K11n22

K11n21

K11n23

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n22 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X5,13,6,12 X2837 X14,9,15,10 X18,11,19,12 X13,7,14,6 X20,16,21,15 X10,17,11,18 X22,20,1,19 X16,22,17,21
Gauss code	1, -4, 2, -1, -3, 7, 4, -2, 5, -9, 6, 3, -7, -5, 8, -11, 9, -6, 10, -8, 11, -10
Dowker-Thistlethwaite code	4 8 -12 2 14 18 -6 20 10 22 16

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n22/ThurstonBennequinNumber Hyperbolic Volume 13.189 A-Polynomial See Data:K11n22/A-polynomial

[edit Notes for K11n22'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 K11n22's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-5 t^2+13 t-17+13 t^{-1} -5 t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6+z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 55, 2 }
Jones polynomial	[math]\displaystyle{ q^9-3 q^8+5 q^7-8 q^6+9 q^5-9 q^4+9 q^3-6 q^2+4 q-1 }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-4} -z^4 a^{-2} +4 z^4 a^{-4} -2 z^4 a^{-6} -z^2 a^{-2} +7 z^2 a^{-4} -5 z^2 a^{-6} +z^2 a^{-8} +4 a^{-4} -4 a^{-6} + a^{-8} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^9 a^{-5} +z^9 a^{-7} +3 z^8 a^{-4} +6 z^8 a^{-6} +3 z^8 a^{-8} +2 z^7 a^{-3} +4 z^7 a^{-5} +5 z^7 a^{-7} +3 z^7 a^{-9} -9 z^6 a^{-4} -17 z^6 a^{-6} -7 z^6 a^{-8} +z^6 a^{-10} -4 z^5 a^{-3} -18 z^5 a^{-5} -24 z^5 a^{-7} -10 z^5 a^{-9} +4 z^4 a^{-2} +17 z^4 a^{-4} +17 z^4 a^{-6} +z^4 a^{-8} -3 z^4 a^{-10} +z^3 a^{-1} +9 z^3 a^{-3} +24 z^3 a^{-5} +25 z^3 a^{-7} +9 z^3 a^{-9} -3 z^2 a^{-2} -12 z^2 a^{-4} -11 z^2 a^{-6} +2 z^2 a^{-10} -z a^{-1} -4 z a^{-3} -10 z a^{-5} -10 z a^{-7} -3 z a^{-9} +4 a^{-4} +4 a^{-6} + a^{-8} }[/math]
The A2 invariant	[math]\displaystyle{ -1+2 q^{-2} - q^{-4} + q^{-6} +3 q^{-8} +3 q^{-12} - q^{-14} - q^{-18} -3 q^{-20} + q^{-22} - q^{-24} + q^{-28} }[/math]
The G2 invariant	Data:K11n22/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n22 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["K11n22"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^3-5 t^2+13 t-17+13 t^{-1} -5 t^{-2} + t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 55, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^9-3 q^8+5 q^7-8 q^6+9 q^5-9 q^4+9 q^3-6 q^2+4 q-1 }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^9 a^{-5} +z^9 a^{-7} +3 z^8 a^{-4} +6 z^8 a^{-6} +3 z^8 a^{-8} +2 z^7 a^{-3} +4 z^7 a^{-5} +5 z^7 a^{-7} +3 z^7 a^{-9} -9 z^6 a^{-4} -17 z^6 a^{-6} -7 z^6 a^{-8} +z^6 a^{-10} -4 z^5 a^{-3} -18 z^5 a^{-5} -24 z^5 a^{-7} -10 z^5 a^{-9} +4 z^4 a^{-2} +17 z^4 a^{-4} +17 z^4 a^{-6} +z^4 a^{-8} -3 z^4 a^{-10} +z^3 a^{-1} +9 z^3 a^{-3} +24 z^3 a^{-5} +25 z^3 a^{-7} +9 z^3 a^{-9} -3 z^2 a^{-2} -12 z^2 a^{-4} -11 z^2 a^{-6} +2 z^2 a^{-10} -z a^{-1} -4 z a^{-3} -10 z a^{-5} -10 z a^{-7} -3 z a^{-9} +4 a^{-4} +4 a^{-6} + a^{-8} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_31, K11n11, K11n112, K11n127,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11n127,}

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["K11n22"];

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^3-5 t^2+13 t-17+13 t^{-1} -5 t^{-2} + t^{-3} }[/math], [math]\displaystyle{ q^9-3 q^8+5 q^7-8 q^6+9 q^5-9 q^4+9 q^3-6 q^2+4 q-1 }[/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]=

{9_31, K11n11, K11n112, K11n127,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11n127,}

Vassiliev invariants

V2 and V3:

(2, 2)

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{ 16 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{124}{3} }[/math] [math]\displaystyle{ \frac{20}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{448}{3} }[/math] [math]\displaystyle{ \frac{64}{3} }[/math] [math]\displaystyle{ 48 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{992}{3} }[/math] [math]\displaystyle{ \frac{160}{3} }[/math] [math]\displaystyle{ \frac{7471}{15} }[/math] [math]\displaystyle{ -\frac{2044}{15} }[/math] [math]\displaystyle{ \frac{19084}{45} }[/math] [math]\displaystyle{ \frac{161}{9} }[/math] [math]\displaystyle{ \frac{991}{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 K11n22. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

K11n21 K11n23