K11a27

K11a26

K11a28

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a27 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral 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:K11a27/ThurstonBennequinNumber Hyperbolic Volume 17.0192 A-Polynomial See Data:K11a27/A-polynomial

[edit Notes for K11a27'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 K11a27's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^3-13 t^2+34 t-45+34 t^{-1} -13 t^{-2} +2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^6-z^4+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 143, -2 }
Jones polynomial	[math]\displaystyle{ -q^2+4 q-8+15 q^{-1} -20 q^{-2} +23 q^{-3} -23 q^{-4} +20 q^{-5} -15 q^{-6} +9 q^{-7} -4 q^{-8} + q^{-9} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^2 a^8-2 z^4 a^6-z^2 a^6+a^6+z^6 a^4-3 z^2 a^4-3 a^4+z^6 a^2+2 z^4 a^2+4 z^2 a^2+3 a^2-z^4-z^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{10}-2 z^4 a^{10}+z^2 a^{10}+4 z^7 a^9-9 z^5 a^9+6 z^3 a^9-z a^9+7 z^8 a^8-15 z^6 a^8+9 z^4 a^8-2 z^2 a^8+6 z^9 a^7-5 z^7 a^7-10 z^5 a^7+9 z^3 a^7-2 z a^7+2 z^{10} a^6+13 z^8 a^6-35 z^6 a^6+22 z^4 a^6-3 z^2 a^6-a^6+12 z^9 a^5-14 z^7 a^5-7 z^5 a^5+10 z^3 a^5-2 z a^5+2 z^{10} a^4+14 z^8 a^4-29 z^6 a^4+11 z^4 a^4+6 z^2 a^4-3 a^4+6 z^9 a^3+2 z^7 a^3-16 z^5 a^3+12 z^3 a^3-2 z a^3+8 z^8 a^2-6 z^6 a^2-6 z^4 a^2+9 z^2 a^2-3 a^2+7 z^7 a-9 z^5 a+4 z^3 a-z a+4 z^6-6 z^4+3 z^2+z^5 a^{-1} -z^3 a^{-1} }[/math]
The A2 invariant	Data:K11a27/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a27/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a27 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["K11a27"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^3-13 t^2+34 t-45+34 t^{-1} -13 t^{-2} +2 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 z^6-z^4+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]=

{ 143, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^2+4 q-8+15 q^{-1} -20 q^{-2} +23 q^{-3} -23 q^{-4} +20 q^{-5} -15 q^{-6} +9 q^{-7} -4 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^8-2 z^4 a^6-z^2 a^6+a^6+z^6 a^4-3 z^2 a^4-3 a^4+z^6 a^2+2 z^4 a^2+4 z^2 a^2+3 a^2-z^4-z^2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^6 a^{10}-2 z^4 a^{10}+z^2 a^{10}+4 z^7 a^9-9 z^5 a^9+6 z^3 a^9-z a^9+7 z^8 a^8-15 z^6 a^8+9 z^4 a^8-2 z^2 a^8+6 z^9 a^7-5 z^7 a^7-10 z^5 a^7+9 z^3 a^7-2 z a^7+2 z^{10} a^6+13 z^8 a^6-35 z^6 a^6+22 z^4 a^6-3 z^2 a^6-a^6+12 z^9 a^5-14 z^7 a^5-7 z^5 a^5+10 z^3 a^5-2 z a^5+2 z^{10} a^4+14 z^8 a^4-29 z^6 a^4+11 z^4 a^4+6 z^2 a^4-3 a^4+6 z^9 a^3+2 z^7 a^3-16 z^5 a^3+12 z^3 a^3-2 z a^3+8 z^8 a^2-6 z^6 a^2-6 z^4 a^2+9 z^2 a^2-3 a^2+7 z^7 a-9 z^5 a+4 z^3 a-z a+4 z^6-6 z^4+3 z^2+z^5 a^{-1} -z^3 a^{-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["K11a27"];

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{ 2 t^3-13 t^2+34 t-45+34 t^{-1} -13 t^{-2} +2 t^{-3} }[/math], [math]\displaystyle{ -q^2+4 q-8+15 q^{-1} -20 q^{-2} +23 q^{-3} -23 q^{-4} +20 q^{-5} -15 q^{-6} +9 q^{-7} -4 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]=

{}

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:

(0, 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{ 0 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{112}{3} }[/math] [math]\displaystyle{ -\frac{160}{3} }[/math] [math]\displaystyle{ -24 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 296 }[/math] [math]\displaystyle{ -\frac{152}{3} }[/math] [math]\displaystyle{ \frac{1096}{3} }[/math] [math]\displaystyle{ -\frac{8}{3} }[/math] [math]\displaystyle{ 56 }[/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 K11a27. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                       1 -1 3                     3   3 1                   5 1   -4 -1                 10 3     7 -3               11 6       -5 -5             12 9         3 -7           11 11           0 -9         9 12             -3 -11       6 11               5 -13     3 9                 -6 -15   1 6                   5 -17   3                     -3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/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.

K11a26 K11a28