L10a135

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,5,15,6 X_16,12,17,11 X_10,13,5,14 X_20,18,11,17 X_8,20,9,19 X_18,8,19,7 X_2,9,3,10 X_4,16,1,15
Gauss code	{1, -9, 2, -10}, {3, -1, 8, -7, 9, -5}, {4, -2, 5, -3, 10, -4, 6, -8, 7, -6}

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

Polynomial invariants

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

χ

13

1

11

3

-3

9

4

1

3

7

6

3

-3

5

8

4

3

6

7

1

8

7

1

-1

4

8

4

-3

3

6

-3

-5

1

5

4

-7

2

-2

-9

1

Integral Khovanov Homology

(db, data source)

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