K11a251

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K11a250

K11a252

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 K11a251's page at Knotilus!

Visit K11a251's page at the original Knot Atlas!

[edit K11a251 Quick Notes]

[edit K11a251 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₆₂₇₁ X₈₄₉₃ X_12,5,13,6 X₂₈₃₇ X_16,10,17,9 X_18,11,19,12 X_4,13,5,14 X_20,15,21,16 X_22,18,1,17 X_14,19,15,20 X_10,21,11,22
Gauss code	1, -4, 2, -7, 3, -1, 4, -2, 5, -11, 6, -3, 7, -10, 8, -5, 9, -6, 10, -8, 11, -9
Dowker-Thistlethwaite code	6 8 12 2 16 18 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:K11a251/ThurstonBennequinNumber
Hyperbolic Volume	16.9235
A-Polynomial	See Data:K11a251/A-polynomial

[edit Notes for K11a251's three dimensional invariants]

[edit] Four dimensional invariants

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

[edit Notes for K11a251's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$t 4 -6 t 3 + 16 t 2 -27 t + 33-27 t -1 + 16 t -2 -6 t -3 + t -4$
Conway polynomial	$z 8 + 2 z 6 - z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 133, 0 }
Jones polynomial	$- q 5 + 4 q 4 -9 q 3 + 15 q 2 -19 q + 22-21 q -1 + 18 q -2 -13 q -3 + 7 q -4 -3 q -5 + q -6$
HOMFLY-PT polynomial (db, data sources)	$z 8 -2 a 2 z 6 - z 6 a -2 + 5 z 6 + a 4 z 4 -8 a 2 z 4 -3 z 4 a -2 + 10 z 4 + 3 a 4 z 2 -11 a 2 z 2 -3 z 2 a -2 + 10 z 2 + 2 a 4 -5 a 2 - a -2 + 5$
Kauffman polynomial (db, data sources)	$2 a 2 z 10 + 2 z 10 + 5 a 3 z 9 + 12 a z 9 + 7 z 9 a -1 + 5 a 4 z 8 + 9 a 2 z 8 + 10 z 8 a -2 + 14 z 8 + 3 a 5 z 7 -8 a 3 z 7 -24 a z 7 -5 z 7 a -1 + 8 z 7 a -3 + a 6 z 6 -12 a 4 z 6 -34 a 2 z 6 -17 z 6 a -2 + 4 z 6 a -4 -42 z 6 -8 a 5 z 5 + a 3 z 5 + 16 a z 5 -6 z 5 a -1 -12 z 5 a -3 + z 5 a -5 -3 a 6 z 4 + 10 a 4 z 4 + 42 a 2 z 4 + 12 z 4 a -2 -5 z 4 a -4 + 46 z 4 + 6 a 5 z 3 + 4 a 3 z 3 - a z 3 + 7 z 3 a -1 + 5 z 3 a -3 - z 3 a -5 + 2 a 6 z 2 -6 a 4 z 2 -24 a 2 z 2 -6 z 2 a -2 + z 2 a -4 -23 z 2 -2 a 5 z -3 a 3 z -2 a z -2 z a -1 - z a -3 + 2 a 4 + 5 a 2 + a -2 + 5$
The A2 invariant	$q 18 + 2 q 12 -4 q 10 + 2 q 8 -2 q 6 -2 q 4 + 4 q 2 -3 + 6 q -2 -2 q -4 + q -6 + 2 q -8 -3 q -10 + 2 q -12 - q -14$
The G2 invariant	Data:K11a251/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a251 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["K11a251"];

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

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

Out[4]= $t 4 -6 t 3 + 16 t 2 -27 t + 33-27 t -1 + 16 t -2 -6 t -3 + t -4$

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]= { 133, 0 }

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

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

Out[8]= $- q 5 + 4 q 4 -9 q 3 + 15 q 2 -19 q + 22-21 q -1 + 18 q -2 -13 q -3 + 7 q -4 -3 q -5 + q -6$

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

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

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

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

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

Out[10]= $2 a 2 z 10 + 2 z 10 + 5 a 3 z 9 + 12 a z 9 + 7 z 9 a -1 + 5 a 4 z 8 + 9 a 2 z 8 + 10 z 8 a -2 + 14 z 8 + 3 a 5 z 7 -8 a 3 z 7 -24 a z 7 -5 z 7 a -1 + 8 z 7 a -3 + a 6 z 6 -12 a 4 z 6 -34 a 2 z 6 -17 z 6 a -2 + 4 z 6 a -4 -42 z 6 -8 a 5 z 5 + a 3 z 5 + 16 a z 5 -6 z 5 a -1 -12 z 5 a -3 + z 5 a -5 -3 a 6 z 4 + 10 a 4 z 4 + 42 a 2 z 4 + 12 z 4 a -2 -5 z 4 a -4 + 46 z 4 + 6 a 5 z 3 + 4 a 3 z 3 - a z 3 + 7 z 3 a -1 + 5 z 3 a -3 - z 3 a -5 + 2 a 6 z 2 -6 a 4 z 2 -24 a 2 z 2 -6 z 2 a -2 + z 2 a -4 -23 z 2 -2 a 5 z -3 a 3 z -2 a z -2 z a -1 - z a -3 + 2 a 4 + 5 a 2 + a -2 + 5$

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

Same Alexander/Conway Polynomial: {K11a253,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11a228, K11a253,}

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["K11a251"];

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]= { $t 4 -6 t 3 + 16 t 2 -27 t + 33-27 t -1 + 16 t -2 -6 t -3 + t -4$ , $- q 5 + 4 q 4 -9 q 3 + 15 q 2 -19 q + 22-21 q -1 + 18 q -2 -13 q -3 + 7 q -4 -3 q -5 + q -6$ }

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]= {K11a253,}

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

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

Out[6]= {K11a228, K11a253,}

[edit] Vassiliev invariants

V₂ and V₃:

(-1, 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 K11a251. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

-5

-4

-1

-3

-5

-7

-6

-9

-11

-2

-13

Integral Khovanov Homology

(db, data source)

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