K11a85

K11a84

K11a86

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a85 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X12,5,13,6 X16,8,17,7 X2,10,3,9 X22,11,1,12 X20,14,21,13 X18,16,19,15 X8,18,9,17 X14,20,15,19 X6,21,7,22
Gauss code	1, -5, 2, -1, 3, -11, 4, -9, 5, -2, 6, -3, 7, -10, 8, -4, 9, -8, 10, -7, 11, -6
Dowker-Thistlethwaite code	4 10 12 16 2 22 20 18 8 14 6

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a85/ThurstonBennequinNumber Hyperbolic Volume 14.0551 A-Polynomial See Data:K11a85/A-polynomial

[edit Notes for K11a85'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 K11a85's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^3-10 t^2+25 t-33+25 t^{-1} -10 t^{-2} +2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^6+2 z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 107, 2 }
Jones polynomial	[math]\displaystyle{ q^9-4 q^8+7 q^7-11 q^6+15 q^5-17 q^4+17 q^3-14 q^2+11 q-6+3 q^{-1} - q^{-2} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-2} +z^6 a^{-4} +3 z^4 a^{-2} +2 z^4 a^{-4} -2 z^4 a^{-6} -z^4+5 z^2 a^{-2} +2 z^2 a^{-4} -3 z^2 a^{-6} +z^2 a^{-8} -2 z^2+3 a^{-2} - a^{-6} -1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{10} a^{-4} +z^{10} a^{-6} +3 z^9 a^{-3} +7 z^9 a^{-5} +4 z^9 a^{-7} +4 z^8 a^{-2} +8 z^8 a^{-4} +10 z^8 a^{-6} +6 z^8 a^{-8} +4 z^7 a^{-1} +z^7 a^{-3} -11 z^7 a^{-5} -4 z^7 a^{-7} +4 z^7 a^{-9} -z^6 a^{-2} -19 z^6 a^{-4} -33 z^6 a^{-6} -17 z^6 a^{-8} +z^6 a^{-10} +3 z^6+a z^5-4 z^5 a^{-1} -4 z^5 a^{-3} +2 z^5 a^{-5} -10 z^5 a^{-7} -11 z^5 a^{-9} -7 z^4 a^{-2} +17 z^4 a^{-4} +33 z^4 a^{-6} +13 z^4 a^{-8} -2 z^4 a^{-10} -6 z^4-2 a z^3-2 z^3 a^{-1} +6 z^3 a^{-5} +12 z^3 a^{-7} +6 z^3 a^{-9} +7 z^2 a^{-2} -5 z^2 a^{-4} -11 z^2 a^{-6} -3 z^2 a^{-8} +4 z^2+a z+2 z a^{-1} +z a^{-3} -3 z a^{-5} -3 z a^{-7} -3 a^{-2} + a^{-6} -1 }[/math]
The A2 invariant	Data:K11a85/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a85/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a85 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["K11a85"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^3-10 t^2+25 t-33+25 t^{-1} -10 t^{-2} +2 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 107, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^9-4 q^8+7 q^7-11 q^6+15 q^5-17 q^4+17 q^3-14 q^2+11 q-6+3 q^{-1} - q^{-2} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^6 a^{-2} +z^6 a^{-4} +3 z^4 a^{-2} +2 z^4 a^{-4} -2 z^4 a^{-6} -z^4+5 z^2 a^{-2} +2 z^2 a^{-4} -3 z^2 a^{-6} +z^2 a^{-8} -2 z^2+3 a^{-2} - a^{-6} -1 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^{10} a^{-4} +z^{10} a^{-6} +3 z^9 a^{-3} +7 z^9 a^{-5} +4 z^9 a^{-7} +4 z^8 a^{-2} +8 z^8 a^{-4} +10 z^8 a^{-6} +6 z^8 a^{-8} +4 z^7 a^{-1} +z^7 a^{-3} -11 z^7 a^{-5} -4 z^7 a^{-7} +4 z^7 a^{-9} -z^6 a^{-2} -19 z^6 a^{-4} -33 z^6 a^{-6} -17 z^6 a^{-8} +z^6 a^{-10} +3 z^6+a z^5-4 z^5 a^{-1} -4 z^5 a^{-3} +2 z^5 a^{-5} -10 z^5 a^{-7} -11 z^5 a^{-9} -7 z^4 a^{-2} +17 z^4 a^{-4} +33 z^4 a^{-6} +13 z^4 a^{-8} -2 z^4 a^{-10} -6 z^4-2 a z^3-2 z^3 a^{-1} +6 z^3 a^{-5} +12 z^3 a^{-7} +6 z^3 a^{-9} +7 z^2 a^{-2} -5 z^2 a^{-4} -11 z^2 a^{-6} -3 z^2 a^{-8} +4 z^2+a z+2 z a^{-1} +z a^{-3} -3 z a^{-5} -3 z a^{-7} -3 a^{-2} + a^{-6} -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["K11a85"];

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^3-10 t^2+25 t-33+25 t^{-1} -10 t^{-2} +2 t^{-3} }[/math], [math]\displaystyle{ q^9-4 q^8+7 q^7-11 q^6+15 q^5-17 q^4+17 q^3-14 q^2+11 q-6+3 q^{-1} - q^{-2} }[/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:

(3, 4)

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{ 32 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 142 }[/math] [math]\displaystyle{ 18 }[/math] [math]\displaystyle{ 384 }[/math] [math]\displaystyle{ \frac{1952}{3} }[/math] [math]\displaystyle{ \frac{416}{3} }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 512 }[/math] [math]\displaystyle{ 1704 }[/math] [math]\displaystyle{ 216 }[/math] [math]\displaystyle{ \frac{30591}{10} }[/math] [math]\displaystyle{ \frac{3014}{15} }[/math] [math]\displaystyle{ \frac{16222}{15} }[/math] [math]\displaystyle{ \frac{1}{6} }[/math] [math]\displaystyle{ \frac{1471}{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]2 is the signature of K11a85. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

K11a84 K11a86