L11a527

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_14,3,15,4 X_12,15,7,16 X_10,21,11,22 X_16,9,17,10 X_22,11,13,12 X_18,6,19,5 X_20,18,21,17 X₂₇₃₈ X_4,13,5,14 X_6,20,1,19
Gauss code	{1, -9, 2, -10, 7, -11}, {9, -1, 5, -4, 6, -3}, {10, -2, 3, -5, 8, -7, 11, -8, 4, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{2}w^{2}-u^{2}v^{2}w+u^{2}vw^{3}-4u^{2}vw^{2}+4u^{2}vw-u^{2}v+2u^{2}w^{2}-3u^{2}w+u^{2}+uv^{2}w^{3}-3uv^{2}w^{2}+3uv^{2}w-2uvw^{3}+7uvw^{2}-7uvw+2uv-3uw^{2}+3uw-u-v^{2}w^{3}+3v^{2}w^{2}-2v^{2}w+vw^{3}-4vw^{2}+4vw-v+w^{2}-w}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{-9}-3q^{-8}+8q^{-7}-13q^{-6}+19q^{-5}-21q^{-4}+23q^{-3}-q^{2}-19q^{-2}+4q+15q^{-1}-9$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z^{2}a^{8}+a^{8}z^{-2}+2a^{8}-3z^{4}a^{6}-8z^{2}a^{6}-2a^{6}z^{-2}-8a^{6}+2z^{6}a^{4}+7z^{4}a^{4}+11z^{2}a^{4}+a^{4}z^{-2}+7a^{4}+z^{6}a^{2}+z^{4}a^{2}-z^{2}a^{2}-a^{2}-z^{4}-z^{2}$ (db)
Kauffman polynomial	$a^{10}z^{6}-3a^{10}z^{4}+3a^{10}z^{2}-a^{10}+3a^{9}z^{7}-7a^{9}z^{5}+4a^{9}z^{3}+5a^{8}z^{8}-10a^{8}z^{6}+6a^{8}z^{4}-5a^{8}z^{2}-a^{8}z^{-2}+4a^{8}+5a^{7}z^{9}-6a^{7}z^{7}-3a^{7}z^{5}+6a^{7}z^{3}-6a^{7}z+2a^{7}z^{-1}+2a^{6}z^{10}+10a^{6}z^{8}-35a^{6}z^{6}+43a^{6}z^{4}-31a^{6}z^{2}-2a^{6}z^{-2}+12a^{6}+12a^{5}z^{9}-21a^{5}z^{7}+8a^{5}z^{5}+7a^{5}z^{3}-8a^{5}z+2a^{5}z^{-1}+2a^{4}z^{10}+15a^{4}z^{8}-45a^{4}z^{6}+52a^{4}z^{4}-31a^{4}z^{2}-a^{4}z^{-2}+10a^{4}+7a^{3}z^{9}-4a^{3}z^{7}-9a^{3}z^{5}+11a^{3}z^{3}-3a^{3}z+10a^{2}z^{8}-17a^{2}z^{6}+13a^{2}z^{4}-7a^{2}z^{2}+2a^{2}+8az^{7}-12az^{5}+z^{5}a^{-1}+5az^{3}-z^{3}a^{-1}-az+4z^{6}-5z^{4}+z^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

5

1

-1

3

1

6

1

-5

-1

9

3

6

-3

11

7

-4

-5

12

8

4

-7

10

12

2

-9

9

11

-2

-11

5

11

6

-13

3

8

-5

-15

1

6

5

-17

2

-2

-19

1

Integral Khovanov Homology

(db, data source)

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