K11a239

K11a238

K11a240

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a239 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number [math]\displaystyle{ \{1,2\} }[/math] 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a239/ThurstonBennequinNumber Hyperbolic Volume 19.5737 A-Polynomial See Data:K11a239/A-polynomial

[edit Notes for K11a239's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a239's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+7 t^3-22 t^2+42 t-51+42 t^{-1} -22 t^{-2} +7 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-z^6+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \left\{3,t^2+1\right\} }[/math]
Determinant and Signature	{ 195, 2 }
Jones polynomial	[math]\displaystyle{ -q^8+5 q^7-12 q^6+20 q^5-28 q^4+32 q^3-31 q^2+28 q-20+12 q^{-1} -5 q^{-2} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^8 a^{-2} -4 z^6 a^{-2} +2 z^6 a^{-4} +z^6-6 z^4 a^{-2} +5 z^4 a^{-4} -z^4 a^{-6} +2 z^4-2 z^2 a^{-2} +3 z^2 a^{-4} -z^2 a^{-6} +z^2+2 a^{-2} - a^{-4} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 4 z^{10} a^{-2} +4 z^{10} a^{-4} +11 z^9 a^{-1} +24 z^9 a^{-3} +13 z^9 a^{-5} +23 z^8 a^{-2} +29 z^8 a^{-4} +17 z^8 a^{-6} +11 z^8+5 a z^7-12 z^7 a^{-1} -33 z^7 a^{-3} -4 z^7 a^{-5} +12 z^7 a^{-7} +a^2 z^6-66 z^6 a^{-2} -72 z^6 a^{-4} -24 z^6 a^{-6} +5 z^6 a^{-8} -22 z^6-8 a z^5-8 z^5 a^{-1} -6 z^5 a^{-3} -21 z^5 a^{-5} -14 z^5 a^{-7} +z^5 a^{-9} -a^2 z^4+45 z^4 a^{-2} +45 z^4 a^{-4} +13 z^4 a^{-6} -3 z^4 a^{-8} +15 z^4+4 a z^3+8 z^3 a^{-1} +10 z^3 a^{-3} +12 z^3 a^{-5} +6 z^3 a^{-7} -8 z^2 a^{-2} -8 z^2 a^{-4} -3 z^2 a^{-6} -3 z^2+z a^{-3} +z a^{-5} -2 a^{-2} - a^{-4} }[/math]
The A2 invariant	[math]\displaystyle{ q^8-3 q^6+4 q^4-3 q^2-1+6 q^{-2} -5 q^{-4} +8 q^{-6} -3 q^{-8} + q^{-10} + q^{-12} -6 q^{-14} +5 q^{-16} -3 q^{-18} +2 q^{-22} - q^{-24} }[/math]
The G2 invariant	Data:K11a239/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a239 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["K11a239"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^4+7 t^3-22 t^2+42 t-51+42 t^{-1} -22 t^{-2} +7 t^{-3} - t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^8-z^6+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{ \left\{3,t^2+1\right\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 195, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^8+5 q^7-12 q^6+20 q^5-28 q^4+32 q^3-31 q^2+28 q-20+12 q^{-1} -5 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^8 a^{-2} -4 z^6 a^{-2} +2 z^6 a^{-4} +z^6-6 z^4 a^{-2} +5 z^4 a^{-4} -z^4 a^{-6} +2 z^4-2 z^2 a^{-2} +3 z^2 a^{-4} -z^2 a^{-6} +z^2+2 a^{-2} - a^{-4} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 4 z^{10} a^{-2} +4 z^{10} a^{-4} +11 z^9 a^{-1} +24 z^9 a^{-3} +13 z^9 a^{-5} +23 z^8 a^{-2} +29 z^8 a^{-4} +17 z^8 a^{-6} +11 z^8+5 a z^7-12 z^7 a^{-1} -33 z^7 a^{-3} -4 z^7 a^{-5} +12 z^7 a^{-7} +a^2 z^6-66 z^6 a^{-2} -72 z^6 a^{-4} -24 z^6 a^{-6} +5 z^6 a^{-8} -22 z^6-8 a z^5-8 z^5 a^{-1} -6 z^5 a^{-3} -21 z^5 a^{-5} -14 z^5 a^{-7} +z^5 a^{-9} -a^2 z^4+45 z^4 a^{-2} +45 z^4 a^{-4} +13 z^4 a^{-6} -3 z^4 a^{-8} +15 z^4+4 a z^3+8 z^3 a^{-1} +10 z^3 a^{-3} +12 z^3 a^{-5} +6 z^3 a^{-7} -8 z^2 a^{-2} -8 z^2 a^{-4} -3 z^2 a^{-6} -3 z^2+z a^{-3} +z a^{-5} -2 a^{-2} - a^{-4} }[/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["K11a239"];

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+7 t^3-22 t^2+42 t-51+42 t^{-1} -22 t^{-2} +7 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ -q^8+5 q^7-12 q^6+20 q^5-28 q^4+32 q^3-31 q^2+28 q-20+12 q^{-1} -5 q^{-2} + q^{-3} }[/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:

(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{62}{3} }[/math] [math]\displaystyle{ \frac{10}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{176}{3} }[/math] [math]\displaystyle{ -\frac{64}{3} }[/math] [math]\displaystyle{ 40 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{248}{3} }[/math] [math]\displaystyle{ \frac{40}{3} }[/math] [math]\displaystyle{ \frac{5071}{30} }[/math] [math]\displaystyle{ -\frac{1382}{15} }[/math] [math]\displaystyle{ \frac{5942}{45} }[/math] [math]\displaystyle{ \frac{593}{18} }[/math] [math]\displaystyle{ \frac{271}{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]2 is the signature of K11a239. 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                     4   4 13                   8 1   -7 11                 12 4     8 9               16 8       -8 7             16 12         4 5           15 16           1 3         13 16             -3 1       8 16               8 -1     4 12                 -8 -3   1 8                   7 -5   4                     -4 -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}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{15} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{16} }[/math] [math]\displaystyle{ {\mathbb Z}^{16} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{16} }[/math] [math]\displaystyle{ {\mathbb Z}^{16} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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.

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.

K11a238 K11a240