L11n335

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_11,16,12,17 X₈₄₉₃ X_2,18,3,17 X_5,14,6,15 X_18,7,19,8 X_15,12,16,5 X_20,14,21,13 X_22,9,13,10 X_10,21,11,22 X_4,19,1,20
Gauss code	{1, -4, 3, -11}, {-5, -1, 6, -3, 9, -10, -2, 7}, {8, 5, -7, 2, 4, -6, 11, -8, 10, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)(w-1)(vw-2v-2w+1)}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{2}+5q-9+13q^{-1}-16q^{-2}+17q^{-3}-14q^{-4}+12q^{-5}-6q^{-6}+3q^{-7}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{-2}+2z^{2}a^{6}-2a^{6}z^{-2}-a^{6}-3z^{4}a^{4}-4z^{2}a^{4}+a^{4}z^{-2}+z^{6}a^{2}+2z^{4}a^{2}+2z^{2}a^{2}-z^{4}+1$ (db)
Kauffman polynomial	$2a^{5}z^{9}+2a^{3}z^{9}+5a^{6}z^{8}+12a^{4}z^{8}+7a^{2}z^{8}+3a^{7}z^{7}+9a^{5}z^{7}+15a^{3}z^{7}+9az^{7}-9a^{6}z^{6}-18a^{4}z^{6}-4a^{2}z^{6}+5z^{6}-3a^{7}z^{5}-23a^{5}z^{5}-36a^{3}z^{5}-15az^{5}+z^{5}a^{-1}+6a^{8}z^{4}+16a^{6}z^{4}+7a^{4}z^{4}-9a^{2}z^{4}-6z^{4}+7a^{7}z^{3}+21a^{5}z^{3}+19a^{3}z^{3}+5az^{3}-11a^{8}z^{2}-16a^{6}z^{2}-3a^{4}z^{2}+2a^{2}z^{2}-9a^{7}z-9a^{5}z+6a^{8}+9a^{6}+3a^{4}+1+2a^{7}z^{-1}+2a^{5}z^{-1}-a^{8}z^{-2}-2a^{6}z^{-2}-a^{4}z^{-2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

5

1

-1

3

4

1

5

1

-4

-1

8

4

-3

10

7

-3

-5

7

6

1

-7

7

10

3

-9

5

7

-2

-11

1

7

6

-13

2

5

-3

-15

3

Integral Khovanov Homology

(db, data source)

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