K11a36

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a36 at Knotilus!
	[edit K11a36 Quick Notes]

[edit K11a36 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₄₉₃ X_14,6,15,5 X₂₈₃₇ X_16,9,17,10 X_20,11,21,12 X_6,14,7,13 X_12,15,13,16 X_22,17,1,18 X_10,19,11,20 X_18,21,19,22
Gauss code	1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -8, 7, -3, 8, -5, 9, -11, 10, -6, 11, -9
Dowker-Thistlethwaite code	4 8 14 2 16 20 6 12 22 10 18

A Braid Representative

A Morse Link Presentation

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	$\{1,2\}$
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a36/ThurstonBennequinNumber
Hyperbolic Volume	15.1597
A-Polynomial	See Data:K11a36/A-polynomial

[edit Notes for K11a36's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$[0,3]$
Rasmussen s-Invariant	0

[edit Notes for K11a36's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+12t^{2}-28t+37-28t^{-1}+12t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 121, 0 }
Jones polynomial	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 -q^5+3 q^4-7 q^3+12 q^2-16 q+20-19 q^{-1} +17 q^{-2} -13 q^{-3} +8 q^{-4} -4 q^{-5} + q^{-6} }
HOMFLY-PT polynomial (db, data sources)	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 -a^2 z^6-z^6+a^4 z^4-2 a^2 z^4+2 z^4 a^{-2} -z^4+a^4 z^2-2 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} +z^2-a^2+ a^{-2} - a^{-4} +2}
Kauffman polynomial (db, data sources)	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 a^2 z^{10}+z^{10}+4 a^3 z^9+7 a z^9+3 z^9 a^{-1} +6 a^4 z^8+12 a^2 z^8+5 z^8 a^{-2} +11 z^8+4 a^5 z^7+a^3 z^7+8 z^7 a^{-1} +5 z^7 a^{-3} +a^6 z^6-13 a^4 z^6-28 a^2 z^6-2 z^6 a^{-2} +3 z^6 a^{-4} -19 z^6-10 a^5 z^5-20 a^3 z^5-25 a z^5-23 z^5 a^{-1} -7 z^5 a^{-3} +z^5 a^{-5} -2 a^6 z^4+6 a^4 z^4+15 a^2 z^4-5 z^4 a^{-2} -5 z^4 a^{-4} +7 z^4+7 a^5 z^3+18 a^3 z^3+25 a z^3+20 z^3 a^{-1} +4 z^3 a^{-3} -2 z^3 a^{-5} +a^6 z^2-3 a^2 z^2+5 z^2 a^{-2} +3 z^2 a^{-4} -a^5 z-5 a^3 z-8 a z-6 z a^{-1} -z a^{-3} +z a^{-5} +a^2- a^{-2} - a^{-4} +2}
The A2 invariant	Data:K11a36/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a36/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a36 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["K11a36"];

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

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

Out[4]= $-2t^{3}+12t^{2}-28t+37-28t^{-1}+12t^{-2}-2t^{-3}$

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

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

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]= { 121, 0 }

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

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

Out[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 -q^5+3 q^4-7 q^3+12 q^2-16 q+20-19 q^{-1} +17 q^{-2} -13 q^{-3} +8 q^{-4} -4 q^{-5} + q^{-6} }

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

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

Out[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 -a^2 z^6-z^6+a^4 z^4-2 a^2 z^4+2 z^4 a^{-2} -z^4+a^4 z^2-2 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} +z^2-a^2+ a^{-2} - a^{-4} +2}

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

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

Out[10]= 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 a^2 z^{10}+z^{10}+4 a^3 z^9+7 a z^9+3 z^9 a^{-1} +6 a^4 z^8+12 a^2 z^8+5 z^8 a^{-2} +11 z^8+4 a^5 z^7+a^3 z^7+8 z^7 a^{-1} +5 z^7 a^{-3} +a^6 z^6-13 a^4 z^6-28 a^2 z^6-2 z^6 a^{-2} +3 z^6 a^{-4} -19 z^6-10 a^5 z^5-20 a^3 z^5-25 a z^5-23 z^5 a^{-1} -7 z^5 a^{-3} +z^5 a^{-5} -2 a^6 z^4+6 a^4 z^4+15 a^2 z^4-5 z^4 a^{-2} -5 z^4 a^{-4} +7 z^4+7 a^5 z^3+18 a^3 z^3+25 a z^3+20 z^3 a^{-1} +4 z^3 a^{-3} -2 z^3 a^{-5} +a^6 z^2-3 a^2 z^2+5 z^2 a^{-2} +3 z^2 a^{-4} -a^5 z-5 a^3 z-8 a z-6 z a^{-1} -z a^{-3} +z a^{-5} +a^2- a^{-2} - a^{-4} +2}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a169,}

Same Jones Polynomial (up to mirroring, 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 q\leftrightarrow q^{-1}} ): {K11a35, K11a316,}

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["K11a36"];

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]= { 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 -2 t^3+12 t^2-28 t+37-28 t^{-1} +12 t^{-2} -2 t^{-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 -q^5+3 q^4-7 q^3+12 q^2-16 q+20-19 q^{-1} +17 q^{-2} -13 q^{-3} +8 q^{-4} -4 q^{-5} + q^{-6} } }

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]= {K11a169,}

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

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

Out[6]= {K11a35, K11a316,}

Vassiliev invariants

V₂ and V₃:

(2, 1)

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 8}

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 \frac{172}{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{44}{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 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{272}{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{64}{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 40}

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{256}{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 32}

{\frac {1376}{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{352}{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{9271}{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{244}{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{13924}{45}}

{\frac {281}{9}}

{\frac {631}{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=

0 is the signature of K11a36. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

2

7

5

1

-4

5

7

2

5

3

9

5

-4

1

11

7

4

-1

9

10

1

-3

8

10

-2

-5

5

9

4

-7

3

8

-5

-9

1

5

4

-11

3

-3

-13

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$	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}^{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 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}\oplus{\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}}
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}_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}}