L10a88

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_12,3,13,4 X_14,6,15,5 X_16,7,17,8 X_18,16,19,15 X_20,13,9,14 X_6,17,7,18 X_4,20,5,19 X_2,9,3,10 X_8,11,1,12
Gauss code	{1, -9, 2, -8, 3, -7, 4, -10}, {9, -1, 10, -2, 6, -3, 5, -4, 7, -5, 8, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{3}v^{2}-2u^{3}v+u^{3}+u^{2}v^{3}-5u^{2}v^{2}+7u^{2}v-2u^{2}-2uv^{3}+7uv^{2}-5uv+u+v^{3}-2v^{2}+v}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{5/2}-4q^{3/2}+7{\sqrt {q}}-{\frac {10}{\sqrt {q}}}+{\frac {12}{q^{3/2}}}-{\frac {13}{q^{5/2}}}+{\frac {11}{q^{7/2}}}-{\frac {9}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}+2z^{3}a^{5}+2za^{5}+a^{5}z^{-1}-z^{5}a^{3}-z^{3}a^{3}-2za^{3}-a^{3}z^{-1}-z^{5}a-z^{3}a-za+z^{3}a^{-1}$ (db)
Kauffman polynomial	$-a^{5}z^{9}-a^{3}z^{9}-3a^{6}z^{8}-7a^{4}z^{8}-4a^{2}z^{8}-3a^{7}z^{7}-7a^{5}z^{7}-11a^{3}z^{7}-7az^{7}-a^{8}z^{6}+6a^{6}z^{6}+11a^{4}z^{6}-3a^{2}z^{6}-7z^{6}+10a^{7}z^{5}+26a^{5}z^{5}+27a^{3}z^{5}+7az^{5}-4z^{5}a^{-1}+3a^{8}z^{4}+a^{4}z^{4}+13a^{2}z^{4}-z^{4}a^{-2}+8z^{4}-10a^{7}z^{3}-26a^{5}z^{3}-20a^{3}z^{3}-az^{3}+3z^{3}a^{-1}-2a^{8}z^{2}-3a^{6}z^{2}-5a^{4}z^{2}-6a^{2}z^{2}-2z^{2}+3a^{7}z+10a^{5}z+7a^{3}z+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

6

1

-1

4

3

2

4

1

-3

0

6

3

-2

7

5

-2

-4

6

5

1

-6

5

7

2

-8

4

6

-2

-10

2

6

4

-12

1

3

-2

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

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