K11a87

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K11a86

K11a88

1 Knot presentations
2 Three dimensional invariants
3 Four dimensional invariants
4 Polynomial invariants
5 "Similar" Knots (within the Atlas)
6 Vassiliev invariants
7 Khovanov Homology
8 Computer Talk
9 Modifying This Page

(Knotscape image)

See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a87's page at Knotilus!

Visit K11a87's page at the original Knot Atlas!

[edit K11a87 Quick Notes]

[edit K11a87 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X_10,3,11,4 X_12,6,13,5 X_16,8,17,7 X_18,10,19,9 X_2,11,3,12 X_20,13,21,14 X_8,16,9,15 X_6,18,7,17 X_22,19,1,20 X_14,21,15,22
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

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	${1,2}$
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]

[edit] Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$0$
Rasmussen s-Invariant	0

[edit Notes for K11a87's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$-2 t 3 + 11 t 2 -28 t + 39-28 t -1 + 11 t -2 -2 t -3$
Conway polynomial	$-2 z 6 - z 4 -2 z 2 + 1$
2nd Alexander ideal (db, data sources)	$\left\{2,t^2+t+1\right\}$
Determinant and Signature	{ 121, 0 }
Jones polynomial	$q 6 -3 q 5 + 7 q 4 -12 q 3 + 16 q 2 -19 q + 20-17 q -1 + 13 q -2 -8 q -3 + 4 q -4 - q -5$
HOMFLY-PT polynomial (db, data sources)	$- z 6 a -2 - z 6 + 2 a 2 z 4 -3 z 4 a -2 + z 4 a -4 - z 4 - a 4 z 2 + 2 a 2 z 2 -6 z 2 a -2 + 2 z 2 a -4 + z 2 -4 a -2 + 2 a -4 + 3$
Kauffman polynomial (db, data sources)	$z 10 a -2 + z 10 + 4 a z 9 + 8 z 9 a -1 + 4 z 9 a -3 + 7 a 2 z 8 + 12 z 8 a -2 + 5 z 8 a -4 + 14 z 8 + 7 a 3 z 7 + 6 a z 7 -8 z 7 a -1 -4 z 7 a -3 + 3 z 7 a -5 + 4 a 4 z 6 -7 a 2 z 6 -37 z 6 a -2 -13 z 6 a -4 + z 6 a -6 -34 z 6 + a 5 z 5 -11 a 3 z 5 -22 a z 5 -8 z 5 a -1 -6 z 5 a -3 -8 z 5 a -5 -6 a 4 z 4 - a 2 z 4 + 41 z 4 a -2 + 12 z 4 a -4 -3 z 4 a -6 + 31 z 4 - a 5 z 3 + 5 a 3 z 3 + 16 a z 3 + 12 z 3 a -1 + 8 z 3 a -3 + 6 z 3 a -5 + 2 a 4 z 2 + a 2 z 2 -21 z 2 a -2 -7 z 2 a -4 + 2 z 2 a -6 -13 z 2 - a 3 z -3 a z -3 z a -1 -3 z a -3 -2 z a -5 + 4 a -2 + 2 a -4 + 3$
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.

In[3]:= K = Knot["K11a87"];

In[4]:= Alexander[K][t]

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

Out[4]= $-2 t 3 + 11 t 2 -28 t + 39-28 t -1 + 11 t -2 -2 t -3$

In[5]:= Conway[K][z]

Out[5]= $-2 z 6 - z 4 -2 z 2 + 1$

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]= $\left\{2,t^2+t+1\right\}$

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 121, 0 }

In[8]:= Jones[K][q]

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

Out[8]= $q 6 -3 q 5 + 7 q 4 -12 q 3 + 16 q 2 -19 q + 20-17 q -1 + 13 q -2 -8 q -3 + 4 q -4 - q -5$

In[9]:= HOMFLYPT[K][a, z]

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

Out[9]= $- z 6 a -2 - z 6 + 2 a 2 z 4 -3 z 4 a -2 + z 4 a -4 - z 4 - a 4 z 2 + 2 a 2 z 2 -6 z 2 a -2 + 2 z 2 a -4 + z 2 -4 a -2 + 2 a -4 + 3$

In[10]:= Kauffman[K][a, z]

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

Out[10]= $z 10 a -2 + z 10 + 4 a z 9 + 8 z 9 a -1 + 4 z 9 a -3 + 7 a 2 z 8 + 12 z 8 a -2 + 5 z 8 a -4 + 14 z 8 + 7 a 3 z 7 + 6 a z 7 -8 z 7 a -1 -4 z 7 a -3 + 3 z 7 a -5 + 4 a 4 z 6 -7 a 2 z 6 -37 z 6 a -2 -13 z 6 a -4 + z 6 a -6 -34 z 6 + a 5 z 5 -11 a 3 z 5 -22 a z 5 -8 z 5 a -1 -6 z 5 a -3 -8 z 5 a -5 -6 a 4 z 4 - a 2 z 4 + 41 z 4 a -2 + 12 z 4 a -4 -3 z 4 a -6 + 31 z 4 - a 5 z 3 + 5 a 3 z 3 + 16 a z 3 + 12 z 3 a -1 + 8 z 3 a -3 + 6 z 3 a -5 + 2 a 4 z 2 + a 2 z 2 -21 z 2 a -2 -7 z 2 a -4 + 2 z 2 a -6 -13 z 2 - a 3 z -3 a z -3 z a -1 -3 z a -3 -2 z a -5 + 4 a -2 + 2 a -4 + 3$

[edit] "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {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.

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]= { $-2 t 3 + 11 t 2 -28 t + 39-28 t -1 + 11 t -2 -2 t -3$ , $q 6 -3 q 5 + 7 q 4 -12 q 3 + 16 q 2 -19 q + 20-17 q -1 + 13 q -2 -8 q -3 + 4 q -4 - q -5$ }

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,}

[edit] Vassiliev invariants

V₂ and V₃:

(-2, -2)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

-8

-16

32

$\frac{116}{3}$

$\frac{76}{3}$

128

$\frac{800}{3}$

$\frac{224}{3}$

80

$-\frac{256}{3}$

128

$-\frac{928}{3}$

$-\frac{608}{3}$

$\frac{1769}{15}$

$\frac{628}{5}$

$-\frac{10204}{45}$

$\frac{1111}{9}$

$-\frac{1591}{15}$

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.

[edit] Khovanov Homology

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

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

-2

-5

-3

-1

-3

-4

-5

-7

-4

-9

-11

-1

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 0$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = 1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 4$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$