L10a76

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_14,9,15,10 X₆₇₁₈ X_20,15,7,16 X_16,6,17,5 X_4,20,5,19 X_10,4,11,3 X_12,17,13,18 X_18,11,19,12 X_2,14,3,13
Gauss code	{1, -10, 7, -6, 5, -3}, {3, -1, 2, -7, 9, -8, 10, -2, 4, -5, 8, -9, 6, -4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{4}-4u^{2}v^{3}+5u^{2}v^{2}-2u^{2}v-uv^{4}+7uv^{3}-11uv^{2}+7uv-u-2v^{3}+5v^{2}-4v+1}{uv^{2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+12{\sqrt {q}}-{\frac {17}{\sqrt {q}}}+{\frac {17}{q^{3/2}}}-{\frac {16}{q^{5/2}}}+{\frac {13}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+4az^{5}-z^{5}a^{-1}+a^{5}z^{3}-5a^{3}z^{3}+7az^{3}-2z^{3}a^{-1}+a^{5}z-4a^{3}z+6az-2za^{-1}-a^{3}z^{-1}+3az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-3a^{3}z^{9}-3az^{9}-7a^{4}z^{8}-15a^{2}z^{8}-8z^{8}-7a^{5}z^{7}-9a^{3}z^{7}-10az^{7}-8z^{7}a^{-1}-4a^{6}z^{6}+8a^{4}z^{6}+29a^{2}z^{6}-4z^{6}a^{-2}+13z^{6}-a^{7}z^{5}+11a^{5}z^{5}+28a^{3}z^{5}+32az^{5}+15z^{5}a^{-1}-z^{5}a^{-3}+6a^{6}z^{4}+a^{4}z^{4}-17a^{2}z^{4}+5z^{4}a^{-2}-7z^{4}+a^{7}z^{3}-5a^{5}z^{3}-23a^{3}z^{3}-28az^{3}-10z^{3}a^{-1}+z^{3}a^{-3}-2a^{6}z^{2}-3a^{4}z^{2}-a^{2}z^{2}+2a^{5}z+7a^{3}z+10az+5za^{-1}+a^{4}+3a^{2}+3-a^{3}z^{-1}-3az^{-1}-2a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

3

-3

4

6

1

5

2

7

4

-3

0

10

5

-2

8

0

-4

8

9

-1

-6

5

8

3

-8

3

8

-5

-10

1

5

4

-12

3

-3

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=0$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$