L11a523

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{2}w^{3}-2u^{2}v^{2}w^{2}+u^{2}v^{2}w-2u^{2}vw^{3}+4u^{2}vw^{2}-2u^{2}vw-2u^{2}w^{2}+u^{2}w-uv^{2}w^{3}+3uv^{2}w^{2}-3uv^{2}w+uv^{2}+2uvw^{3}-7uvw^{2}+7uvw-2uv-uw^{3}+3uw^{2}-3uw+u-v^{2}w^{2}+2v^{2}w+2vw^{2}-4vw+2v-w^{2}+2w-1}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{-6}-q^{5}-3q^{-5}+4q^{4}+8q^{-4}-8q^{3}-12q^{-3}+14q^{2}+18q^{-2}-18q-20q^{-1}+21$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{4}+3a^{4}z^{2}+a^{4}z^{-2}+3a^{4}-2a^{2}z^{6}-z^{6}a^{-2}-8a^{2}z^{4}-3z^{4}a^{-2}-11a^{2}z^{2}-2z^{2}a^{-2}-2a^{2}z^{-2}-6a^{2}+a^{-2}+z^{8}+5z^{6}+9z^{4}+6z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$a^{6}z^{6}-3a^{6}z^{4}+2a^{6}z^{2}+3a^{5}z^{7}-7a^{5}z^{5}+z^{5}a^{-5}+3a^{5}z^{3}-z^{3}a^{-5}+6a^{4}z^{8}-18a^{4}z^{6}+4z^{6}a^{-4}+23a^{4}z^{4}-6z^{4}a^{-4}-19a^{4}z^{2}+2z^{2}a^{-4}-a^{4}z^{-2}+7a^{4}+5a^{3}z^{9}-7a^{3}z^{7}+7z^{7}a^{-3}-5a^{3}z^{5}-10z^{5}a^{-3}+13a^{3}z^{3}+3z^{3}a^{-3}-9a^{3}z+2a^{3}z^{-1}+2a^{2}z^{10}+8a^{2}z^{8}+8z^{8}a^{-2}-31a^{2}z^{6}-10z^{6}a^{-2}+39a^{2}z^{4}+z^{4}a^{-2}-27a^{2}z^{2}+3z^{2}a^{-2}-2a^{2}z^{-2}+11a^{2}-2a^{-2}+11az^{9}+6z^{9}a^{-1}-21az^{7}-4z^{7}a^{-1}+10az^{5}-3z^{5}a^{-1}+8az^{3}+2z^{3}a^{-1}-9az+2az^{-1}+2z^{10}+10z^{8}-26z^{6}+20z^{4}-5z^{2}-z^{-2}+3$ (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

4

5

χ

11

1

-1

9

3

7

5

1

-4

5

9

3

6

3

10

6

-4

1

11

8

3

-1

11

12

1

-3

7

9

-2

-5

5

11

6

-7

3

7

-4

-9

5

-11

1

3

-2

-13

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-6$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$