L10a134

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_16,7,17,8 X_20,9,11,10 X_18,12,19,11 X_8,15,9,16 X_10,19,5,20 X_14,18,15,17 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -6, 4, -7}, {5, -2, 10, -8, 6, -3, 8, -5, 7, -4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{2}w^{2}-2uv^{2}w+uv^{2}-3uvw^{2}+5uvw-2uv+3uw^{2}-3uw+u-v^{2}w^{2}+3v^{2}w-3v^{2}+2vw^{2}-5vw+3v-w^{2}+2w-1}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$q^{-6}-2q^{-5}+q^{4}+7q^{-4}-4q^{3}-9q^{-3}+8q^{2}+13q^{-2}-11q-14q^{-1}+14$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-3a^{4}z^{2}-2a^{4}z^{-2}-4a^{4}+3a^{2}z^{4}+z^{4}a^{-2}+6a^{2}z^{2}+a^{2}z^{-2}+z^{2}a^{-2}+4a^{2}+a^{-2}-z^{6}-3z^{4}-5z^{2}-2$ (db)
Kauffman polynomial	$a^{6}z^{6}-4a^{6}z^{4}+6a^{6}z^{2}+a^{6}z^{-2}-4a^{6}+2a^{5}z^{7}-4a^{5}z^{5}+4a^{5}z-2a^{5}z^{-1}+3a^{4}z^{8}-5a^{4}z^{6}-a^{4}z^{4}+z^{4}a^{-4}+6a^{4}z^{2}+2a^{4}z^{-2}-6a^{4}+a^{3}z^{9}+9a^{3}z^{7}-28a^{3}z^{5}+4z^{5}a^{-3}+20a^{3}z^{3}-2z^{3}a^{-3}-2a^{3}z^{-1}+8a^{2}z^{8}-9a^{2}z^{6}+8z^{6}a^{-2}-9a^{2}z^{4}-9z^{4}a^{-2}+9a^{2}z^{2}+4z^{2}a^{-2}+a^{2}z^{-2}-3a^{2}-a^{-2}+az^{9}+16az^{7}+9z^{7}a^{-1}-39az^{5}-11z^{5}a^{-1}+27az^{3}+5z^{3}a^{-1}-6az-2za^{-1}+5z^{8}+5z^{6}-22z^{4}+13z^{2}-1$ (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

χ

9

1

7

3

-3

5

1

4

3

6

3

-3

1

8

5

3

-1

8

0

-3

5

6

-1

-5

5

9

4

-7

2

4

-2

-9

5

-11

1

2

-1

-13

1

Integral Khovanov Homology

(db, data source)

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