K11a21

K11a20

K11a22

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a21 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X12,6,13,5 X2837 X18,9,19,10 X6,12,7,11 X22,14,1,13 X20,16,21,15 X10,17,11,18 X16,20,17,19 X14,22,15,21
Gauss code	1, -4, 2, -1, 3, -6, 4, -2, 5, -9, 6, -3, 7, -11, 8, -10, 9, -5, 10, -8, 11, -7
Dowker-Thistlethwaite code	4 8 12 2 18 6 22 20 10 16 14

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

Three dimensional invariants

Symmetry type Reversible Unknotting number [math]\displaystyle{ \{1,2\} }[/math] 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a21/ThurstonBennequinNumber Hyperbolic Volume 12.6394 A-Polynomial See Data:K11a21/A-polynomial

[edit Notes for K11a21'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 K11a21's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -5 t^2+19 t-27+19 t^{-1} -5 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ -5 z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 75, 2 }
Jones polynomial	[math]\displaystyle{ -q^{10}+3 q^9-5 q^8+8 q^7-10 q^6+11 q^5-12 q^4+10 q^3-7 q^2+5 q-2+ q^{-1} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^4 a^{-2} -2 z^4 a^{-4} -2 z^4 a^{-6} -2 z^2 a^{-4} -3 z^2 a^{-6} +3 z^2 a^{-8} +z^2+ a^{-2} - a^{-4} -2 a^{-6} +3 a^{-8} - a^{-10} +1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{10} a^{-6} +z^{10} a^{-8} +3 z^9 a^{-5} +6 z^9 a^{-7} +3 z^9 a^{-9} +3 z^8 a^{-4} +4 z^8 a^{-6} +4 z^8 a^{-8} +3 z^8 a^{-10} +3 z^7 a^{-3} -7 z^7 a^{-5} -21 z^7 a^{-7} -10 z^7 a^{-9} +z^7 a^{-11} +3 z^6 a^{-2} -2 z^6 a^{-4} -21 z^6 a^{-6} -29 z^6 a^{-8} -13 z^6 a^{-10} +2 z^5 a^{-1} -z^5 a^{-3} +12 z^5 a^{-5} +24 z^5 a^{-7} +5 z^5 a^{-9} -4 z^5 a^{-11} -3 z^4 a^{-2} -2 z^4 a^{-4} +28 z^4 a^{-6} +42 z^4 a^{-8} +16 z^4 a^{-10} +z^4-2 z^3 a^{-1} -2 z^3 a^{-3} -15 z^3 a^{-5} -16 z^3 a^{-7} +3 z^3 a^{-9} +4 z^3 a^{-11} +2 z^2 a^{-2} +3 z^2 a^{-4} -16 z^2 a^{-6} -22 z^2 a^{-8} -7 z^2 a^{-10} -2 z^2+2 z a^{-3} +7 z a^{-5} +6 z a^{-7} -z a^{-11} - a^{-2} - a^{-4} +2 a^{-6} +3 a^{-8} + a^{-10} +1 }[/math]
The A2 invariant	Data:K11a21/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a21/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a21 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["K11a21"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -5 t^2+19 t-27+19 t^{-1} -5 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -5 z^4-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]=

{ 75, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^{10}+3 q^9-5 q^8+8 q^7-10 q^6+11 q^5-12 q^4+10 q^3-7 q^2+5 q-2+ q^{-1} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^4 a^{-2} -2 z^4 a^{-4} -2 z^4 a^{-6} -2 z^2 a^{-4} -3 z^2 a^{-6} +3 z^2 a^{-8} +z^2+ a^{-2} - a^{-4} -2 a^{-6} +3 a^{-8} - a^{-10} +1 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^{10} a^{-6} +z^{10} a^{-8} +3 z^9 a^{-5} +6 z^9 a^{-7} +3 z^9 a^{-9} +3 z^8 a^{-4} +4 z^8 a^{-6} +4 z^8 a^{-8} +3 z^8 a^{-10} +3 z^7 a^{-3} -7 z^7 a^{-5} -21 z^7 a^{-7} -10 z^7 a^{-9} +z^7 a^{-11} +3 z^6 a^{-2} -2 z^6 a^{-4} -21 z^6 a^{-6} -29 z^6 a^{-8} -13 z^6 a^{-10} +2 z^5 a^{-1} -z^5 a^{-3} +12 z^5 a^{-5} +24 z^5 a^{-7} +5 z^5 a^{-9} -4 z^5 a^{-11} -3 z^4 a^{-2} -2 z^4 a^{-4} +28 z^4 a^{-6} +42 z^4 a^{-8} +16 z^4 a^{-10} +z^4-2 z^3 a^{-1} -2 z^3 a^{-3} -15 z^3 a^{-5} -16 z^3 a^{-7} +3 z^3 a^{-9} +4 z^3 a^{-11} +2 z^2 a^{-2} +3 z^2 a^{-4} -16 z^2 a^{-6} -22 z^2 a^{-8} -7 z^2 a^{-10} -2 z^2+2 z a^{-3} +7 z a^{-5} +6 z a^{-7} -z a^{-11} - a^{-2} - a^{-4} +2 a^{-6} +3 a^{-8} + a^{-10} +1 }[/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["K11a21"];

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{ -5 t^2+19 t-27+19 t^{-1} -5 t^{-2} }[/math], [math]\displaystyle{ -q^{10}+3 q^9-5 q^8+8 q^7-10 q^6+11 q^5-12 q^4+10 q^3-7 q^2+5 q-2+ 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]=

{}

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{226}{3} }[/math] [math]\displaystyle{ \frac{134}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{1360}{3} }[/math] [math]\displaystyle{ \frac{352}{3} }[/math] [math]\displaystyle{ 152 }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -\frac{904}{3} }[/math] [math]\displaystyle{ -\frac{536}{3} }[/math] [math]\displaystyle{ \frac{27089}{30} }[/math] [math]\displaystyle{ \frac{422}{15} }[/math] [math]\displaystyle{ \frac{16378}{45} }[/math] [math]\displaystyle{ \frac{2863}{18} }[/math] [math]\displaystyle{ -\frac{1231}{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 K11a21. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

K11a20 K11a22