 a^{-9} -4 z^2 a^{-2} -16 z^2 a^{-4} -15 z^2 a^{-6} -3 z^2 a^{-8} -z a^{-1} -4 z a^{-3} -8 z a^{-5} -5 z a^{-7} +3 a^{-4} +2 a^{-6} }[/math]
The A2 invariant	Data:K11a327/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a327/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a327 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["K11a327"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 3 t^3-17 t^2+44 t-59+44 t^{-1} -17 t^{-2} +3 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 3 z^6+z^4+3 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]=

{ 187, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^9-5 q^8+11 q^7-19 q^6+26 q^5-30 q^4+31 q^3-26 q^2+20 q-12+5 q^{-1} - q^{-2} }[/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} +2 z^6 a^{-4} +5 z^4 a^{-4} -3 z^4 a^{-6} -z^4-z^2 a^{-2} +7 z^2 a^{-4} -4 z^2 a^{-6} +z^2 a^{-8} +3 a^{-4} -2 a^{-6} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 4 z^{10} a^{-4} +4 z^{10} a^{-6} +13 z^9 a^{-3} +23 z^9 a^{-5} +10 z^9 a^{-7} +17 z^8 a^{-2} +26 z^8 a^{-4} +19 z^8 a^{-6} +10 z^8 a^{-8} +12 z^7 a^{-1} -8 z^7 a^{-3} -36 z^7 a^{-5} -11 z^7 a^{-7} +5 z^7 a^{-9} -26 z^6 a^{-2} -71 z^6 a^{-4} -61 z^6 a^{-6} -20 z^6 a^{-8} +z^6 a^{-10} +5 z^6+a z^5-14 z^5 a^{-1} -16 z^5 a^{-3} -3 z^5 a^{-5} -11 z^5 a^{-7} -9 z^5 a^{-9} +14 z^4 a^{-2} +54 z^4 a^{-4} +50 z^4 a^{-6} +12 z^4 a^{-8} -z^4 a^{-10} -3 z^4+4 z^3 a^{-1} +15 z^3 a^{-3} +23 z^3 a^{-5} +16 z^3 a^{-7} +4 z^3 a^{-9} -4 z^2 a^{-2} -16 z^2 a^{-4} -15 z^2 a^{-6} -3 z^2 a^{-8} -z a^{-1} -4 z a^{-3} -8 z a^{-5} -5 z a^{-7} +3 a^{-4} +2 a^{-6} }[/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["K11a327"];

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{ 3 t^3-17 t^2+44 t-59+44 t^{-1} -17 t^{-2} +3 t^{-3} }[/math], [math]\displaystyle{ q^9-5 q^8+11 q^7-19 q^6+26 q^5-30 q^4+31 q^3-26 q^2+20 q-12+5 q^{-1} - q^{-2} }[/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:

(3, 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{ 12 }[/math] [math]\displaystyle{ 40 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 174 }[/math] [math]\displaystyle{ 26 }[/math] [math]\displaystyle{ 480 }[/math] [math]\displaystyle{ \frac{2608}{3} }[/math] [math]\displaystyle{ \frac{448}{3} }[/math] [math]\displaystyle{ 136 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 800 }[/math] [math]\displaystyle{ 2088 }[/math] [math]\displaystyle{ 312 }[/math] [math]\displaystyle{ \frac{42751}{10} }[/math] [math]\displaystyle{ -\frac{1106}{15} }[/math] [math]\displaystyle{ \frac{9714}{5} }[/math] [math]\displaystyle{ \frac{11}{2} }[/math] [math]\displaystyle{ \frac{2751}{10} }[/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 K11a327. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 19                       1 1 17                     4   -4 15                   7 1   6 13                 12 4     -8 11               14 7       7 9             16 12         -4 7           15 14           1 5         11 16             5 3       9 15               -6 1     4 12                 8 -1   1 8                   -7 -3   4                     4 -5 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=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{15} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{16} }[/math] [math]\displaystyle{ {\mathbb Z}^{16} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=8 }[/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.

K11a326 K11a328