L11a524

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(v-1)\left(u^{2}vw^{3}-3u^{2}vw^{2}+2u^{2}vw-u^{2}w^{3}+2u^{2}w^{2}-u^{2}w-uvw^{3}+4uvw^{2}-5uvw+uv+uw^{3}-5uw^{2}+4uw-u-vw^{2}+2vw-v+2w^{2}-3w+1\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{6}-4q^{5}-q^{-5}+10q^{4}+5q^{-4}-16q^{3}-11q^{-3}+24q^{2}+18q^{-2}-26q-24q^{-1}+28$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{4}a^{-4}+2z^{2}a^{-4}+a^{-4}z^{-2}+a^{-4}-a^{2}z^{6}-2z^{6}a^{-2}-2a^{2}z^{4}-6z^{4}a^{-2}-5z^{2}a^{-2}-2a^{-2}z^{-2}+a^{2}-2a^{-2}+z^{8}+4z^{6}+5z^{4}+z^{2}+z^{-2}$ (db)
Kauffman polynomial	$z^{6}a^{-6}-2z^{4}a^{-6}+z^{2}a^{-6}+4z^{7}a^{-5}+a^{5}z^{5}-8z^{5}a^{-5}+4z^{3}a^{-5}+8z^{8}a^{-4}+5a^{4}z^{6}-18z^{6}a^{-4}-4a^{4}z^{4}+16z^{4}a^{-4}-10z^{2}a^{-4}-a^{-4}z^{-2}+4a^{-4}+8z^{9}a^{-3}+11a^{3}z^{7}-11z^{7}a^{-3}-14a^{3}z^{5}-z^{5}a^{-3}+4a^{3}z^{3}+6z^{3}a^{-3}-5za^{-3}+2a^{-3}z^{-1}+3z^{10}a^{-2}+14a^{2}z^{8}+16z^{8}a^{-2}-19a^{2}z^{6}-52z^{6}a^{-2}+6a^{2}z^{4}+50z^{4}a^{-2}-22z^{2}a^{-2}-2a^{-2}z^{-2}-a^{2}+6a^{-2}+10az^{9}+18z^{9}a^{-1}-2az^{7}-28z^{7}a^{-1}-20az^{5}+2z^{5}a^{-1}+13az^{3}+11z^{3}a^{-1}-5za^{-1}+2a^{-1}z^{-1}+3z^{10}+22z^{8}-57z^{6}+42z^{4}-11z^{2}-z^{-2}+2$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

13

1

11

3

-3

9

7

1

6

7

10

4

-6

5

14

6

8

3

13

11

-2

1

15

13

2

-1

10

14

4

-3

8

14

-6

-5

4

11

7

-7

1

7

-6

-9

4

-11

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=-1$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{15}$
$r=1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{14}$
$r=3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$