L10a128

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}+t(1)t(2)^{2}-2t(1)t(3)t(2)^{2}+3t(3)t(2)^{2}-t(2)^{2}-2t(1)t(3)^{2}t(2)+3t(3)^{2}t(2)-3t(1)t(2)+4t(1)t(3)t(2)-4t(3)t(2)+2t(2)+t(1)t(3)^{2}-t(3)^{2}+2t(1)-3t(1)t(3)+2t(3)-1}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{-6}-2q^{-5}+q^{4}+6q^{-4}-4q^{3}-8q^{-3}+7q^{2}+12q^{-2}-10q-12q^{-1}+13$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-3a^{4}z^{2}-2a^{4}z^{-2}-5a^{4}+3a^{2}z^{4}+z^{4}a^{-2}+7a^{2}z^{2}+a^{2}z^{-2}+z^{2}a^{-2}+5a^{2}-z^{6}-3z^{4}-4z^{2}-1$ (db)
Kauffman polynomial	$a^{6}z^{6}-4a^{6}z^{4}+6a^{6}z^{2}+a^{6}z^{-2}-4a^{6}+2a^{5}z^{7}-5a^{5}z^{5}+2a^{5}z^{3}+3a^{5}z-2a^{5}z^{-1}+2a^{4}z^{8}-13a^{4}z^{4}+z^{4}a^{-4}+18a^{4}z^{2}+2a^{4}z^{-2}-9a^{4}+a^{3}z^{9}+5a^{3}z^{7}-14a^{3}z^{5}+4z^{5}a^{-3}+4a^{3}z^{3}-3z^{3}a^{-3}+5a^{3}z-2a^{3}z^{-1}+6a^{2}z^{8}-4a^{2}z^{6}+7z^{6}a^{-2}-16a^{2}z^{4}-8z^{4}a^{-2}+19a^{2}z^{2}+3z^{2}a^{-2}+a^{2}z^{-2}-8a^{2}+az^{9}+10az^{7}+7z^{7}a^{-1}-19az^{5}-6z^{5}a^{-1}+4az^{3}-z^{3}a^{-1}+3az+za^{-1}+4z^{8}+4z^{6}-16z^{4}+10z^{2}-2$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

9

1

7

3

-3

5

4

1

3

6

3

-3

1

7

4

3

-1

6

7

1

-3

6

0

-5

4

8

4

-7

2

4

-2

-9

1

5

4

-11

1

-1

-13

1

Integral Khovanov Homology

(db, data source)

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