K11a118

From Knot Atlas

Jump to: navigation, search

K11a117

K11a119

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 K11a118's page at Knotilus!

Visit K11a118's page at the original Knot Atlas!

[edit K11a118 Quick Notes]

[edit K11a118 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X_10,3,11,4 X_14,6,15,5 X_18,7,19,8 X_12,10,13,9 X_2,11,3,12 X_8,14,9,13 X_20,16,21,15 X_22,18,1,17 X_6,19,7,20 X_16,22,17,21
Gauss code	1, -6, 2, -1, 3, -10, 4, -7, 5, -2, 6, -5, 7, -3, 8, -11, 9, -4, 10, -8, 11, -9
Dowker-Thistlethwaite code	4 10 14 18 12 2 8 20 22 6 16

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	${1,2}$
3-genus	3
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a118/ThurstonBennequinNumber
Hyperbolic Volume	13.3497
A-Polynomial	See Data:K11a118/A-polynomial

[edit Notes for K11a118's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$3$
Rasmussen s-Invariant	-2

[edit Notes for K11a118's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$2 t 3 -10 t 2 + 20 t -23 + 20 t -1 -10 t -2 + 2 t -3$
Conway polynomial	$2 z 6 + 2 z 4 -2 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 87, 2 }
Jones polynomial	$- q 8 + 3 q 7 -6 q 6 + 10 q 5 -12 q 4 + 14 q 3 -14 q 2 + 11 q -8 + 5 q -1 -2 q -2 + q -3$
HOMFLY-PT polynomial (db, data sources)	$z 6 a -2 + z 6 a -4 + 2 z 4 a -2 + 3 z 4 a -4 - z 4 a -6 -2 z 4 + a 2 z 2 + 4 z 2 a -4 -2 z 2 a -6 -5 z 2 + 2 a 2 - a -2 + 3 a -4 - a -6 -2$
Kauffman polynomial (db, data sources)	$z 10 a -2 + z 10 a -4 + 2 z 9 a -1 + 6 z 9 a -3 + 4 z 9 a -5 + 2 z 8 a -2 + 6 z 8 a -4 + 6 z 8 a -6 + 2 z 8 + 2 a z 7 -3 z 7 a -1 -19 z 7 a -3 -9 z 7 a -5 + 5 z 7 a -7 + a 2 z 6 -9 z 6 a -2 -26 z 6 a -4 -17 z 6 a -6 + 3 z 6 a -8 -2 z 6 -6 a z 5 + 3 z 5 a -1 + 31 z 5 a -3 + 10 z 5 a -5 -11 z 5 a -7 + z 5 a -9 -4 a 2 z 4 + 13 z 4 a -2 + 43 z 4 a -4 + 21 z 4 a -6 -6 z 4 a -8 -7 z 4 + 4 a z 3 -9 z 3 a -1 -23 z 3 a -3 -2 z 3 a -5 + 6 z 3 a -7 -2 z 3 a -9 + 5 a 2 z 2 -10 z 2 a -2 -24 z 2 a -4 -10 z 2 a -6 + z 2 a -8 + 8 z 2 + 5 z a -1 + 7 z a -3 + z a -5 - z a -7 -2 a 2 + a -2 + 3 a -4 + a -6 -2$
The A2 invariant	Data:K11a118/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a118/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a118 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["K11a118"];

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

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

Out[4]= $2 t 3 -10 t 2 + 20 t -23 + 20 t -1 -10 t -2 + 2 t -3$

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

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

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]= ${1}$

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

Out[7]= { 87, 2 }

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

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

Out[8]= $- q 8 + 3 q 7 -6 q 6 + 10 q 5 -12 q 4 + 14 q 3 -14 q 2 + 11 q -8 + 5 q -1 -2 q -2 + q -3$

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

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

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

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

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

Out[10]= $z 10 a -2 + z 10 a -4 + 2 z 9 a -1 + 6 z 9 a -3 + 4 z 9 a -5 + 2 z 8 a -2 + 6 z 8 a -4 + 6 z 8 a -6 + 2 z 8 + 2 a z 7 -3 z 7 a -1 -19 z 7 a -3 -9 z 7 a -5 + 5 z 7 a -7 + a 2 z 6 -9 z 6 a -2 -26 z 6 a -4 -17 z 6 a -6 + 3 z 6 a -8 -2 z 6 -6 a z 5 + 3 z 5 a -1 + 31 z 5 a -3 + 10 z 5 a -5 -11 z 5 a -7 + z 5 a -9 -4 a 2 z 4 + 13 z 4 a -2 + 43 z 4 a -4 + 21 z 4 a -6 -6 z 4 a -8 -7 z 4 + 4 a z 3 -9 z 3 a -1 -23 z 3 a -3 -2 z 3 a -5 + 6 z 3 a -7 -2 z 3 a -9 + 5 a 2 z 2 -10 z 2 a -2 -24 z 2 a -4 -10 z 2 a -6 + z 2 a -8 + 8 z 2 + 5 z a -1 + 7 z a -3 + z a -5 - z a -7 -2 a 2 + a -2 + 3 a -4 + a -6 -2$

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

Same Alexander/Conway Polynomial: {K11a90,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11a45,}

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["K11a118"];

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]= { $2 t 3 -10 t 2 + 20 t -23 + 20 t -1 -10 t -2 + 2 t -3$ , $- q 8 + 3 q 7 -6 q 6 + 10 q 5 -12 q 4 + 14 q 3 -14 q 2 + 11 q -8 + 5 q -1 -2 q -2 + q -3$ }

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]= {K11a90,}

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

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

Out[6]= {K11a45,}

[edit] Vassiliev invariants

V₂ and V₃:

(-2, 1)

[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 K11a118. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-4

-3

-2

-1

-3

-2

-3

-1

-3

-5

-1

-7

Integral Khovanov Homology

(db, data source)

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