K11a86

K11a85

K11a87

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a86 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a86/ThurstonBennequinNumber Hyperbolic Volume 13.5574 A-Polynomial See Data:K11a86/A-polynomial

[edit Notes for K11a86's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 4 }[/math] Rasmussen s-Invariant -2

[edit Notes for K11a86's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+5 t^3-12 t^2+18 t-19+18 t^{-1} -12 t^{-2} +5 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-3 z^6-2 z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 91, 2 }
Jones polynomial	[math]\displaystyle{ q^7-3 q^6+6 q^5-10 q^4+13 q^3-14 q^2+14 q-12+9 q^{-1} -5 q^{-2} +3 q^{-3} - q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^8 a^{-2} -6 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-14 z^4 a^{-2} +4 z^4 a^{-4} +9 z^4-3 a^2 z^2-15 z^2 a^{-2} +5 z^2 a^{-4} +12 z^2-a^2-5 a^{-2} +2 a^{-4} +5 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{10} a^{-2} +z^{10}+3 a z^9+7 z^9 a^{-1} +4 z^9 a^{-3} +3 a^2 z^8+9 z^8 a^{-2} +6 z^8 a^{-4} +6 z^8+a^3 z^7-9 a z^7-20 z^7 a^{-1} -4 z^7 a^{-3} +6 z^7 a^{-5} -13 a^2 z^6-36 z^6 a^{-2} -9 z^6 a^{-4} +5 z^6 a^{-6} -35 z^6-4 a^3 z^5+3 a z^5+14 z^5 a^{-1} -2 z^5 a^{-3} -6 z^5 a^{-5} +3 z^5 a^{-7} +17 a^2 z^4+45 z^4 a^{-2} +7 z^4 a^{-4} -5 z^4 a^{-6} +z^4 a^{-8} +49 z^4+4 a^3 z^3+5 a z^3-2 z^3 a^{-1} +z^3 a^{-3} +z^3 a^{-5} -3 z^3 a^{-7} -8 a^2 z^2-26 z^2 a^{-2} -5 z^2 a^{-4} +2 z^2 a^{-6} -z^2 a^{-8} -26 z^2-a^3 z-2 a z-z a^{-1} +z a^{-5} +z a^{-7} +a^2+5 a^{-2} +2 a^{-4} +5 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{12}+q^8+3 q^4-q^2+1+ q^{-2} -2 q^{-4} +3 q^{-6} -3 q^{-8} + q^{-10} - q^{-12} - q^{-14} +2 q^{-16} - q^{-18} + q^{-20} }[/math]
The G2 invariant	[math]\displaystyle{ q^{60}-2 q^{58}+5 q^{56}-9 q^{54}+10 q^{52}-11 q^{50}+3 q^{48}+14 q^{46}-35 q^{44}+57 q^{42}-66 q^{40}+48 q^{38}-8 q^{36}-54 q^{34}+115 q^{32}-153 q^{30}+144 q^{28}-77 q^{26}-25 q^{24}+131 q^{22}-196 q^{20}+197 q^{18}-123 q^{16}+13 q^{14}+97 q^{12}-163 q^{10}+159 q^8-78 q^6-23 q^4+116 q^2-144+96 q^{-2} -2 q^{-4} -111 q^{-6} +188 q^{-8} -199 q^{-10} +135 q^{-12} -10 q^{-14} -126 q^{-16} +233 q^{-18} -263 q^{-20} +201 q^{-22} -81 q^{-24} -64 q^{-26} +176 q^{-28} -222 q^{-30} +186 q^{-32} -84 q^{-34} -29 q^{-36} +115 q^{-38} -137 q^{-40} +82 q^{-42} -82 q^{-46} +119 q^{-48} -98 q^{-50} +32 q^{-52} +51 q^{-54} -118 q^{-56} +147 q^{-58} -124 q^{-60} +63 q^{-62} +7 q^{-64} -72 q^{-66} +110 q^{-68} -116 q^{-70} +101 q^{-72} -58 q^{-74} +14 q^{-76} +30 q^{-78} -62 q^{-80} +71 q^{-82} -66 q^{-84} +48 q^{-86} -20 q^{-88} -4 q^{-90} +22 q^{-92} -30 q^{-94} +28 q^{-96} -20 q^{-98} +11 q^{-100} -2 q^{-102} -4 q^{-104} +5 q^{-106} -6 q^{-108} +4 q^{-110} -2 q^{-112} + q^{-114} }[/math]

Further Quantum Invariants

K11a86 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["K11a86"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^4+5 t^3-12 t^2+18 t-19+18 t^{-1} -12 t^{-2} +5 t^{-3} - t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^8-3 z^6-2 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]=

{ 91, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^7-3 q^6+6 q^5-10 q^4+13 q^3-14 q^2+14 q-12+9 q^{-1} -5 q^{-2} +3 q^{-3} - q^{-4} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^8 a^{-2} -6 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-14 z^4 a^{-2} +4 z^4 a^{-4} +9 z^4-3 a^2 z^2-15 z^2 a^{-2} +5 z^2 a^{-4} +12 z^2-a^2-5 a^{-2} +2 a^{-4} +5 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^{10} a^{-2} +z^{10}+3 a z^9+7 z^9 a^{-1} +4 z^9 a^{-3} +3 a^2 z^8+9 z^8 a^{-2} +6 z^8 a^{-4} +6 z^8+a^3 z^7-9 a z^7-20 z^7 a^{-1} -4 z^7 a^{-3} +6 z^7 a^{-5} -13 a^2 z^6-36 z^6 a^{-2} -9 z^6 a^{-4} +5 z^6 a^{-6} -35 z^6-4 a^3 z^5+3 a z^5+14 z^5 a^{-1} -2 z^5 a^{-3} -6 z^5 a^{-5} +3 z^5 a^{-7} +17 a^2 z^4+45 z^4 a^{-2} +7 z^4 a^{-4} -5 z^4 a^{-6} +z^4 a^{-8} +49 z^4+4 a^3 z^3+5 a z^3-2 z^3 a^{-1} +z^3 a^{-3} +z^3 a^{-5} -3 z^3 a^{-7} -8 a^2 z^2-26 z^2 a^{-2} -5 z^2 a^{-4} +2 z^2 a^{-6} -z^2 a^{-8} -26 z^2-a^3 z-2 a z-z a^{-1} +z a^{-5} +z a^{-7} +a^2+5 a^{-2} +2 a^{-4} +5 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a205,}

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["K11a86"];

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+5 t^3-12 t^2+18 t-19+18 t^{-1} -12 t^{-2} +5 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^7-3 q^6+6 q^5-10 q^4+13 q^3-14 q^2+14 q-12+9 q^{-1} -5 q^{-2} +3 q^{-3} - q^{-4} }[/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]=

{K11a205,}

Vassiliev invariants

V2 and V3:

(-1, -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{ -4 }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{34}{3} }[/math] [math]\displaystyle{ \frac{62}{3} }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ \frac{608}{3} }[/math] [math]\displaystyle{ \frac{224}{3} }[/math] [math]\displaystyle{ 80 }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ -\frac{136}{3} }[/math] [math]\displaystyle{ -\frac{248}{3} }[/math] [math]\displaystyle{ \frac{13169}{30} }[/math] [math]\displaystyle{ \frac{714}{5} }[/math] [math]\displaystyle{ \frac{898}{45} }[/math] [math]\displaystyle{ \frac{1999}{18} }[/math] [math]\displaystyle{ -\frac{2191}{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 K11a86. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

K11a85 K11a87