K11n109

K11n108

K11n110

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n109 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X5,14,6,15 X16,7,17,8 X18,9,19,10 X2,11,3,12 X20,13,21,14 X15,22,16,1 X8,17,9,18 X12,19,13,20 X21,7,22,6
Gauss code	1, -6, 2, -1, -3, 11, 4, -9, 5, -2, 6, -10, 7, 3, -8, -4, 9, -5, 10, -7, -11, 8
Dowker-Thistlethwaite code	4 10 -14 16 18 2 20 -22 8 12 -6

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n109/ThurstonBennequinNumber Hyperbolic Volume 13.0869 A-Polynomial See Data:K11n109/A-polynomial

[edit Notes for K11n109's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n109's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+7 t^2-13 t+15-13 t^{-1} +7 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6+z^4+6 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 57, -4 }
Jones polynomial	[math]\displaystyle{ 2 q^{-2} -4 q^{-3} +7 q^{-4} -9 q^{-5} +10 q^{-6} -9 q^{-7} +8 q^{-8} -5 q^{-9} +2 q^{-10} - q^{-11} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^{10}-2 a^{10}+2 z^4 a^8+5 z^2 a^8+3 a^8-z^6 a^6-3 z^4 a^6-3 z^2 a^6-2 a^6+2 z^4 a^4+5 z^2 a^4+2 a^4 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^5 a^{13}-3 z^3 a^{13}+2 z a^{13}+2 z^6 a^{12}-4 z^4 a^{12}+z^2 a^{12}+3 z^7 a^{11}-6 z^5 a^{11}+4 z^3 a^{11}-2 z a^{11}+3 z^8 a^{10}-7 z^6 a^{10}+11 z^4 a^{10}-10 z^2 a^{10}+2 a^{10}+z^9 a^9+3 z^7 a^9-10 z^5 a^9+12 z^3 a^9-4 z a^9+5 z^8 a^8-13 z^6 a^8+20 z^4 a^8-13 z^2 a^8+3 a^8+z^9 a^7+z^7 a^7-2 z^5 a^7+2 z^3 a^7+2 z^8 a^6-4 z^6 a^6+8 z^4 a^6-8 z^2 a^6+2 a^6+z^7 a^5+z^5 a^5-3 z^3 a^5+3 z^4 a^4-6 z^2 a^4+2 a^4 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{34}-q^{32}-2 q^{28}+2 q^{26}+q^{24}+2 q^{20}-2 q^{18}+2 q^{16}-q^{14}+2 q^{10}-q^8+2 q^6 }[/math]
The G2 invariant	Data:K11n109/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n109 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["K11n109"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+7 t^2-13 t+15-13 t^{-1} +7 t^{-2} - t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 57, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ 2 q^{-2} -4 q^{-3} +7 q^{-4} -9 q^{-5} +10 q^{-6} -9 q^{-7} +8 q^{-8} -5 q^{-9} +2 q^{-10} - q^{-11} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^2 a^{10}-2 a^{10}+2 z^4 a^8+5 z^2 a^8+3 a^8-z^6 a^6-3 z^4 a^6-3 z^2 a^6-2 a^6+2 z^4 a^4+5 z^2 a^4+2 a^4 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n137,}

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["K11n109"];

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^3+7 t^2-13 t+15-13 t^{-1} +7 t^{-2} - t^{-3} }[/math], [math]\displaystyle{ 2 q^{-2} -4 q^{-3} +7 q^{-4} -9 q^{-5} +10 q^{-6} -9 q^{-7} +8 q^{-8} -5 q^{-9} +2 q^{-10} - q^{-11} }[/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]=

{K11n137,}

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:

(6, -16)

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{ 24 }[/math] [math]\displaystyle{ -128 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 828 }[/math] [math]\displaystyle{ 132 }[/math] [math]\displaystyle{ -3072 }[/math] [math]\displaystyle{ -\frac{17792}{3} }[/math] [math]\displaystyle{ -\frac{3104}{3} }[/math] [math]\displaystyle{ -864 }[/math] [math]\displaystyle{ 2304 }[/math] [math]\displaystyle{ 8192 }[/math] [math]\displaystyle{ 19872 }[/math] [math]\displaystyle{ 3168 }[/math] [math]\displaystyle{ \frac{215311}{5} }[/math] [math]\displaystyle{ \frac{1476}{5} }[/math] [math]\displaystyle{ \frac{269564}{15} }[/math] [math]\displaystyle{ \frac{1201}{3} }[/math] [math]\displaystyle{ \frac{11951}{5} }[/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 K11n109. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                   2 2 -5                 3 1 -2 -7               4 1   3 -9             5 3     -2 -11           5 4       1 -13         4 5         1 -15       4 5           -1 -17     1 4             3 -19   1 4               -3 -21   1                 1 -23 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=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/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}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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.

K11n108 K11n110