L11a508

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_18,12,19,11 X_10,4,11,3 X_2,20,3,19 X_16,8,17,7 X_20,9,21,10 X_12,18,7,17 X_4,21,5,22 X_14,6,15,5 X_22,15,13,16 X_6,14,1,13
Gauss code	{1, -4, 3, -8, 9, -11}, {5, -1, 6, -3, 2, -7}, {11, -9, 10, -5, 7, -2, 4, -6, 8, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(w-1)\left(u^{2}v^{2}w+u^{2}\left(-v^{2}\right)+u^{2}vw^{2}-2u^{2}vw+u^{2}v-u^{2}w^{2}+u^{2}w-2uv^{2}w+2uv^{2}-2uvw^{2}+3uvw-2uv+2uw^{2}-2uw+v^{2}w-v^{2}+vw^{2}-2vw+v-w^{2}+w\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{7}+4q^{6}-7q^{5}+13q^{4}+q^{-4}-16q^{3}-4q^{-3}+20q^{2}+8q^{-2}-19q-13q^{-1}+18$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-2}-z^{6}+a^{2}z^{4}-z^{4}a^{-2}+2z^{4}a^{-4}-2z^{4}+a^{2}z^{2}+z^{2}a^{-2}+2z^{2}a^{-4}-z^{2}a^{-6}-2z^{2}+2a^{-2}-2a^{-4}+a^{-2}z^{-2}-2a^{-4}z^{-2}+a^{-6}z^{-2}$ (db)
Kauffman polynomial	$z^{7}a^{-7}-3z^{5}a^{-7}+2z^{3}a^{-7}+4z^{8}a^{-6}-15z^{6}a^{-6}+17z^{4}a^{-6}-5z^{2}a^{-6}+a^{-6}z^{-2}-2a^{-6}+5z^{9}a^{-5}-15z^{7}a^{-5}+10z^{5}a^{-5}+2za^{-5}-2a^{-5}z^{-1}+2z^{10}a^{-4}+7z^{8}a^{-4}-41z^{6}a^{-4}+a^{4}z^{4}+47z^{4}a^{-4}-14z^{2}a^{-4}+2a^{-4}z^{-2}-3a^{-4}+12z^{9}a^{-3}-30z^{7}a^{-3}+4a^{3}z^{5}+14z^{5}a^{-3}-2a^{3}z^{3}+2z^{3}a^{-3}+2za^{-3}-2a^{-3}z^{-1}+2z^{10}a^{-2}+14z^{8}a^{-2}+8a^{2}z^{6}-49z^{6}a^{-2}-7a^{2}z^{4}+42z^{4}a^{-2}+2a^{2}z^{2}-11z^{2}a^{-2}+a^{-2}z^{-2}-2a^{-2}+7z^{9}a^{-1}+11az^{7}-3z^{7}a^{-1}-13az^{5}-16z^{5}a^{-1}+4az^{3}+10z^{3}a^{-1}+11z^{8}-15z^{6}+4z^{4}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

15

1

-1

13

3

11

4

1

-3

9

3

6

7

8

5

-3

5

12

8

4

3

9

10

1

9

10

-1

5

10

5

-3

3

8

-5

1

5

4

-7

3

-3

-9

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-4$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{12}$
$r=3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$