K11n58

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K11n57

K11n59

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 K11n58's page at Knotilus!

Visit K11n58's page at the original Knot Atlas!

[edit K11n58 Quick Notes]

[edit K11n58 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₄₉₃ X_14,5,15,6 X₂₈₃₇ X_9,17,10,16 X_11,18,12,19 X_13,20,14,21 X_6,15,7,16 X_17,1,18,22 X_19,12,20,13 X_21,10,22,11
Gauss code	1, -4, 2, -1, 3, -8, 4, -2, -5, 11, -6, 10, -7, -3, 8, 5, -9, 6, -10, 7, -11, 9
Dowker-Thistlethwaite code	4 8 14 2 -16 -18 -20 6 -22 -12 -10

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:K11n58/ThurstonBennequinNumber
Hyperbolic Volume	9.97833
A-Polynomial	See Data:K11n58/A-polynomial

[edit Notes for K11n58'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 K11n58's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$t 3 -4 t 2 + 8 t -9 + 8 t -1 -4 t -2 + t -3$
Conway polynomial	$z 6 + 2 z 4 + z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 35, -2 }
Jones polynomial	$- q 4 + 2 q 3 -3 q 2 + 5 q -5 + 6 q -1 -5 q -2 + 4 q -3 -3 q -4 + q -5$
HOMFLY-PT polynomial (db, data sources)	$z 6 -2 a 2 z 4 - z 4 a -2 + 5 z 4 + a 4 z 2 -6 a 2 z 2 -3 z 2 a -2 + 9 z 2 + a 4 -4 a 2 -2 a -2 + 6$
Kauffman polynomial (db, data sources)	$a z 9 + z 9 a -1 + 2 a 2 z 8 + 2 z 8 a -2 + 4 z 8 + 2 a 3 z 7 - a z 7 -2 z 7 a -1 + z 7 a -3 + a 4 z 6 -6 a 2 z 6 -10 z 6 a -2 -17 z 6 -5 a 3 z 5 -5 a z 5 -5 z 5 a -1 -5 z 5 a -3 + 8 a 2 z 4 + 15 z 4 a -2 + 23 z 4 + 3 a 5 z 3 + 6 a 3 z 3 + 6 a z 3 + 10 z 3 a -1 + 7 z 3 a -3 + a 6 z 2 -10 a 2 z 2 -8 z 2 a -2 -17 z 2 - a 5 z -3 a 3 z -4 a z -4 z a -1 -2 z a -3 + a 4 + 4 a 2 + 2 a -2 + 6$
The A2 invariant	$q 16 - q 12 -2 q 8 + q 2 + 3 + q -2 + 2 q -4 - q -8 - q -12$
The G2 invariant	Data:K11n58/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n58 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["K11n58"];

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

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

Out[4]= $t 3 -4 t 2 + 8 t -9 + 8 t -1 -4 t -2 + t -3$

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

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

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]= { 35, -2 }

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

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

Out[8]= $- q 4 + 2 q 3 -3 q 2 + 5 q -5 + 6 q -1 -5 q -2 + 4 q -3 -3 q -4 + q -5$

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

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

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

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

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

Out[10]= $a z 9 + z 9 a -1 + 2 a 2 z 8 + 2 z 8 a -2 + 4 z 8 + 2 a 3 z 7 - a z 7 -2 z 7 a -1 + z 7 a -3 + a 4 z 6 -6 a 2 z 6 -10 z 6 a -2 -17 z 6 -5 a 3 z 5 -5 a z 5 -5 z 5 a -1 -5 z 5 a -3 + 8 a 2 z 4 + 15 z 4 a -2 + 23 z 4 + 3 a 5 z 3 + 6 a 3 z 3 + 6 a z 3 + 10 z 3 a -1 + 7 z 3 a -3 + a 6 z 2 -10 a 2 z 2 -8 z 2 a -2 -17 z 2 - a 5 z -3 a 3 z -4 a z -4 z a -1 -2 z a -3 + a 4 + 4 a 2 + 2 a -2 + 6$

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

Same Alexander/Conway Polynomial: {8_16, 10_156, K11n15, K11n56,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11n56,}

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["K11n58"];

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]= { $t 3 -4 t 2 + 8 t -9 + 8 t -1 -4 t -2 + t -3$ , $- q 4 + 2 q 3 -3 q 2 + 5 q -5 + 6 q -1 -5 q -2 + 4 q -3 -3 q -4 + q -5$ }

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_16, 10_156, K11n15, K11n56,}

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

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

Out[6]= {K11n56,}

[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 K11n58. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-4

-3

-2

-1

-3

-5

-1

-7

-9

-2

-11

Integral Khovanov Homology

(db, data source)

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