L11a469

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(w-1)\left(2uv^{2}w-uv^{2}+2uvw^{2}-4uvw+uv-2uw^{2}+uw+v^{2}w^{2}-2v^{2}w+vw^{3}-4vw^{2}+2vw-w^{3}+2w^{2}\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{-10}+3q^{-9}-6q^{-8}+11q^{-7}-13q^{-6}+17q^{-5}-16q^{-4}+15q^{-3}-11q^{-2}+q+7q^{-1}-3$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{10}+3a^{8}z^{2}+a^{8}z^{-2}+2a^{8}-2a^{6}z^{4}-a^{6}z^{2}-2a^{6}z^{-2}-2a^{6}-3a^{4}z^{4}-3a^{4}z^{2}+a^{4}z^{-2}-a^{4}-a^{2}z^{4}+2a^{2}z^{2}+2a^{2}+z^{2}$ (db)
Kauffman polynomial	$a^{11}z^{7}-4a^{11}z^{5}+5a^{11}z^{3}-2a^{11}z+3a^{10}z^{8}-12a^{10}z^{6}+16a^{10}z^{4}-9a^{10}z^{2}+3a^{10}+3a^{9}z^{9}-6a^{9}z^{7}-7a^{9}z^{5}+18a^{9}z^{3}-7a^{9}z+a^{8}z^{10}+8a^{8}z^{8}-38a^{8}z^{6}+51a^{8}z^{4}-34a^{8}z^{2}-a^{8}z^{-2}+12a^{8}+7a^{7}z^{9}-11a^{7}z^{7}-13a^{7}z^{5}+25a^{7}z^{3}-14a^{7}z+2a^{7}z^{-1}+a^{6}z^{10}+12a^{6}z^{8}-38a^{6}z^{6}+39a^{6}z^{4}-28a^{6}z^{2}-2a^{6}z^{-2}+12a^{6}+4a^{5}z^{9}+4a^{5}z^{7}-24a^{5}z^{5}+22a^{5}z^{3}-10a^{5}z+2a^{5}z^{-1}+7a^{4}z^{8}-6a^{4}z^{6}-4a^{4}z^{4}+4a^{4}z^{2}-a^{4}z^{-2}+2a^{4}+8a^{3}z^{7}-11a^{3}z^{5}+8a^{3}z^{3}-a^{3}z+6a^{2}z^{6}-7a^{2}z^{4}+6a^{2}z^{2}-2a^{2}+3az^{5}-2az^{3}+z^{4}-z^{2}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

3

1

2

-2

-1

5

1

4

-3

7

3

-4

-5

8

4

-7

9

8

-1

-9

8

7

1

-11

6

10

4

-13

5

7

-2

-15

2

7

5

-17

1

4

-3

-19

2

-21

1

-1

Integral Khovanov Homology

(db, data source)

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