K11n42

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K11n41

K11n43

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 K11n42's page at Knotilus!

Visit K11n42's page at the original Knot Atlas!

[edit K11n42 Quick Notes]

K11n42 is the mirror of the "Kinoshita-Terasaka" knot; it is a mutant of the (mirror of the) Conway knot K11n34. See also Heegaard Floer Knot Homology.

[edit K11n42 Further Notes and Views]

K11n42 is not $k$ -colourable for any $k$ . See The Determinant and the Signature.

Knot K11n42.

A graph, knot K11n42.

A part of a knot and a part of a graph.

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₄₉₃ X_12,5,13,6 X₂₈₃₇ X_9,18,10,19 X_11,21,12,20 X_6,13,7,14 X_15,10,16,11 X_17,22,18,1 X_19,15,20,14 X_21,16,22,17
Gauss code	1, -4, 2, -1, 3, -7, 4, -2, -5, 8, -6, -3, 7, 10, -8, 11, -9, 5, -10, 6, -11, 9
Dowker-Thistlethwaite code	4 8 12 2 -18 -20 6 -10 -22 -14 -16

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Chiral
Unknotting number	1
3-genus	2
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11n42/ThurstonBennequinNumber
Hyperbolic Volume	11.2191
A-Polynomial	See Data:K11n42/A-polynomial

[edit Notes for K11n42'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 K11n42's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	1
Conway polynomial	1
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 1, 0 }
Jones polynomial	$- q 4 + 2 q 3 -2 q 2 + 2 q + q -2 -2 q -3 + 2 q -4 -2 q -5 + q -6$
HOMFLY-PT polynomial (db, data sources)	$- a 2 z 6 + z 6 + a 4 z 4 -6 a 2 z 4 - z 4 a -2 + 6 z 4 + 3 a 4 z 2 -11 a 2 z 2 -3 z 2 a -2 + 11 z 2 + 2 a 4 -6 a 2 -2 a -2 + 7$
Kauffman polynomial (db, data sources)	$a z 9 + z 9 a -1 + a 4 z 8 + 2 a 2 z 8 + 2 z 8 a -2 + 3 z 8 + 2 a 5 z 7 + 2 a 3 z 7 -5 a z 7 -4 z 7 a -1 + z 7 a -3 + a 6 z 6 -4 a 4 z 6 -14 a 2 z 6 -11 z 6 a -2 -20 z 6 -9 a 5 z 5 -12 a 3 z 5 -2 z 5 a -1 -5 z 5 a -3 -4 a 6 z 4 + 2 a 4 z 4 + 26 a 2 z 4 + 16 z 4 a -2 + 36 z 4 + 9 a 5 z 3 + 16 a 3 z 3 + 12 a z 3 + 11 z 3 a -1 + 6 z 3 a -3 + 3 a 6 z 2 -2 a 4 z 2 -20 a 2 z 2 -9 z 2 a -2 -24 z 2 -3 a 5 z -7 a 3 z -7 a z -5 z a -1 -2 z a -3 + 2 a 4 + 6 a 2 + 2 a -2 + 7$
The A2 invariant	$q 18 + q 14 - q 12 - q 10 - q 8 -2 q 6 + q 4 + 3 + 2 q -2 + q -4 + q -6 - q -8 - q -12$
The G2 invariant	Data:K11n42/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n42 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["K11n42"];

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

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

Out[4]= 1

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

Out[5]= 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]= { 1, 0 }

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

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

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

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

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

Out[9]= $- a 2 z 6 + z 6 + a 4 z 4 -6 a 2 z 4 - z 4 a -2 + 6 z 4 + 3 a 4 z 2 -11 a 2 z 2 -3 z 2 a -2 + 11 z 2 + 2 a 4 -6 a 2 -2 a -2 + 7$

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

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

Out[10]= $a z 9 + z 9 a -1 + a 4 z 8 + 2 a 2 z 8 + 2 z 8 a -2 + 3 z 8 + 2 a 5 z 7 + 2 a 3 z 7 -5 a z 7 -4 z 7 a -1 + z 7 a -3 + a 6 z 6 -4 a 4 z 6 -14 a 2 z 6 -11 z 6 a -2 -20 z 6 -9 a 5 z 5 -12 a 3 z 5 -2 z 5 a -1 -5 z 5 a -3 -4 a 6 z 4 + 2 a 4 z 4 + 26 a 2 z 4 + 16 z 4 a -2 + 36 z 4 + 9 a 5 z 3 + 16 a 3 z 3 + 12 a z 3 + 11 z 3 a -1 + 6 z 3 a -3 + 3 a 6 z 2 -2 a 4 z 2 -20 a 2 z 2 -9 z 2 a -2 -24 z 2 -3 a 5 z -7 a 3 z -7 a z -5 z a -1 -2 z a -3 + 2 a 4 + 6 a 2 + 2 a -2 + 7$

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

Same Alexander/Conway Polynomial: {0_1, K11n34,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11n34,}

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["K11n42"];

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

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]= {0_1, K11n34,}

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

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

Out[6]= {K11n34,}

[edit] Vassiliev invariants

V₂ and V₃:

(0, 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 =

0 is the signature of K11n42. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

-3

-5

-1

-7

-9

-11

-1

-13

Integral Khovanov Homology

(db, data source)

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