K11a261

K11a260

K11a262

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a261 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X8394 X14,5,15,6 X16,8,17,7 X4,9,5,10 X18,11,19,12 X20,13,21,14 X2,15,3,16 X22,18,1,17 X12,19,13,20 X10,21,11,22
Gauss code	1, -8, 2, -5, 3, -1, 4, -2, 5, -11, 6, -10, 7, -3, 8, -4, 9, -6, 10, -7, 11, -9
Dowker-Thistlethwaite code	6 8 14 16 4 18 20 2 22 12 10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a261/ThurstonBennequinNumber Hyperbolic Volume 15.9423 A-Polynomial See Data:K11a261/A-polynomial

[edit Notes for K11a261's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a261's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-6 t^3+16 t^2-26 t+31-26 t^{-1} +16 t^{-2} -6 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+2 z^6+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 129, -4 }
Jones polynomial	[math]\displaystyle{ -q+4-7 q^{-1} +13 q^{-2} -17 q^{-3} +20 q^{-4} -21 q^{-5} +18 q^{-6} -14 q^{-7} +9 q^{-8} -4 q^{-9} + q^{-10} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^4 a^8+2 z^2 a^8+a^8-2 z^6 a^6-7 z^4 a^6-7 z^2 a^6-2 a^6+z^8 a^4+5 z^6 a^4+9 z^4 a^4+6 z^2 a^4-z^6 a^2-3 z^4 a^2-z^2 a^2+2 a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^4 a^{12}+4 z^5 a^{11}-z^3 a^{11}+9 z^6 a^{10}-8 z^4 a^{10}+3 z^2 a^{10}+13 z^7 a^9-18 z^5 a^9+10 z^3 a^9-2 z a^9+12 z^8 a^8-15 z^6 a^8+3 z^4 a^8-2 z^2 a^8+a^8+7 z^9 a^7+z^7 a^7-24 z^5 a^7+14 z^3 a^7-2 z a^7+2 z^{10} a^6+15 z^8 a^6-48 z^6 a^6+39 z^4 a^6-15 z^2 a^6+2 a^6+12 z^9 a^5-28 z^7 a^5+12 z^5 a^5+2 z a^5+2 z^{10} a^4+7 z^8 a^4-39 z^6 a^4+44 z^4 a^4-14 z^2 a^4+5 z^9 a^3-15 z^7 a^3+11 z^5 a^3-z^3 a^3+2 z a^3+4 z^8 a^2-15 z^6 a^2+17 z^4 a^2-4 z^2 a^2-2 a^2+z^7 a-3 z^5 a+2 z^3 a }[/math]
The A2 invariant	[math]\displaystyle{ q^{30}-q^{28}+2 q^{24}-3 q^{22}+3 q^{20}-2 q^{18}-q^{16}+q^{14}-5 q^{12}+4 q^{10}-2 q^8+3 q^6+3 q^4-q^2+2- q^{-2} }[/math]
The G2 invariant	Data:K11a261/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a261 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["K11a261"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-6 t^3+16 t^2-26 t+31-26 t^{-1} +16 t^{-2} -6 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+2 z^6+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]=

{ 129, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q+4-7 q^{-1} +13 q^{-2} -17 q^{-3} +20 q^{-4} -21 q^{-5} +18 q^{-6} -14 q^{-7} +9 q^{-8} -4 q^{-9} + q^{-10} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^4 a^{12}+4 z^5 a^{11}-z^3 a^{11}+9 z^6 a^{10}-8 z^4 a^{10}+3 z^2 a^{10}+13 z^7 a^9-18 z^5 a^9+10 z^3 a^9-2 z a^9+12 z^8 a^8-15 z^6 a^8+3 z^4 a^8-2 z^2 a^8+a^8+7 z^9 a^7+z^7 a^7-24 z^5 a^7+14 z^3 a^7-2 z a^7+2 z^{10} a^6+15 z^8 a^6-48 z^6 a^6+39 z^4 a^6-15 z^2 a^6+2 a^6+12 z^9 a^5-28 z^7 a^5+12 z^5 a^5+2 z a^5+2 z^{10} a^4+7 z^8 a^4-39 z^6 a^4+44 z^4 a^4-14 z^2 a^4+5 z^9 a^3-15 z^7 a^3+11 z^5 a^3-z^3 a^3+2 z a^3+4 z^8 a^2-15 z^6 a^2+17 z^4 a^2-4 z^2 a^2-2 a^2+z^7 a-3 z^5 a+2 z^3 a }[/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["K11a261"];

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^4-6 t^3+16 t^2-26 t+31-26 t^{-1} +16 t^{-2} -6 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ -q+4-7 q^{-1} +13 q^{-2} -17 q^{-3} +20 q^{-4} -21 q^{-5} +18 q^{-6} -14 q^{-7} +9 q^{-8} -4 q^{-9} + q^{-10} }[/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, 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{ 0 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -64 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{352}{3} }[/math] [math]\displaystyle{ -\frac{128}{3} }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 160 }[/math] [math]\displaystyle{ \frac{320}{3} }[/math] [math]\displaystyle{ 192 }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ 32 }[/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]-4 is the signature of K11a261. 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 3                       1 -1 1                     3   3 -1                   4 1   -3 -3                 9 3     6 -5               9 5       -4 -7             11 8         3 -9           10 9           -1 -11         8 11             -3 -13       6 10               4 -15     3 8                 -5 -17   1 6                   5 -19   3                     -3 -21 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/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}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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.

K11a260 K11a262