L11a416

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a416 at Knotilus!
	[edit L11a416 Quick Notes]

[edit L11a416 Further Notes and Views]

Link Presentations

[edit Notes on L11a416's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_22,14,11,13 X_10,15,5,16 X_20,17,21,18 X_16,7,17,8 X_8,20,9,19 X_18,10,19,9 X_14,22,15,21 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 6, -7, 8, -4}, {11, -2, 3, -9, 4, -6, 5, -8, 7, -5, 9, -3}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}+2t(1)t(2)^{2}-3t(1)t(3)t(2)^{2}+4t(3)t(2)^{2}-2t(2)^{2}-3t(1)t(3)^{2}t(2)+4t(3)^{2}t(2)-4t(1)t(2)+8t(1)t(3)t(2)-8t(3)t(2)+3t(2)+2t(1)t(3)^{2}-2t(3)^{2}+2t(1)-4t(1)t(3)+3t(3)-1}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{-8}+3q^{-7}-6q^{-6}+12q^{-5}-15q^{-4}+q^{3}+19q^{-3}-4q^{2}-18q^{-2}+8q+17q^{-1}-12$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{8}+3a^{6}z^{2}+a^{6}z^{-2}+3a^{6}-3a^{4}z^{4}-5a^{4}z^{2}-2a^{4}z^{-2}-5a^{4}+a^{2}z^{6}+2a^{2}z^{4}+4a^{2}z^{2}+a^{2}z^{-2}+z^{2}a^{-2}+3a^{2}-2z^{4}-2z^{2}$ (db)
Kauffman polynomial	$a^{9}z^{5}-2a^{9}z^{3}+a^{9}z+3a^{8}z^{6}-6a^{8}z^{4}+5a^{8}z^{2}-2a^{8}+4a^{7}z^{7}-3a^{7}z^{5}-3a^{7}z^{3}+3a^{7}z+4a^{6}z^{8}+2a^{6}z^{6}-15a^{6}z^{4}+18a^{6}z^{2}+a^{6}z^{-2}-8a^{6}+3a^{5}z^{9}+4a^{5}z^{7}-9a^{5}z^{5}+a^{5}z^{3}+5a^{5}z-2a^{5}z^{-1}+a^{4}z^{10}+9a^{4}z^{8}-13a^{4}z^{6}-4a^{4}z^{4}+16a^{4}z^{2}+2a^{4}z^{-2}-9a^{4}+7a^{3}z^{9}-5a^{3}z^{7}-8a^{3}z^{5}+4a^{3}z^{3}+3a^{3}z-2a^{3}z^{-1}+a^{2}z^{10}+11a^{2}z^{8}-27a^{2}z^{6}+z^{6}a^{-2}+15a^{2}z^{4}-2z^{4}a^{-2}+a^{2}z^{2}+z^{2}a^{-2}+a^{2}z^{-2}-4a^{2}+4az^{9}-az^{7}+4z^{7}a^{-1}-13az^{5}-10z^{5}a^{-1}+8az^{3}+6z^{3}a^{-1}+6z^{8}-14z^{6}+8z^{4}-z^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

3

-3

3

5

1

4

1

7

3

-4

-1

10

5

-3

9

8

-1

-5

10

9

1

-7

7

11

4

-9

5

8

-3

-11

2

8

6

-13

1

4

-3

-15

2

-17

1

-1

Integral Khovanov Homology

(db, data source)

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