K11n76

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K11n75

K11n77

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 K11n76's page at Knotilus!

Visit K11n76's page at the original Knot Atlas!

[edit K11n76 Quick Notes]

[edit K11n76 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₄₉₃ X_14,5,15,6 X₂₈₃₇ X_9,20,10,21 X_16,11,17,12 X_18,13,19,14 X_6,15,7,16 X_12,17,13,18 X_19,22,20,1 X_21,10,22,11
Gauss code	1, -4, 2, -1, 3, -8, 4, -2, -5, 11, 6, -9, 7, -3, 8, -6, 9, -7, -10, 5, -11, 10
Dowker-Thistlethwaite code	4 8 14 2 -20 16 18 6 12 -22 -10

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	3
3-genus	4
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11n76/ThurstonBennequinNumber
Hyperbolic Volume	12.3125
A-Polynomial	See Data:K11n76/A-polynomial

[edit Notes for K11n76's three dimensional invariants]

[edit] Four dimensional invariants

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

[edit Notes for K11n76's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$t 4 -3 t 3 + 6 t 2 -8 t + 9-8 t -1 + 6 t -2 -3 t -3 + t -4$
Conway polynomial	$z 8 + 5 z 6 + 8 z 4 + 5 z 2 + 1$
2nd Alexander ideal (db, data sources)	$\left\{t^2-t+1\right\}$
Determinant and Signature	{ 45, -4 }
Jones polynomial	$- q + 2-4 q -1 + 7 q -2 -6 q -3 + 8 q -4 -7 q -5 + 5 q -6 -4 q -7 + q -8$
HOMFLY-PT polynomial (db, data sources)	$z 2 a 8 + 2 a 8 - z 6 a 6 -6 z 4 a 6 -13 z 2 a 6 -10 a 6 + z 8 a 4 + 7 z 6 a 4 + 19 z 4 a 4 + 25 z 2 a 4 + 13 a 4 - z 6 a 2 -5 z 4 a 2 -8 z 2 a 2 -4 a 2$
Kauffman polynomial (db, data sources)	$z 2 a 10 + 4 z 3 a 9 -2 z a 9 + 2 z 6 a 8 -2 z 4 a 8 + z 2 a 8 + 2 a 8 + 5 z 7 a 7 -18 z 5 a 7 + 26 z 3 a 7 -14 z a 7 + 4 z 8 a 6 -13 z 6 a 6 + 16 z 4 a 6 -17 z 2 a 6 + 10 a 6 + z 9 a 5 + 5 z 7 a 5 -31 z 5 a 5 + 42 z 3 a 5 -21 z a 5 + 6 z 8 a 4 -24 z 6 a 4 + 31 z 4 a 4 -26 z 2 a 4 + 13 a 4 + z 9 a 3 + z 7 a 3 -18 z 5 a 3 + 28 z 3 a 3 -13 z a 3 + 2 z 8 a 2 -9 z 6 a 2 + 13 z 4 a 2 -9 z 2 a 2 + 4 a 2 + z 7 a -5 z 5 a + 8 z 3 a -4 z a$
The A2 invariant	$q 28 + q 24 -3 q 22 -3 q 20 -3 q 18 -2 q 16 + 4 q 14 + 2 q 12 + 6 q 10 + 2 q 8 + q 6 -2 q 2 - q -2$
The G2 invariant	Data:K11n76/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n76 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["K11n76"];

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

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

Out[4]= $t 4 -3 t 3 + 6 t 2 -8 t + 9-8 t -1 + 6 t -2 -3 t -3 + t -4$

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

Out[5]= $z 8 + 5 z 6 + 8 z 4 + 5 z 2 + 1$

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]= $\left\{t^2-t+1\right\}$

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

Out[7]= { 45, -4 }

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

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

Out[8]= $- q + 2-4 q -1 + 7 q -2 -6 q -3 + 8 q -4 -7 q -5 + 5 q -6 -4 q -7 + q -8$

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

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

Out[9]= $z 2 a 8 + 2 a 8 - z 6 a 6 -6 z 4 a 6 -13 z 2 a 6 -10 a 6 + z 8 a 4 + 7 z 6 a 4 + 19 z 4 a 4 + 25 z 2 a 4 + 13 a 4 - z 6 a 2 -5 z 4 a 2 -8 z 2 a 2 -4 a 2$

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

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

Out[10]= $z 2 a 10 + 4 z 3 a 9 -2 z a 9 + 2 z 6 a 8 -2 z 4 a 8 + z 2 a 8 + 2 a 8 + 5 z 7 a 7 -18 z 5 a 7 + 26 z 3 a 7 -14 z a 7 + 4 z 8 a 6 -13 z 6 a 6 + 16 z 4 a 6 -17 z 2 a 6 + 10 a 6 + z 9 a 5 + 5 z 7 a 5 -31 z 5 a 5 + 42 z 3 a 5 -21 z a 5 + 6 z 8 a 4 -24 z 6 a 4 + 31 z 4 a 4 -26 z 2 a 4 + 13 a 4 + z 9 a 3 + z 7 a 3 -18 z 5 a 3 + 28 z 3 a 3 -13 z a 3 + 2 z 8 a 2 -9 z 6 a 2 + 13 z 4 a 2 -9 z 2 a 2 + 4 a 2 + z 7 a -5 z 5 a + 8 z 3 a -4 z a$

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

Same Alexander/Conway Polynomial: {10_62, K11n78,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11n78,}

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["K11n76"];

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]= { $t 4 -3 t 3 + 6 t 2 -8 t + 9-8 t -1 + 6 t -2 -3 t -3 + t -4$ , $- q + 2-4 q -1 + 7 q -2 -6 q -3 + 8 q -4 -7 q -5 + 5 q -6 -4 q -7 + q -8$ }

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]= {10_62, K11n78,}

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

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

Out[6]= {K11n78,}

[edit] Vassiliev invariants

V₂ and V₃:

(5, -7)

[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 =

-4 is the signature of K11n76. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

-2

-3

-5

-7

-9

-11

-2

-13

-15

-3

-17

Integral Khovanov Homology

(db, data source)

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