K11n71

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n71 at Knotilus!
	[edit K11n71 Quick Notes]

[edit K11n71 Further Notes and Views]

Knot presentations

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

A Braid Representative

A Morse Link Presentation

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	$\{1,2\}$
3-genus	3
Bridge index	4
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11n71/ThurstonBennequinNumber
Hyperbolic Volume	13.5931
A-Polynomial	See Data:K11n71/A-polynomial

[edit Notes for K11n71's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$1$
Rasmussen s-Invariant	-2

[edit Notes for K11n71's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-7t^{2}+14t-17+14t^{-1}-7t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+5z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{t^{2}-t+1\right\}$
Determinant and Signature	{ 63, 2 }
Jones polynomial	$q^{9}-3q^{8}+5q^{7}-9q^{6}+10q^{5}-10q^{4}+11q^{3}-7q^{2}+5q-2$
HOMFLY-PT polynomial (db, data sources)	$2z^{6}a^{-4}-2z^{4}a^{-2}+10z^{4}a^{-4}-3z^{4}a^{-6}-5z^{2}a^{-2}+18z^{2}a^{-4}-10z^{2}a^{-6}+z^{2}a^{-8}-3a^{-2}+11a^{-4}-9a^{-6}+2a^{-8}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-5}+z^{9}a^{-7}+4z^{8}a^{-4}+7z^{8}a^{-6}+3z^{8}a^{-8}+4z^{7}a^{-3}+8z^{7}a^{-5}+7z^{7}a^{-7}+3z^{7}a^{-9}+z^{6}a^{-2}-10z^{6}a^{-4}-18z^{6}a^{-6}-6z^{6}a^{-8}+z^{6}a^{-10}-10z^{5}a^{-3}-31z^{5}a^{-5}-31z^{5}a^{-7}-10z^{5}a^{-9}+4z^{4}a^{-2}+19z^{4}a^{-4}+17z^{4}a^{-6}-z^{4}a^{-8}-3z^{4}a^{-10}+3z^{3}a^{-1}+19z^{3}a^{-3}+42z^{3}a^{-5}+36z^{3}a^{-7}+10z^{3}a^{-9}-6z^{2}a^{-2}-20z^{2}a^{-4}-15z^{2}a^{-6}+z^{2}a^{-8}+2z^{2}a^{-10}-3za^{-1}-11za^{-3}-21za^{-5}-17za^{-7}-4za^{-9}+3a^{-2}+11a^{-4}+9a^{-6}+2a^{-8}$
The A2 invariant	Data:K11n71/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n71/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n71 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["K11n71"];

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

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

Out[4]= $2t^{3}-7t^{2}+14t-17+14t^{-1}-7t^{-2}+2t^{-3}$

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

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

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]= { 63, 2 }

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

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

Out[8]= $q^{9}-3q^{8}+5q^{7}-9q^{6}+10q^{5}-10q^{4}+11q^{3}-7q^{2}+5q-2$

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

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

Out[9]= $2z^{6}a^{-4}-2z^{4}a^{-2}+10z^{4}a^{-4}-3z^{4}a^{-6}-5z^{2}a^{-2}+18z^{2}a^{-4}-10z^{2}a^{-6}+z^{2}a^{-8}-3a^{-2}+11a^{-4}-9a^{-6}+2a^{-8}$

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

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

Out[10]= $z^{9}a^{-5}+z^{9}a^{-7}+4z^{8}a^{-4}+7z^{8}a^{-6}+3z^{8}a^{-8}+4z^{7}a^{-3}+8z^{7}a^{-5}+7z^{7}a^{-7}+3z^{7}a^{-9}+z^{6}a^{-2}-10z^{6}a^{-4}-18z^{6}a^{-6}-6z^{6}a^{-8}+z^{6}a^{-10}-10z^{5}a^{-3}-31z^{5}a^{-5}-31z^{5}a^{-7}-10z^{5}a^{-9}+4z^{4}a^{-2}+19z^{4}a^{-4}+17z^{4}a^{-6}-z^{4}a^{-8}-3z^{4}a^{-10}+3z^{3}a^{-1}+19z^{3}a^{-3}+42z^{3}a^{-5}+36z^{3}a^{-7}+10z^{3}a^{-9}-6z^{2}a^{-2}-20z^{2}a^{-4}-15z^{2}a^{-6}+z^{2}a^{-8}+2z^{2}a^{-10}-3za^{-1}-11za^{-3}-21za^{-5}-17za^{-7}-4za^{-9}+3a^{-2}+11a^{-4}+9a^{-6}+2a^{-8}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_65, 10_77, K11n75,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11n75,}

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["K11n71"];

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]= { $2t^{3}-7t^{2}+14t-17+14t^{-1}-7t^{-2}+2t^{-3}$ , $q^{9}-3q^{8}+5q^{7}-9q^{6}+10q^{5}-10q^{4}+11q^{3}-7q^{2}+5q-2$ }

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_65, 10_77, K11n75,}

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

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

Out[6]= {K11n75,}

Vassiliev invariants

V₂ and V₃:

(4, 5)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

16

40

128

{\frac {536}{3}}

{\frac {64}{3}}

640

{\frac {2416}{3}}

{\frac {352}{3}}

104

{\frac {2048}{3}}

800

{\frac {8576}{3}}

{\frac {1024}{3}}

{\frac {58622}{15}}

{\frac {3632}{15}}

{\frac {53888}{45}}

{\frac {370}{9}}

{\frac {1982}{15}}

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j-2r=s+1

or

j-2r=s-1

, where

s=

2 is the signature of K11n71. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-1

0

1

2

3

4

5

6

7

8

χ

19

1

17

2

-2

15

3

1

2

13

6

2

-4

11

4

3

1

9

6

0

7

5

4

1

5

2

6

4

3

5

-2

1

3

-1

2

-2

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-1$	${\mathbb {Z} }^{2}$
$r=0$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{3}$
$r=1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=7$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=8$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$