L10a112

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X₈₄₉₃ X_14,6,15,5 X_18,7,19,8 X_16,10,17,9 X_10,11,1,12 X_6,14,7,13 X_4,19,5,20 X_20,15,11,16 X_2,18,3,17
Gauss code	{1, -10, 2, -8, 3, -7, 4, -2, 5, -6}, {6, -1, 7, -3, 9, -5, 10, -4, 8, -9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(u^{2}v^{2}-u^{2}v+u^{2}-2uv^{2}+3uv-2u+v^{2}-v+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {4}{q^{9/2}}}-4q^{7/2}+{\frac {9}{q^{7/2}}}+8q^{5/2}-{\frac {14}{q^{5/2}}}-13q^{3/2}+{\frac {16}{q^{3/2}}}+{\frac {1}{q^{11/2}}}+16{\sqrt {q}}-{\frac {18}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-a^{3}z^{5}+4az^{5}-2z^{5}a^{-1}-2a^{3}z^{3}+7az^{3}-5z^{3}a^{-1}+z^{3}a^{-3}-2a^{3}z+5az-4za^{-1}+za^{-3}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}+4a^{5}z^{5}-a^{5}z^{3}+9a^{4}z^{6}+z^{6}a^{-4}-8a^{4}z^{4}-2z^{4}a^{-4}+3a^{4}z^{2}+z^{2}a^{-4}+13a^{3}z^{7}+4z^{7}a^{-3}-20a^{3}z^{5}-10z^{5}a^{-3}+13a^{3}z^{3}+8z^{3}a^{-3}-4a^{3}z-2za^{-3}+10a^{2}z^{8}+6z^{8}a^{-2}-10a^{2}z^{6}-13z^{6}a^{-2}-3a^{2}z^{4}+6z^{4}a^{-2}+3a^{2}z^{2}+z^{2}a^{-2}+3az^{9}+3z^{9}a^{-1}+16az^{7}+7z^{7}a^{-1}-47az^{5}-33z^{5}a^{-1}+34az^{3}+28z^{3}a^{-1}-10az-8za^{-1}+az^{-1}+a^{-1}z^{-1}+16z^{8}-33z^{6}+14z^{4}-1$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

3

6

5

1

-4

4

8

3

5

2

8

5

-3

0

10

8

2

-2

8

10

2

-4

6

8

-2

-6

3

8

5

-8

1

6

-5

-10

3

-12

1

-1

Integral Khovanov Homology

(db, data source)

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