L11a412

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{2}w^{3}-3uv^{2}w^{2}+3uv^{2}w-uv^{2}+uvw^{4}-5uvw^{3}+9uvw^{2}-6uvw+2uv-uw^{4}+4uw^{3}-5uw^{2}+3uw-u+v^{2}w^{4}-3v^{2}w^{3}+5v^{2}w^{2}-4v^{2}w+v^{2}-2vw^{4}+6vw^{3}-9vw^{2}+5vw-v+w^{4}-3w^{3}+3w^{2}-w}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{2}+5q-12+20q^{-1}-25q^{-2}+31q^{-3}-28q^{-4}+25q^{-5}-18q^{-6}+10q^{-7}-4q^{-8}+q^{-9}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{2}+a^{8}-3a^{6}z^{4}-5a^{6}z^{2}+a^{6}z^{-2}-3a^{6}+2a^{4}z^{6}+5a^{4}z^{4}+6a^{4}z^{2}-2a^{4}z^{-2}+a^{4}+a^{2}z^{6}-a^{2}z^{2}+a^{2}z^{-2}+a^{2}-z^{4}$ (db)
Kauffman polynomial	$a^{10}z^{6}-2a^{10}z^{4}+a^{10}z^{2}+4a^{9}z^{7}-8a^{9}z^{5}+6a^{9}z^{3}-2a^{9}z+8a^{8}z^{8}-16a^{8}z^{6}+13a^{8}z^{4}-7a^{8}z^{2}+2a^{8}+8a^{7}z^{9}-6a^{7}z^{7}-14a^{7}z^{5}+18a^{7}z^{3}-7a^{7}z+3a^{6}z^{10}+21a^{6}z^{8}-64a^{6}z^{6}+59a^{6}z^{4}-24a^{6}z^{2}+a^{6}z^{-2}+4a^{6}+20a^{5}z^{9}-26a^{5}z^{7}-13a^{5}z^{5}+25a^{5}z^{3}-7a^{5}z-2a^{5}z^{-1}+3a^{4}z^{10}+30a^{4}z^{8}-79a^{4}z^{6}+63a^{4}z^{4}-21a^{4}z^{2}+2a^{4}z^{-2}+3a^{4}+12a^{3}z^{9}-4a^{3}z^{7}-22a^{3}z^{5}+17a^{3}z^{3}-3a^{3}z-2a^{3}z^{-1}+17a^{2}z^{8}-27a^{2}z^{6}+16a^{2}z^{4}-5a^{2}z^{2}+a^{2}z^{-2}+12az^{7}-14az^{5}+z^{5}a^{-1}+4az^{3}-az+5z^{6}-3z^{4}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

5

1

-1

3

4

1

8

1

-7

-1

12

4

8

-3

14

9

-5

17

11

6

-7

13

16

3

-9

12

15

-3

-11

7

14

7

-13

3

11

-8

-15

1

7

6

-17

3

-3

-19

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-5$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-4$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{12}$
$r=-3$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=-2$	${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{15}$	${\mathbb {Z} }^{17}$
$r=-1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{12}$
$r=1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$