L11a382

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a382 at Knotilus!
	[edit L11a382 Quick Notes]

[edit L11a382 Further Notes and Views]

Link Presentations

[edit Notes on L11a382's Link Presentations]

Planar diagram presentation	X_12,1,13,2 X_2,13,3,14 X_14,3,15,4 X_16,7,17,8 X_18,10,19,9 X_20,6,21,5 X_6,22,7,21 X_22,18,11,17 X_10,20,1,19 X_8,11,9,12 X_4,15,5,16
Gauss code	{1, -2, 3, -11, 6, -7, 4, -10, 5, -9}, {10, -1, 2, -3, 11, -4, 8, -5, 9, -6, 7, -8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {\left(t(1)^{2}t(2)^{2}-t(1)t(2)^{2}+2t(1)t(2)-t(2)-t(1)+1\right)\left(t(2)^{2}t(1)^{2}-t(2)t(1)^{2}-t(2)^{2}t(1)+2t(2)t(1)-t(1)+1\right)}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+6q^{5/2}-10q^{3/2}+13{\sqrt {q}}-{\frac {16}{\sqrt {q}}}+{\frac {15}{q^{3/2}}}-{\frac {14}{q^{5/2}}}+{\frac {10}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-az^{9}+a^{3}z^{7}-7az^{7}+z^{7}a^{-1}+5a^{3}z^{5}-18az^{5}+5z^{5}a^{-1}+8a^{3}z^{3}-20az^{3}+8z^{3}a^{-1}+4a^{3}z-9az+4za^{-1}+a^{3}z^{-1}-az^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-2a^{7}z^{3}+a^{7}z+3a^{6}z^{6}-6a^{6}z^{4}+3a^{6}z^{2}+4a^{5}z^{7}-5a^{5}z^{5}-a^{5}z^{3}+a^{5}z+4a^{4}z^{8}-3a^{4}z^{6}+z^{6}a^{-4}-3a^{4}z^{4}-3z^{4}a^{-4}+a^{4}z^{2}+z^{2}a^{-4}+4a^{3}z^{9}-8a^{3}z^{7}+3z^{7}a^{-3}+14a^{3}z^{5}-9z^{5}a^{-3}-14a^{3}z^{3}+5z^{3}a^{-3}+5a^{3}z-za^{-3}-a^{3}z^{-1}+2a^{2}z^{10}+5z^{8}a^{-2}-6a^{2}z^{6}-16z^{6}a^{-2}+12a^{2}z^{4}+14z^{4}a^{-2}-6a^{2}z^{2}-5z^{2}a^{-2}+a^{2}+9az^{9}+5z^{9}a^{-1}-31az^{7}-16z^{7}a^{-1}+47az^{5}+18z^{5}a^{-1}-30az^{3}-10z^{3}a^{-1}+9az+3za^{-1}-az^{-1}+2z^{10}+z^{8}-17z^{6}+26z^{4}-10z^{2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

2

6

4

1

-3

4

6

2

4

2

7

4

-3

0

9

6

3

-2

7

8

1

-4

7

8

-1

-6

4

8

4

-8

2

6

-4

-10

1

4

3

-12

2

-2

-14

1

Integral Khovanov Homology

(db, data source)

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