L11a333

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a333 at Knotilus!
	[edit L11a333 Quick Notes]

[edit L11a333 Further Notes and Views]

Link Presentations

[edit Notes on L11a333's Link Presentations]

Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_20,5,21,6 X_16,7,17,8 X_8,9,1,10 X_18,12,19,11 X_6,15,7,16 X_4,14,5,13 X_22,18,9,17 X_2,19,3,20 X_14,22,15,21
Gauss code	{1, -10, 2, -8, 3, -7, 4, -5}, {5, -1, 6, -2, 8, -11, 7, -4, 9, -6, 10, -3, 11, -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{(t(1) t(2)+1) \left(t(1)^2 t(2)^4-t(1) t(2)^4-3 t(1)^2 t(2)^3+5 t(1) t(2)^3-t(2)^3+3 t(1)^2 t(2)^2-7 t(1) t(2)^2+3 t(2)^2-t(1)^2 t(2)+5 t(1) t(2)-3 t(2)-t(1)+1\right)}{t(1)^{3/2} t(2)^{5/2}}} (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^{9/2}-4 q^{7/2}+9 q^{5/2}-14 q^{3/2}+19 \sqrt{q}-\frac{23}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{20}{q^{5/2}}+\frac{14}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}}} (db)
Signature	-1 (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 a^3 z^7+4 a^3 z^5+5 a^3 z^3+3 a^3 z+2 a^3 z^{-1} -a z^9-6 a z^7+z^7 a^{-1} -13 a z^5+4 z^5 a^{-1} -13 a z^3+5 z^3 a^{-1} -8 a z+3 z a^{-1} -3 a z^{-1} + a^{-1} z^{-1} } (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 -3 a^2 z^{10}-3 z^{10}-8 a^3 z^9-15 a z^9-7 z^9 a^{-1} -10 a^4 z^8-9 a^2 z^8-7 z^8 a^{-2} -6 z^8-8 a^5 z^7+11 a^3 z^7+37 a z^7+14 z^7 a^{-1} -4 z^7 a^{-3} -4 a^6 z^6+18 a^4 z^6+32 a^2 z^6+16 z^6 a^{-2} -z^6 a^{-4} +27 z^6-a^7 z^5+13 a^5 z^5-5 a^3 z^5-37 a z^5-9 z^5 a^{-1} +9 z^5 a^{-3} +5 a^6 z^4-13 a^4 z^4-34 a^2 z^4-9 z^4 a^{-2} +2 z^4 a^{-4} -27 z^4+a^7 z^3-6 a^5 z^3+a^3 z^3+19 a z^3+7 z^3 a^{-1} -4 z^3 a^{-3} +4 a^4 z^2+13 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} +13 z^2+2 a^5 z-4 a^3 z-9 a z-3 z a^{-1} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 a z^{-1} + a^{-1} z^{-1} } (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{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} ).

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

3

6

1

-5

4

8

3

5

2

11

6

-5

0

12

8

4

-2

11

12

1

-4

9

11

-2

-6

5

11

6

-8

4

9

-5

-10

1

6

5

-12

3

-3

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{4}$
$r=-3$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{12}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	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}}