K11n111

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K11n110

K11n112

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 K11n111's page at Knotilus!

Visit K11n111's page at the original Knot Atlas!

[edit K11n111 Quick Notes]

[edit K11n111 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X_10,3,11,4 X_5,14,6,15 X_7,17,8,16 X_9,19,10,18 X_2,11,3,12 X_13,21,14,20 X_15,22,16,1 X_17,9,18,8 X_19,13,20,12 X_21,7,22,6
Gauss code	1, -6, 2, -1, -3, 11, -4, 9, -5, -2, 6, 10, -7, 3, -8, 4, -9, 5, -10, 7, -11, 8
Dowker-Thistlethwaite code	4 10 -14 -16 -18 2 -20 -22 -8 -12 -6

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	3
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11n111/ThurstonBennequinNumber
Hyperbolic Volume	8.61249
A-Polynomial	See Data:K11n111/A-polynomial

[edit Notes for K11n111'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 K11n111's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$- t 3 + t 2 + 3 t -5 + 3 t -1 + t -2 - t -3$
Conway polynomial	$- z 6 -5 z 4 -2 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 7, 2 }
Jones polynomial	$q 7 -2 q 6 + 2 q 5 -2 q 4 + 2 q 3 - q 2 + 1- q -1 + q -2$
HOMFLY-PT polynomial (db, data sources)	$- z 6 a -2 -6 z 4 a -2 + z 4 -9 z 2 a -2 + 2 z 2 a -4 + z 2 a -6 + 4 z 2 -4 a -2 + 2 a -4 + 3$
Kauffman polynomial (db, data sources)	$z 9 a -1 + z 9 a -3 + 2 z 8 a -2 + z 8 a -4 + z 8 -7 z 7 a -1 -7 z 7 a -3 -15 z 6 a -2 -7 z 6 a -4 + z 6 a -6 -7 z 6 + 14 z 5 a -1 + 13 z 5 a -3 + z 5 a -5 + 2 z 5 a -7 + 32 z 4 a -2 + 14 z 4 a -4 -2 z 4 a -6 + z 4 a -8 + 15 z 4 -9 z 3 a -1 -6 z 3 a -3 -2 z 3 a -5 -5 z 3 a -7 -22 z 2 a -2 -8 z 2 a -4 -2 z 2 a -8 -12 z 2 + z a -1 + z a -3 + z a -5 + z a -7 + 4 a -2 + 2 a -4 + 3$
The A2 invariant	$q 6 + q 4 + q 2 + q -2 - q -4 - q -6 - q -10 + q -12 + q -16 - q -20 + q -22$
The G2 invariant	Data:K11n111/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n111 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["K11n111"];

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

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

Out[4]= $- t 3 + t 2 + 3 t -5 + 3 t -1 + t -2 - t -3$

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

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

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]= { 7, 2 }

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

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

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

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

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

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

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

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

Out[10]= $z 9 a -1 + z 9 a -3 + 2 z 8 a -2 + z 8 a -4 + z 8 -7 z 7 a -1 -7 z 7 a -3 -15 z 6 a -2 -7 z 6 a -4 + z 6 a -6 -7 z 6 + 14 z 5 a -1 + 13 z 5 a -3 + z 5 a -5 + 2 z 5 a -7 + 32 z 4 a -2 + 14 z 4 a -4 -2 z 4 a -6 + z 4 a -8 + 15 z 4 -9 z 3 a -1 -6 z 3 a -3 -2 z 3 a -5 -5 z 3 a -7 -22 z 2 a -2 -8 z 2 a -4 -2 z 2 a -8 -12 z 2 + z a -1 + z a -3 + z a -5 + z a -7 + 4 a -2 + 2 a -4 + 3$

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

Same Alexander/Conway Polynomial: {}

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

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["K11n111"];

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]= { $- t 3 + t 2 + 3 t -5 + 3 t -1 + t -2 - t -3$ , $q 7 -2 q 6 + 2 q 5 -2 q 4 + 2 q 3 - q 2 + 1- q -1 + q -2$ }

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]= {}

[edit] Vassiliev invariants

V₂ and V₃:

(-2, -2)

[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 K11n111. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-4

-3

-2

-1

-3

-5

Integral Khovanov Homology

(db, data source)

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