K11n99

K11n98

K11n100

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n99 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X5,14,6,15 X7,12,8,13 X9,19,10,18 X2,11,3,12 X13,6,14,7 X15,20,16,21 X17,22,18,1 X19,9,20,8 X21,16,22,17
Gauss code	1, -6, 2, -1, -3, 7, -4, 10, -5, -2, 6, 4, -7, 3, -8, 11, -9, 5, -10, 8, -11, 9
Dowker-Thistlethwaite code	4 10 -14 -12 -18 2 -6 -20 -22 -8 -16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number [math]\displaystyle{ \{1,2,3\} }[/math] 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n99/ThurstonBennequinNumber Hyperbolic Volume 11.6344 A-Polynomial See Data:K11n99/A-polynomial

[edit Notes for K11n99's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n99's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -3 t^2+10 t-13+10 t^{-1} -3 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -3 z^4-2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 39, -2 }
Jones polynomial	[math]\displaystyle{ 3 q^{-1} -4 q^{-2} +6 q^{-3} -7 q^{-4} +6 q^{-5} -6 q^{-6} +4 q^{-7} -2 q^{-8} + q^{-9} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^2 a^8+a^8-z^4 a^6-z^2 a^6-2 z^4 a^4-5 z^2 a^4-4 a^4+3 z^2 a^2+4 a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{10}-4 z^4 a^{10}+4 z^2 a^{10}+2 z^7 a^9-7 z^5 a^9+6 z^3 a^9-z a^9+2 z^8 a^8-6 z^6 a^8+5 z^4 a^8-4 z^2 a^8+a^8+z^9 a^7-2 z^7 a^7+4 z^5 a^7-11 z^3 a^7+5 z a^7+3 z^8 a^6-9 z^6 a^6+12 z^4 a^6-9 z^2 a^6+z^9 a^5-4 z^7 a^5+14 z^5 a^5-21 z^3 a^5+11 z a^5+z^8 a^4-2 z^6 a^4+3 z^4 a^4+4 z^2 a^4-4 a^4+3 z^5 a^3-4 z^3 a^3+5 z a^3+5 z^2 a^2-4 a^2 }[/math]
The A2 invariant	Data:K11n99/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n99/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n99 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["K11n99"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -3 t^2+10 t-13+10 t^{-1} -3 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -3 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]=

{ 39, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^6 a^{10}-4 z^4 a^{10}+4 z^2 a^{10}+2 z^7 a^9-7 z^5 a^9+6 z^3 a^9-z a^9+2 z^8 a^8-6 z^6 a^8+5 z^4 a^8-4 z^2 a^8+a^8+z^9 a^7-2 z^7 a^7+4 z^5 a^7-11 z^3 a^7+5 z a^7+3 z^8 a^6-9 z^6 a^6+12 z^4 a^6-9 z^2 a^6+z^9 a^5-4 z^7 a^5+14 z^5 a^5-21 z^3 a^5+11 z a^5+z^8 a^4-2 z^6 a^4+3 z^4 a^4+4 z^2 a^4-4 a^4+3 z^5 a^3-4 z^3 a^3+5 z a^3+5 z^2 a^2-4 a^2 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_144,}

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["K11n99"];

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

{10_144,}

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, 6)

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{ 48 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -\frac{172}{3} }[/math] [math]\displaystyle{ \frac{124}{3} }[/math] [math]\displaystyle{ -384 }[/math] [math]\displaystyle{ -576 }[/math] [math]\displaystyle{ -192 }[/math] [math]\displaystyle{ -240 }[/math] [math]\displaystyle{ -\frac{256}{3} }[/math] [math]\displaystyle{ 1152 }[/math] [math]\displaystyle{ \frac{1376}{3} }[/math] [math]\displaystyle{ -\frac{992}{3} }[/math] [math]\displaystyle{ \frac{53849}{15} }[/math] [math]\displaystyle{ \frac{1244}{15} }[/math] [math]\displaystyle{ \frac{67316}{45} }[/math] [math]\displaystyle{ \frac{2935}{9} }[/math] [math]\displaystyle{ \frac{329}{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]-2 is the signature of K11n99. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

K11n98 K11n100