K11a155

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K11a154

K11a156

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 K11a155's page at Knotilus!

Visit K11a155's page at the original Knot Atlas!

[edit K11a155 Quick Notes]

[edit K11a155 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X_10,4,11,3 X_18,5,19,6 X_12,7,13,8 X_2,10,3,9 X_16,11,17,12 X_20,13,21,14 X_8,15,9,16 X_22,17,1,18 X_14,19,15,20 X_6,21,7,22
Gauss code	1, -5, 2, -1, 3, -11, 4, -8, 5, -2, 6, -4, 7, -10, 8, -6, 9, -3, 10, -7, 11, -9
Dowker-Thistlethwaite code	4 10 18 12 2 16 20 8 22 14 6

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	${1,2}$
3-genus	3
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a155/ThurstonBennequinNumber
Hyperbolic Volume	18.5384
A-Polynomial	See Data:K11a155/A-polynomial

[edit Notes for K11a155'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 K11a155's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$3 t 3 -16 t 2 + 40 t -53 + 40 t -1 -16 t -2 + 3 t -3$
Conway polynomial	$3 z 6 + 2 z 4 + 3 z 2 + 1$
2nd Alexander ideal (db, data sources)	${3, t + 1}$
Determinant and Signature	{ 171, -2 }
Jones polynomial	$- q 2 + 5 q -12 + 19 q -1 -24 q -2 + 29 q -3 -27 q -4 + 23 q -5 -17 q -6 + 9 q -7 -4 q -8 + q -9$
HOMFLY-PT polynomial (db, data sources)	$z 2 a 8 + a 8 -3 z 4 a 6 -6 z 2 a 6 -5 a 6 + 2 z 6 a 4 + 6 z 4 a 4 + 10 z 2 a 4 + 6 a 4 + z 6 a 2 -2 z 2 a 2 - a 2 - z 4$
Kauffman polynomial (db, data sources)	$z 6 a 10 -2 z 4 a 10 + z 2 a 10 + 4 z 7 a 9 -9 z 5 a 9 + 7 z 3 a 9 -2 z a 9 + 7 z 8 a 8 -13 z 6 a 8 + 7 z 4 a 8 -2 z 2 a 8 + a 8 + 7 z 9 a 7 -5 z 7 a 7 -14 z 5 a 7 + 20 z 3 a 7 -9 z a 7 + 3 z 10 a 6 + 15 z 8 a 6 -45 z 6 a 6 + 38 z 4 a 6 -15 z 2 a 6 + 5 a 6 + 18 z 9 a 5 -25 z 7 a 5 -7 z 5 a 5 + 22 z 3 a 5 -10 z a 5 + 3 z 10 a 4 + 24 z 8 a 4 -61 z 6 a 4 + 45 z 4 a 4 -16 z 2 a 4 + 6 a 4 + 11 z 9 a 3 -4 z 7 a 3 -18 z 5 a 3 + 13 z 3 a 3 -4 z a 3 + 16 z 8 a 2 -25 z 6 a 2 + 13 z 4 a 2 -4 z 2 a 2 + a 2 + 12 z 7 a -15 z 5 a + 4 z 3 a - z a + 5 z 6 -3 z 4 + z 5 a -1$
The A2 invariant	Data:K11a155/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a155/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a155 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["K11a155"];

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

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

Out[4]= $3 t 3 -16 t 2 + 40 t -53 + 40 t -1 -16 t -2 + 3 t -3$

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

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

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]= ${3, t + 1}$

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 171, -2 }

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

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

Out[8]= $- q 2 + 5 q -12 + 19 q -1 -24 q -2 + 29 q -3 -27 q -4 + 23 q -5 -17 q -6 + 9 q -7 -4 q -8 + q -9$

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

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

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

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

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

Out[10]= $z 6 a 10 -2 z 4 a 10 + z 2 a 10 + 4 z 7 a 9 -9 z 5 a 9 + 7 z 3 a 9 -2 z a 9 + 7 z 8 a 8 -13 z 6 a 8 + 7 z 4 a 8 -2 z 2 a 8 + a 8 + 7 z 9 a 7 -5 z 7 a 7 -14 z 5 a 7 + 20 z 3 a 7 -9 z a 7 + 3 z 10 a 6 + 15 z 8 a 6 -45 z 6 a 6 + 38 z 4 a 6 -15 z 2 a 6 + 5 a 6 + 18 z 9 a 5 -25 z 7 a 5 -7 z 5 a 5 + 22 z 3 a 5 -10 z a 5 + 3 z 10 a 4 + 24 z 8 a 4 -61 z 6 a 4 + 45 z 4 a 4 -16 z 2 a 4 + 6 a 4 + 11 z 9 a 3 -4 z 7 a 3 -18 z 5 a 3 + 13 z 3 a 3 -4 z a 3 + 16 z 8 a 2 -25 z 6 a 2 + 13 z 4 a 2 -4 z 2 a 2 + a 2 + 12 z 7 a -15 z 5 a + 4 z 3 a - z a + 5 z 6 -3 z 4 + z 5 a -1$

[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["K11a155"];

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]= { $3 t 3 -16 t 2 + 40 t -53 + 40 t -1 -16 t -2 + 3 t -3$ , $- q 2 + 5 q -12 + 19 q -1 -24 q -2 + 29 q -3 -27 q -4 + 23 q -5 -17 q -6 + 9 q -7 -4 q -8 + q -9$ }

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:

(3, -4)

[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 K11a155. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-8

-7

-6

-5

-4

-3

-2

-1

-7

-1

-3

-5

-7

-9

-4

-11

-13

-8

-15

-17

-3

-19

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -3$	$i = -1$
$r = -8$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -5$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -4$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = -3$	${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = -2$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{15}$	${\mathbb Z}^{15}$
$r = -1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{14}$
$r = 0$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{11}$
$r = 1$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$