K11n24

From Knot Atlas

Jump to: navigation, search

K11n23

K11n25

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 K11n24's page at Knotilus!

Visit K11n24's page at the original Knot Atlas!

[edit K11n24 Quick Notes]

[edit K11n24 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₄₉₃ X_12,5,13,6 X₂₈₃₇ X_14,9,15,10 X_11,18,12,19 X_6,13,7,14 X_15,21,16,20 X_17,1,18,22 X_19,10,20,11 X_21,17,22,16
Gauss code	1, -4, 2, -1, 3, -7, 4, -2, 5, 10, -6, -3, 7, -5, -8, 11, -9, 6, -10, 8, -11, 9
Dowker-Thistlethwaite code	4 8 12 2 14 -18 6 -20 -22 -10 -16

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Chiral
Unknotting number	${1,2}$
3-genus	3
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11n24/ThurstonBennequinNumber
Hyperbolic Volume	9.90549
A-Polynomial	See Data:K11n24/A-polynomial

[edit Notes for K11n24's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$3$
Rasmussen s-Invariant	0

[edit Notes for K11n24's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$t 3 -3 t 2 + 5 t -5 + 5 t -1 -3 t -2 + t -3$
Conway polynomial	$z 6 + 3 z 4 + 2 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 23, 2 }
Jones polynomial	$- q 4 + 2 q 3 -3 q 2 + 4 q -3 + 4 q -1 -3 q -2 + 2 q -3 - q -4$
HOMFLY-PT polynomial (db, data sources)	$z 6 - a 2 z 4 - z 4 a -2 + 5 z 4 -3 a 2 z 2 -3 z 2 a -2 + 8 z 2 -2 a 2 -2 a -2 + 5$
Kauffman polynomial (db, data sources)	$a z 9 + z 9 a -1 + 2 a 2 z 8 + 2 z 8 a -2 + 4 z 8 + a 3 z 7 -2 a z 7 -2 z 7 a -1 + z 7 a -3 -10 a 2 z 6 -10 z 6 a -2 -20 z 6 -5 a 3 z 5 -8 a z 5 -8 z 5 a -1 -5 z 5 a -3 + 14 a 2 z 4 + 14 z 4 a -2 + 28 z 4 + 7 a 3 z 3 + 16 a z 3 + 16 z 3 a -1 + 7 z 3 a -3 -8 a 2 z 2 -8 z 2 a -2 -16 z 2 -3 a 3 z -7 a z -7 z a -1 -3 z a -3 + 2 a 2 + 2 a -2 + 5$
The A2 invariant	$- q 12 - q 8 + q 4 + q 2 + 3 + q -2 + q -4 - q -8 - q -12$
The G2 invariant	Data:K11n24/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n24 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["K11n24"];

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

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

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

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

Out[5]= $z 6 + 3 z 4 + 2 z 2 + 1$

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]= { 23, 2 }

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

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

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

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

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

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

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

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

Out[10]= $a z 9 + z 9 a -1 + 2 a 2 z 8 + 2 z 8 a -2 + 4 z 8 + a 3 z 7 -2 a z 7 -2 z 7 a -1 + z 7 a -3 -10 a 2 z 6 -10 z 6 a -2 -20 z 6 -5 a 3 z 5 -8 a z 5 -8 z 5 a -1 -5 z 5 a -3 + 14 a 2 z 4 + 14 z 4 a -2 + 28 z 4 + 7 a 3 z 3 + 16 a z 3 + 16 z 3 a -1 + 7 z 3 a -3 -8 a 2 z 2 -8 z 2 a -2 -16 z 2 -3 a 3 z -7 a z -7 z a -1 -3 z a -3 + 2 a 2 + 2 a -2 + 5$

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

Same Alexander/Conway Polynomial: {8_7,}

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["K11n24"];

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

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]= {8_7,}

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, 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 K11n24. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

-3

-5

-1

-7

-9

-1

Integral Khovanov Homology

(db, data source)

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