K11n81

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n81 at Knotilus!
	[edit K11n81 Quick Notes]

[edit K11n81 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₄₉₃ X_5,16,6,17 X₂₈₃₇ X_9,21,10,20 X_11,1,12,22 X_13,19,14,18 X_15,6,16,7 X_17,13,18,12 X_19,15,20,14 X_21,11,22,10
Gauss code	1, -4, 2, -1, -3, 8, 4, -2, -5, 11, -6, 9, -7, 10, -8, 3, -9, 7, -10, 5, -11, 6
Dowker-Thistlethwaite code	4 8 -16 2 -20 -22 -18 -6 -12 -14 -10

A Braid Representative

A Morse Link Presentation

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	3
3-genus	4
Bridge index	4
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11n81/ThurstonBennequinNumber
Hyperbolic Volume	10.956
A-Polynomial	See Data:K11n81/A-polynomial

[edit Notes for K11n81's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$4$
Rasmussen s-Invariant	-6

[edit Notes for K11n81's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+3t^{3}-4t^{2}+4t-3+4t^{-1}-4t^{-2}+3t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-5z^{6}-6z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{t^{2}-t+1\right\}$
Determinant and Signature	{ 27, 6 }
Jones polynomial	$2q^{9}-3q^{8}+3q^{7}-5q^{6}+4q^{5}-4q^{4}+4q^{3}-q^{2}+q$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-6}+z^{6}a^{-4}-7z^{6}a^{-6}+z^{6}a^{-8}+6z^{4}a^{-4}-18z^{4}a^{-6}+6z^{4}a^{-8}+12z^{2}a^{-4}-23z^{2}a^{-6}+11z^{2}a^{-8}-z^{2}a^{-10}+8a^{-4}-13a^{-6}+7a^{-8}-a^{-10}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-5}+z^{9}a^{-7}+z^{8}a^{-4}+5z^{8}a^{-6}+4z^{8}a^{-8}-4z^{7}a^{-5}+2z^{7}a^{-7}+6z^{7}a^{-9}-7z^{6}a^{-4}-28z^{6}a^{-6}-18z^{6}a^{-8}+3z^{6}a^{-10}-30z^{5}a^{-7}-30z^{5}a^{-9}+18z^{4}a^{-4}+50z^{4}a^{-6}+20z^{4}a^{-8}-12z^{4}a^{-10}+13z^{3}a^{-5}+53z^{3}a^{-7}+41z^{3}a^{-9}+z^{3}a^{-11}-20z^{2}a^{-4}-38z^{2}a^{-6}-10z^{2}a^{-8}+8z^{2}a^{-10}-11za^{-5}-26za^{-7}-18za^{-9}-3za^{-11}+8a^{-4}+13a^{-6}+7a^{-8}+a^{-10}$
The A2 invariant	$q^{-4}+q^{-6}+3q^{-8}+3q^{-10}+2q^{-12}-4q^{-16}-3q^{-18}-5q^{-20}+q^{-24}+2q^{-26}+2q^{-28}+q^{-32}-q^{-34}$
The G2 invariant	Data:K11n81/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n81 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["K11n81"];

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

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

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

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

Out[5]= $-z^{8}-5z^{6}-6z^{4}-z^{2}+1$

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

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

Out[7]= { 27, 6 }

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

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

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

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

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

Out[9]= $-z^{8}a^{-6}+z^{6}a^{-4}-7z^{6}a^{-6}+z^{6}a^{-8}+6z^{4}a^{-4}-18z^{4}a^{-6}+6z^{4}a^{-8}+12z^{2}a^{-4}-23z^{2}a^{-6}+11z^{2}a^{-8}-z^{2}a^{-10}+8a^{-4}-13a^{-6}+7a^{-8}-a^{-10}$

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

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

Out[10]= $z^{9}a^{-5}+z^{9}a^{-7}+z^{8}a^{-4}+5z^{8}a^{-6}+4z^{8}a^{-8}-4z^{7}a^{-5}+2z^{7}a^{-7}+6z^{7}a^{-9}-7z^{6}a^{-4}-28z^{6}a^{-6}-18z^{6}a^{-8}+3z^{6}a^{-10}-30z^{5}a^{-7}-30z^{5}a^{-9}+18z^{4}a^{-4}+50z^{4}a^{-6}+20z^{4}a^{-8}-12z^{4}a^{-10}+13z^{3}a^{-5}+53z^{3}a^{-7}+41z^{3}a^{-9}+z^{3}a^{-11}-20z^{2}a^{-4}-38z^{2}a^{-6}-10z^{2}a^{-8}+8z^{2}a^{-10}-11za^{-5}-26za^{-7}-18za^{-9}-3za^{-11}+8a^{-4}+13a^{-6}+7a^{-8}+a^{-10}$

"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["K11n81"];

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

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, -6)

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

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 -4}

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 -48}

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 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{302}{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{158}{3}}

192

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 416}

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 256}

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 272}

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 1152}

{\frac {1208}{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{632}{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{123809}{30}}

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{7502}{15}}

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{83218}{45}}

{\frac {6559}{18}}

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{991}{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 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 t^rq^j} are shown, along with their alternating sums 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 \chi} (fixed 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 j} , alternation over 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} ). The squares with yellow highlighting are those on the "critical diagonals", where

j-2r=s+1

or 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 j-2r=s-1} , where 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 s=} 6 is the signature of K11n81. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-2

-1

0

1

2

3

4

5

6

χ

19

2

17

1

-1

15

3

0

13

3

1

-2

11

1

3

1

-1

9

3

0

7

1

0

5

1

4

3

0

1

Integral Khovanov Homology

(db, data source)

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