K11a87

K11a86

K11a88

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a87 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 X8,16,9,15 X6,18,7,17 X22,19,1,20 X14,21,15,22
Gauss code	1, -6, 2, -1, 3, -9, 4, -8, 5, -2, 6, -3, 7, -11, 8, -4, 9, -5, 10, -7, 11, -10
Dowker-Thistlethwaite code	4 10 12 16 18 2 20 8 6 22 14

A Braid Representative	{{{braid_table}}}
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:K11a87/ThurstonBennequinNumber Hyperbolic Volume 15.1381 A-Polynomial See Data:K11a87/A-polynomial

[edit Notes for K11a87's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a87's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t^3+11 t^2-28 t+39-28 t^{-1} +11 t^{-2} -2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -2 z^6-z^4-2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \left\{2,t^2+t+1\right\} }[/math]
Determinant and Signature	{ 121, 0 }
Jones polynomial	[math]\displaystyle{ q^6-3 q^5+7 q^4-12 q^3+16 q^2-19 q+20-17 q^{-1} +13 q^{-2} -8 q^{-3} +4 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6 a^{-2} -z^6+2 a^2 z^4-3 z^4 a^{-2} +z^4 a^{-4} -z^4-a^4 z^2+2 a^2 z^2-6 z^2 a^{-2} +2 z^2 a^{-4} +z^2-4 a^{-2} +2 a^{-4} +3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{10} a^{-2} +z^{10}+4 a z^9+8 z^9 a^{-1} +4 z^9 a^{-3} +7 a^2 z^8+12 z^8 a^{-2} +5 z^8 a^{-4} +14 z^8+7 a^3 z^7+6 a z^7-8 z^7 a^{-1} -4 z^7 a^{-3} +3 z^7 a^{-5} +4 a^4 z^6-7 a^2 z^6-37 z^6 a^{-2} -13 z^6 a^{-4} +z^6 a^{-6} -34 z^6+a^5 z^5-11 a^3 z^5-22 a z^5-8 z^5 a^{-1} -6 z^5 a^{-3} -8 z^5 a^{-5} -6 a^4 z^4-a^2 z^4+41 z^4 a^{-2} +12 z^4 a^{-4} -3 z^4 a^{-6} +31 z^4-a^5 z^3+5 a^3 z^3+16 a z^3+12 z^3 a^{-1} +8 z^3 a^{-3} +6 z^3 a^{-5} +2 a^4 z^2+a^2 z^2-21 z^2 a^{-2} -7 z^2 a^{-4} +2 z^2 a^{-6} -13 z^2-a^3 z-3 a z-3 z a^{-1} -3 z a^{-3} -2 z a^{-5} +4 a^{-2} +2 a^{-4} +3 }[/math]
The A2 invariant	Data:K11a87/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a87/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a87 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["K11a87"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -2 t^3+11 t^2-28 t+39-28 t^{-1} +11 t^{-2} -2 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -2 z^6-z^4-2 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{ \left\{2,t^2+t+1\right\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 121, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^6-3 q^5+7 q^4-12 q^3+16 q^2-19 q+20-17 q^{-1} +13 q^{-2} -8 q^{-3} +4 q^{-4} - q^{-5} }[/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+2 a^2 z^4-3 z^4 a^{-2} +z^4 a^{-4} -z^4-a^4 z^2+2 a^2 z^2-6 z^2 a^{-2} +2 z^2 a^{-4} +z^2-4 a^{-2} +2 a^{-4} +3 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^{10} a^{-2} +z^{10}+4 a z^9+8 z^9 a^{-1} +4 z^9 a^{-3} +7 a^2 z^8+12 z^8 a^{-2} +5 z^8 a^{-4} +14 z^8+7 a^3 z^7+6 a z^7-8 z^7 a^{-1} -4 z^7 a^{-3} +3 z^7 a^{-5} +4 a^4 z^6-7 a^2 z^6-37 z^6 a^{-2} -13 z^6 a^{-4} +z^6 a^{-6} -34 z^6+a^5 z^5-11 a^3 z^5-22 a z^5-8 z^5 a^{-1} -6 z^5 a^{-3} -8 z^5 a^{-5} -6 a^4 z^4-a^2 z^4+41 z^4 a^{-2} +12 z^4 a^{-4} -3 z^4 a^{-6} +31 z^4-a^5 z^3+5 a^3 z^3+16 a z^3+12 z^3 a^{-1} +8 z^3 a^{-3} +6 z^3 a^{-5} +2 a^4 z^2+a^2 z^2-21 z^2 a^{-2} -7 z^2 a^{-4} +2 z^2 a^{-6} -13 z^2-a^3 z-3 a z-3 z a^{-1} -3 z a^{-3} -2 z a^{-5} +4 a^{-2} +2 a^{-4} +3 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a28, K11a96,}

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["K11a87"];

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+11 t^2-28 t+39-28 t^{-1} +11 t^{-2} -2 t^{-3} }[/math], [math]\displaystyle{ q^6-3 q^5+7 q^4-12 q^3+16 q^2-19 q+20-17 q^{-1} +13 q^{-2} -8 q^{-3} +4 q^{-4} - q^{-5} }[/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]=

{K11a28, K11a96,}

Vassiliev invariants

V2 and V3:

(-2, -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{ -8 }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{116}{3} }[/math] [math]\displaystyle{ \frac{76}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{800}{3} }[/math] [math]\displaystyle{ \frac{224}{3} }[/math] [math]\displaystyle{ 80 }[/math] [math]\displaystyle{ -\frac{256}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ -\frac{928}{3} }[/math] [math]\displaystyle{ -\frac{608}{3} }[/math] [math]\displaystyle{ \frac{1769}{15} }[/math] [math]\displaystyle{ \frac{628}{5} }[/math] [math]\displaystyle{ -\frac{10204}{45} }[/math] [math]\displaystyle{ \frac{1111}{9} }[/math] [math]\displaystyle{ -\frac{1591}{15} }[/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 K11a87. 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 13                       1 1 11                     2   -2 9                   5 1   4 7                 7 2     -5 5               9 5       4 3             10 7         -3 1           10 9           1 -1         8 11             3 -3       5 9               -4 -5     3 8                 5 -7   1 5                   -4 -9   3                     3 -11 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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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.

K11a86 K11a88