K11a286

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K11a285

K11a287

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 K11a286's page at Knotilus!

Visit K11a286's page at the original Knot Atlas!

[edit K11a286 Quick Notes]

[edit K11a286 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₆₂₇₁ X_10,3,11,4 X_16,6,17,5 X_18,7,19,8 X_14,10,15,9 X_2,11,3,12 X_20,14,21,13 X_4,16,5,15 X_22,17,1,18 X_12,20,13,19 X_8,21,9,22
Gauss code	1, -6, 2, -8, 3, -1, 4, -11, 5, -2, 6, -10, 7, -5, 8, -3, 9, -4, 10, -7, 11, -9
Dowker-Thistlethwaite code	6 10 16 18 14 2 20 4 22 12 8

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Chiral
Unknotting number	${1,2}$
3-genus	4
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a286/ThurstonBennequinNumber
Hyperbolic Volume	17.8936
A-Polynomial	See Data:K11a286/A-polynomial

[edit Notes for K11a286's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$1$
Rasmussen s-Invariant	-2

[edit Notes for K11a286's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$- t 4 + 6 t 3 -17 t 2 + 31 t -37 + 31 t -1 -17 t -2 + 6 t -3 - t -4$
Conway polynomial	$- z 8 -2 z 6 - z 4 + z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 147, 2 }
Jones polynomial	$q 7 -4 q 6 + 9 q 5 -16 q 4 + 21 q 3 -23 q 2 + 24 q -20 + 15 q -1 -9 q -2 + 4 q -3 - q -4$
HOMFLY-PT polynomial (db, data sources)	$- z 8 a -2 -5 z 6 a -2 + z 6 a -4 + 2 z 6 - a 2 z 4 -10 z 4 a -2 + 3 z 4 a -4 + 7 z 4 -2 a 2 z 2 -8 z 2 a -2 + 3 z 2 a -4 + 8 z 2 - a 2 - a -2 + 3$
Kauffman polynomial (db, data sources)	$3 z 10 a -2 + 3 z 10 + 6 a z 9 + 17 z 9 a -1 + 11 z 9 a -3 + 4 a 2 z 8 + 20 z 8 a -2 + 17 z 8 a -4 + 7 z 8 + a 3 z 7 -16 a z 7 -41 z 7 a -1 -9 z 7 a -3 + 15 z 7 a -5 -13 a 2 z 6 -72 z 6 a -2 -31 z 6 a -4 + 9 z 6 a -6 -45 z 6 -3 a 3 z 5 + 8 a z 5 + 15 z 5 a -1 -21 z 5 a -3 -21 z 5 a -5 + 4 z 5 a -7 + 14 a 2 z 4 + 64 z 4 a -2 + 20 z 4 a -4 -6 z 4 a -6 + z 4 a -8 + 51 z 4 + 3 a 3 z 3 + 5 a z 3 + 11 z 3 a -1 + 21 z 3 a -3 + 11 z 3 a -5 - z 3 a -7 -6 a 2 z 2 -21 z 2 a -2 -6 z 2 a -4 + z 2 a -6 -20 z 2 - a 3 z -3 a z -5 z a -1 -4 z a -3 - z a -5 + a 2 + a -2 + 3$
The A2 invariant	$- q 12 + q 10 -2 q 6 + 4 q 4 -3 q 2 + 2 + 2 q -2 -2 q -4 + 6 q -6 -4 q -8 + 3 q -10 -2 q -12 -3 q -14 + 3 q -16 -2 q -18 + q -20$
The G2 invariant	Data:K11a286/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a286 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["K11a286"];

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

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

Out[4]= $- t 4 + 6 t 3 -17 t 2 + 31 t -37 + 31 t -1 -17 t -2 + 6 t -3 - t -4$

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

Out[5]= $- z 8 -2 z 6 - z 4 + z 2 + 1$

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]= ${1}$

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

Out[7]= { 147, 2 }

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

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

Out[8]= $q 7 -4 q 6 + 9 q 5 -16 q 4 + 21 q 3 -23 q 2 + 24 q -20 + 15 q -1 -9 q -2 + 4 q -3 - q -4$

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

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

Out[9]= $- z 8 a -2 -5 z 6 a -2 + z 6 a -4 + 2 z 6 - a 2 z 4 -10 z 4 a -2 + 3 z 4 a -4 + 7 z 4 -2 a 2 z 2 -8 z 2 a -2 + 3 z 2 a -4 + 8 z 2 - a 2 - a -2 + 3$

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

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

Out[10]= $3 z 10 a -2 + 3 z 10 + 6 a z 9 + 17 z 9 a -1 + 11 z 9 a -3 + 4 a 2 z 8 + 20 z 8 a -2 + 17 z 8 a -4 + 7 z 8 + a 3 z 7 -16 a z 7 -41 z 7 a -1 -9 z 7 a -3 + 15 z 7 a -5 -13 a 2 z 6 -72 z 6 a -2 -31 z 6 a -4 + 9 z 6 a -6 -45 z 6 -3 a 3 z 5 + 8 a z 5 + 15 z 5 a -1 -21 z 5 a -3 -21 z 5 a -5 + 4 z 5 a -7 + 14 a 2 z 4 + 64 z 4 a -2 + 20 z 4 a -4 -6 z 4 a -6 + z 4 a -8 + 51 z 4 + 3 a 3 z 3 + 5 a z 3 + 11 z 3 a -1 + 21 z 3 a -3 + 11 z 3 a -5 - z 3 a -7 -6 a 2 z 2 -21 z 2 a -2 -6 z 2 a -4 + z 2 a -6 -20 z 2 - a 3 z -3 a z -5 z a -1 -4 z a -3 - z a -5 + a 2 + a -2 + 3$

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

Same Alexander/Conway Polynomial: {K11a196, K11a216,}

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

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["K11a286"];

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]= { $- t 4 + 6 t 3 -17 t 2 + 31 t -37 + 31 t -1 -17 t -2 + 6 t -3 - t -4$ , $q 7 -4 q 6 + 9 q 5 -16 q 4 + 21 q 3 -23 q 2 + 24 q -20 + 15 q -1 -9 q -2 + 4 q -3 - q -4$ }

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]= {K11a196, K11a216,}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]= {}

[edit] Vassiliev invariants

V₂ and V₃:

(1, 0)

[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 =

2 is the signature of K11a286. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

-3

-7

-2

-1

-5

-3

-5

-7

-9

-1

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 1$	$i = 3$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 0$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{12}$
$r = 1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$