L10a79

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2u^{2}v^{3}-5u^{2}v^{2}+4u^{2}v-u^{2}+uv^{4}-5uv^{3}+9uv^{2}-5uv+u-v^{4}+4v^{3}-5v^{2}+2v}{uv^{2}}}$ (db)
Jones polynomial	$q^{5/2}-4q^{3/2}+8{\sqrt {q}}-{\frac {12}{\sqrt {q}}}+{\frac {14}{q^{3/2}}}-{\frac {16}{q^{5/2}}}+{\frac {13}{q^{7/2}}}-{\frac {11}{q^{9/2}}}+{\frac {7}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+3z^{3}a^{5}+5za^{5}+3a^{5}z^{-1}-2z^{5}a^{3}-5z^{3}a^{3}-6za^{3}-2a^{3}z^{-1}-z^{5}a+za+z^{3}a^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-3a^{8}z^{4}+3a^{8}z^{2}-a^{8}+3a^{7}z^{7}-8a^{7}z^{5}+7a^{7}z^{3}-3a^{7}z+a^{7}z^{-1}+4a^{6}z^{8}-7a^{6}z^{6}-2a^{6}z^{4}+7a^{6}z^{2}-3a^{6}+2a^{5}z^{9}+6a^{5}z^{7}-26a^{5}z^{5}+24a^{5}z^{3}-10a^{5}z+3a^{5}z^{-1}+11a^{4}z^{8}-20a^{4}z^{6}+3a^{4}z^{4}+6a^{4}z^{2}-3a^{4}+2a^{3}z^{9}+13a^{3}z^{7}-36a^{3}z^{5}+27a^{3}z^{3}-9a^{3}z+2a^{3}z^{-1}+7a^{2}z^{8}-4a^{2}z^{6}-7a^{2}z^{4}+z^{4}a^{-2}+4a^{2}z^{2}+10az^{7}-14az^{5}+4z^{5}a^{-1}+8az^{3}-2z^{3}a^{-1}-2az+8z^{6}-8z^{4}+2z^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

6

1

-1

4

3

2

5

1

-4

0

7

3

4

-2

8

6

-2

-4

8

6

2

-6

6

9

3

-8

5

7

-2

-10

2

6

4

-12

1

5

-4

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

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