K11n82

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n82 at Knotilus!
	[edit K11n82 Quick Notes]

[edit K11n82 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₃₉₄ X_5,17,6,16 X_7,12,8,13 X_2,9,3,10 X_11,18,12,19 X_13,21,14,20 X_15,1,16,22 X_17,10,18,11 X_19,7,20,6 X_21,15,22,14
Gauss code	1, -5, 2, -1, -3, 10, -4, -2, 5, 9, -6, 4, -7, 11, -8, 3, -9, 6, -10, 7, -11, 8
Dowker-Thistlethwaite code	4 8 -16 -12 2 -18 -20 -22 -10 -6 -14

A Braid Representative

A Morse Link Presentation

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11n82/ThurstonBennequinNumber
Hyperbolic Volume	8.68999
A-Polynomial	See Data:K11n82/A-polynomial

[edit Notes for K11n82's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$3$
Rasmussen s-Invariant	-2

[edit Notes for K11n82's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-3t^{2}+4t-3+4t^{-1}-3t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+3z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 19, 2 }
Jones polynomial	$-q^{4}+2q^{3}-2q^{2}+3q-3+3q^{-1}-2q^{-2}+2q^{-3}-q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}-a^{2}z^{4}-z^{4}a^{-2}+5z^{4}-3a^{2}z^{2}-3z^{2}a^{-2}+7z^{2}-a^{2}-a^{-2}+3$
Kauffman polynomial (db, data sources)	$az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+z^{8}a^{-2}+3z^{8}+a^{3}z^{7}-4az^{7}-5z^{7}a^{-1}-11a^{2}z^{6}-5z^{6}a^{-2}-16z^{6}-5a^{3}z^{5}+2az^{5}+8z^{5}a^{-1}+z^{5}a^{-3}+17a^{2}z^{4}+8z^{4}a^{-2}+25z^{4}+6a^{3}z^{3}+2az^{3}-7z^{3}a^{-1}-3z^{3}a^{-3}-8a^{2}z^{2}-6z^{2}a^{-2}+z^{2}a^{-4}-15z^{2}-a^{3}z-az+za^{-1}+2za^{-3}+za^{-5}+a^{2}+a^{-2}+3$
The A2 invariant	$-q^{12}+q^{6}+q^{4}+1+q^{-4}+q^{-6}-q^{-12}$
The G2 invariant	Data:K11n82/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n82 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["K11n82"];

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

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

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

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

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

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]= { 19, 2 }

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

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

Out[8]= $-q^{4}+2q^{3}-2q^{2}+3q-3+3q^{-1}-2q^{-2}+2q^{-3}-q^{-4}$

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

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

Out[9]= $z^{6}-a^{2}z^{4}-z^{4}a^{-2}+5z^{4}-3a^{2}z^{2}-3z^{2}a^{-2}+7z^{2}-a^{2}-a^{-2}+3$

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

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

Out[10]= $az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+z^{8}a^{-2}+3z^{8}+a^{3}z^{7}-4az^{7}-5z^{7}a^{-1}-11a^{2}z^{6}-5z^{6}a^{-2}-16z^{6}-5a^{3}z^{5}+2az^{5}+8z^{5}a^{-1}+z^{5}a^{-3}+17a^{2}z^{4}+8z^{4}a^{-2}+25z^{4}+6a^{3}z^{3}+2az^{3}-7z^{3}a^{-1}-3z^{3}a^{-3}-8a^{2}z^{2}-6z^{2}a^{-2}+z^{2}a^{-4}-15z^{2}-a^{3}z-az+za^{-1}+2za^{-3}+za^{-5}+a^{2}+a^{-2}+3$

"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["K11n82"];

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]= { $t^{3}-3t^{2}+4t-3+4t^{-1}-3t^{-2}+t^{-3}$ , $-q^{4}+2q^{3}-2q^{2}+3q-3+3q^{-1}-2q^{-2}+2q^{-3}-q^{-4}$ }

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]= {}

Vassiliev invariants

V₂ and V₃:

(1, 0)

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

4

0

8

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{34}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{62}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -32}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 32}

-64

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{32}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{136}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{248}{3}}

-{\frac {2609}{30}}

{\frac {806}{5}}

-{\frac {10978}{45}}

-{\frac {559}{18}}

-{\frac {2129}{30}}

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 K11n82. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

0

1

2

3

χ

9

1

-1

7

1

5

1

0

3

2

1

2

0

-1

1

0

-3

1

2

1

-5

1

0

-7

1

-9

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=0$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$
$r=1$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}