K11a252

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K11a251

K11a253

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 K11a252's page at Knotilus!

Visit K11a252's page at the original Knot Atlas!

[edit K11a252 Quick Notes]

[edit K11a252 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₆₂₇₁ X₈₄₉₃ X_12,5,13,6 X₂₈₃₇ X_18,9,19,10 X_16,12,17,11 X_4,13,5,14 X_20,16,21,15 X_22,17,1,18 X_14,20,15,19 X_10,21,11,22
Gauss code	1, -4, 2, -7, 3, -1, 4, -2, 5, -11, 6, -3, 7, -10, 8, -6, 9, -5, 10, -8, 11, -9
Dowker-Thistlethwaite code	6 8 12 2 18 16 4 20 22 14 10

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:K11a252/ThurstonBennequinNumber
Hyperbolic Volume	16.8559
A-Polynomial	See Data:K11a252/A-polynomial

[edit Notes for K11a252's three dimensional invariants]

[edit] Four dimensional invariants

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

[edit Notes for K11a252's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$- t 4 + 6 t 3 -16 t 2 + 27 t -31 + 27 t -1 -16 t -2 + 6 t -3 - t -4$
Conway polynomial	$- z 8 -2 z 6 + z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 131, 2 }
Jones polynomial	$q 7 -4 q 6 + 9 q 5 -15 q 4 + 19 q 3 -21 q 2 + 21 q -17 + 13 q -1 -7 q -2 + 3 q -3 - q -4$
HOMFLY-PT polynomial (db, data sources)	$- z 8 a -2 -5 z 6 a -2 + z 6 a -4 + 2 z 6 - a 2 z 4 -10 z 4 a -2 + 3 z 4 a -4 + 8 z 4 -3 a 2 z 2 -10 z 2 a -2 + 3 z 2 a -4 + 11 z 2 -2 a 2 -4 a -2 + a -4 + 6$
Kauffman polynomial (db, data sources)	$2 z 10 a -2 + 2 z 10 + 4 a z 9 + 12 z 9 a -1 + 8 z 9 a -3 + 3 a 2 z 8 + 17 z 8 a -2 + 14 z 8 a -4 + 6 z 8 + a 3 z 7 -10 a z 7 -26 z 7 a -1 - z 7 a -3 + 14 z 7 a -5 -11 a 2 z 6 -55 z 6 a -2 -22 z 6 a -4 + 9 z 6 a -6 -35 z 6 -4 a 3 z 5 + 2 a z 5 + 3 z 5 a -1 -27 z 5 a -3 -20 z 5 a -5 + 4 z 5 a -7 + 14 a 2 z 4 + 47 z 4 a -2 + 10 z 4 a -4 -7 z 4 a -6 + z 4 a -8 + 43 z 4 + 5 a 3 z 3 + 9 a z 3 + 15 z 3 a -1 + 23 z 3 a -3 + 11 z 3 a -5 - z 3 a -7 -8 a 2 z 2 -19 z 2 a -2 -3 z 2 a -4 + 2 z 2 a -6 -22 z 2 -2 a 3 z -5 a z -7 z a -1 -6 z a -3 -2 z a -5 + 2 a 2 + 4 a -2 + a -4 + 6$
The A2 invariant	$- q 12 -2 q 6 + 4 q 4 - q 2 + 3 + 3 q -2 -3 q -4 + 4 q -6 -5 q -8 + 2 q -10 - q -12 -2 q -14 + 3 q -16 -2 q -18 + q -20$
The G2 invariant	Data:K11a252/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a252 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["K11a252"];

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

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

Out[4]= $- t 4 + 6 t 3 -16 t 2 + 27 t -31 + 27 t -1 -16 t -2 + 6 t -3 - t -4$

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

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

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]= { 131, 2 }

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

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

Out[8]= $q 7 -4 q 6 + 9 q 5 -15 q 4 + 19 q 3 -21 q 2 + 21 q -17 + 13 q -1 -7 q -2 + 3 q -3 - q -4$

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

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

Out[9]= $- z 8 a -2 -5 z 6 a -2 + z 6 a -4 + 2 z 6 - a 2 z 4 -10 z 4 a -2 + 3 z 4 a -4 + 8 z 4 -3 a 2 z 2 -10 z 2 a -2 + 3 z 2 a -4 + 11 z 2 -2 a 2 -4 a -2 + a -4 + 6$

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

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

Out[10]= $2 z 10 a -2 + 2 z 10 + 4 a z 9 + 12 z 9 a -1 + 8 z 9 a -3 + 3 a 2 z 8 + 17 z 8 a -2 + 14 z 8 a -4 + 6 z 8 + a 3 z 7 -10 a z 7 -26 z 7 a -1 - z 7 a -3 + 14 z 7 a -5 -11 a 2 z 6 -55 z 6 a -2 -22 z 6 a -4 + 9 z 6 a -6 -35 z 6 -4 a 3 z 5 + 2 a z 5 + 3 z 5 a -1 -27 z 5 a -3 -20 z 5 a -5 + 4 z 5 a -7 + 14 a 2 z 4 + 47 z 4 a -2 + 10 z 4 a -4 -7 z 4 a -6 + z 4 a -8 + 43 z 4 + 5 a 3 z 3 + 9 a z 3 + 15 z 3 a -1 + 23 z 3 a -3 + 11 z 3 a -5 - z 3 a -7 -8 a 2 z 2 -19 z 2 a -2 -3 z 2 a -4 + 2 z 2 a -6 -22 z 2 -2 a 3 z -5 a z -7 z a -1 -6 z a -3 -2 z a -5 + 2 a 2 + 4 a -2 + a -4 + 6$

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

Same Alexander/Conway Polynomial: {K11a131, K11a254,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11a254,}

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["K11a252"];

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]= { $- t 4 + 6 t 3 -16 t 2 + 27 t -31 + 27 t -1 -16 t -2 + 6 t -3 - t -4$ , $q 7 -4 q 6 + 9 q 5 -15 q 4 + 19 q 3 -21 q 2 + 21 q -17 + 13 q -1 -7 q -2 + 3 q -3 - q -4$ }

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]= {K11a131, K11a254,}

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

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

Out[6]= {K11a254,}

[edit] Vassiliev invariants

V₂ and V₃:

(1, -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 K11a252. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

-3

-6

-2

-1

-4

-3

-5

-4

-7

-9

-1

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 1$	$i = 3$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = 1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$