L11n366

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n366 at Knotilus!
	[edit L11n366 Quick Notes]

[edit L11n366 Further Notes and Views]

Link Presentations

[edit Notes on L11n366's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_15,17,16,22 X_18,10,19,9 X_8,18,9,17 X_13,21,14,20 X_21,15,22,14 X_19,5,20,16 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {6, -5, -9, 7, -8, 4}, {10, -1, 3, -6, 5, -2, 11, -3, -7, 8, -4, 9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in 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 u} , 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 v} , 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 w} , ...)	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{(u-1) (w-1) \left(v^2 w^3-1\right)}{\sqrt{u} v w^2}} (db)
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^9-q^8+2 q^7-2 q^6+3 q^5-2 q^4+3 q^3-q^2+q} (db)
Signature	6 (db)
HOMFLY-PT 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 -z^8 a^{-6} +z^6 a^{-4} -7 z^6 a^{-6} +z^6 a^{-8} +6 z^4 a^{-4} -17 z^4 a^{-6} +6 z^4 a^{-8} +11 z^2 a^{-4} -20 z^2 a^{-6} +10 z^2 a^{-8} -z^2 a^{-10} +7 a^{-4} -12 a^{-6} +6 a^{-8} - a^{-10} + a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2} } (db)
Kauffman 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 a^{-12} -z a^{-11} +z^4 a^{-10} -5 z^2 a^{-10} +2 a^{-10} +2 z^7 a^{-9} -10 z^5 a^{-9} +11 z^3 a^{-9} -3 z a^{-9} +3 z^8 a^{-8} -18 z^6 a^{-8} +33 z^4 a^{-8} -26 z^2 a^{-8} - a^{-8} z^{-2} +9 a^{-8} +z^9 a^{-7} -3 z^7 a^{-7} -6 z^5 a^{-7} +19 z^3 a^{-7} -12 z a^{-7} +2 a^{-7} z^{-1} +4 z^8 a^{-6} -25 z^6 a^{-6} +49 z^4 a^{-6} -39 z^2 a^{-6} -2 a^{-6} z^{-2} +15 a^{-6} +z^9 a^{-5} -5 z^7 a^{-5} +4 z^5 a^{-5} +8 z^3 a^{-5} -10 z a^{-5} +2 a^{-5} z^{-1} +z^8 a^{-4} -7 z^6 a^{-4} +17 z^4 a^{-4} -18 z^2 a^{-4} - a^{-4} z^{-2} +8 a^{-4} } (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

\		r
	\
j		\

-2

-1

0

1

2

3

4

5

6

χ

19

1

17

2

0

15

1

13

2

0

11

1

9

1

2

1

7

2

1

5

1

3

2

3

0

1

Integral Khovanov Homology

(db, data source)

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