K11a5

K11a4

K11a6

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a5 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X10,6,11,5 X14,7,15,8 X2,9,3,10 X20,12,21,11 X16,14,17,13 X6,15,7,16 X22,17,1,18 X12,20,13,19 X18,21,19,22
Gauss code	1, -5, 2, -1, 3, -8, 4, -2, 5, -3, 6, -10, 7, -4, 8, -7, 9, -11, 10, -6, 11, -9
Dowker-Thistlethwaite code	4 8 10 14 2 20 16 6 22 12 18

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number [math]\displaystyle{ \{1,2\} }[/math] 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a5/ThurstonBennequinNumber Hyperbolic Volume 15.5012 A-Polynomial See Data:K11a5/A-polynomial

[edit Notes for K11a5's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 1 }[/math] Rasmussen s-Invariant 0

[edit Notes for K11a5's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+9 t^2-30 t+45-30 t^{-1} +9 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6+3 z^4-3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 125, 0 }
Jones polynomial	[math]\displaystyle{ -q^5+4 q^4-8 q^3+14 q^2-18 q+20-20 q^{-1} +17 q^{-2} -12 q^{-3} +7 q^{-4} -3 q^{-5} + q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^6-3 z^2 a^4-2 a^4+3 z^4 a^2+4 z^2 a^2+3 a^2-z^6-2 z^4-5 z^2-3+2 z^4 a^{-2} +2 z^2 a^{-2} +2 a^{-2} -z^2 a^{-4} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^2 z^{10}+z^{10}+3 a^3 z^9+7 a z^9+4 z^9 a^{-1} +4 a^4 z^8+10 a^2 z^8+7 z^8 a^{-2} +13 z^8+3 a^5 z^7+3 a^3 z^7+a z^7+8 z^7 a^{-1} +7 z^7 a^{-3} +a^6 z^6-6 a^4 z^6-19 a^2 z^6-4 z^6 a^{-2} +4 z^6 a^{-4} -20 z^6-8 a^5 z^5-18 a^3 z^5-23 a z^5-24 z^5 a^{-1} -10 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4-2 a^4 z^4+3 a^2 z^4-8 z^4 a^{-2} -6 z^4 a^{-4} +7 a^5 z^3+15 a^3 z^3+18 a z^3+16 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +3 a^6 z^2+6 a^4 z^2+7 a^2 z^2+8 z^2 a^{-2} +3 z^2 a^{-4} +9 z^2-2 a^5 z-3 a^3 z-4 a z-4 z a^{-1} -z a^{-3} -a^6-2 a^4-3 a^2-2 a^{-2} -3 }[/math]
The A2 invariant	Data:K11a5/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a5/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a5 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["K11a5"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+9 t^2-30 t+45-30 t^{-1} +9 t^{-2} - t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^6+3 z^4-3 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]=

{ 125, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^5+4 q^4-8 q^3+14 q^2-18 q+20-20 q^{-1} +17 q^{-2} -12 q^{-3} +7 q^{-4} -3 q^{-5} + q^{-6} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^2 z^{10}+z^{10}+3 a^3 z^9+7 a z^9+4 z^9 a^{-1} +4 a^4 z^8+10 a^2 z^8+7 z^8 a^{-2} +13 z^8+3 a^5 z^7+3 a^3 z^7+a z^7+8 z^7 a^{-1} +7 z^7 a^{-3} +a^6 z^6-6 a^4 z^6-19 a^2 z^6-4 z^6 a^{-2} +4 z^6 a^{-4} -20 z^6-8 a^5 z^5-18 a^3 z^5-23 a z^5-24 z^5 a^{-1} -10 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4-2 a^4 z^4+3 a^2 z^4-8 z^4 a^{-2} -6 z^4 a^{-4} +7 a^5 z^3+15 a^3 z^3+18 a z^3+16 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +3 a^6 z^2+6 a^4 z^2+7 a^2 z^2+8 z^2 a^{-2} +3 z^2 a^{-4} +9 z^2-2 a^5 z-3 a^3 z-4 a z-4 z a^{-1} -z a^{-3} -a^6-2 a^4-3 a^2-2 a^{-2} -3 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a112,}

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["K11a5"];

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+9 t^2-30 t+45-30 t^{-1} +9 t^{-2} - t^{-3} }[/math], [math]\displaystyle{ -q^5+4 q^4-8 q^3+14 q^2-18 q+20-20 q^{-1} +17 q^{-2} -12 q^{-3} +7 q^{-4} -3 q^{-5} + q^{-6} }[/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]=

{K11a112,}

Vassiliev invariants

V2 and V3:

(-3, 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{ -12 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 50 }[/math] [math]\displaystyle{ 14 }[/math] [math]\displaystyle{ -192 }[/math] [math]\displaystyle{ -\frac{704}{3} }[/math] [math]\displaystyle{ -\frac{224}{3} }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ -288 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ -600 }[/math] [math]\displaystyle{ -168 }[/math] [math]\displaystyle{ -\frac{1471}{10} }[/math] [math]\displaystyle{ -\frac{454}{15} }[/math] [math]\displaystyle{ -\frac{1022}{15} }[/math] [math]\displaystyle{ \frac{127}{6} }[/math] [math]\displaystyle{ -\frac{191}{10} }[/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]0 is the signature of K11a5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   5 1   -4 5                 9 3     6 3               9 5       -4 1             11 9         2 -1           10 10           0 -3         7 10             -3 -5       5 10               5 -7     2 7                 -5 -9   1 5                   4 -11   2                     -2 -13 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=1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/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.

K11a4 K11a6