L10a85

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_16,5,17,6 X_18,10,19,9 X_10,20,11,19 X_14,18,15,17 X_2,11,3,12 X_12,3,13,4 X₄₇₅₈ X_20,14,7,13 X_6,15,1,16
Gauss code	{1, -6, 7, -8, 2, -10}, {8, -1, 3, -4, 6, -7, 9, -5, 10, -2, 5, -3, 4, -9}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}-3t(1)^{2}t(2)^{3}+6t(1)t(2)^{3}-3t(2)^{3}+4t(1)^{2}t(2)^{2}-9t(1)t(2)^{2}+4t(2)^{2}-3t(1)^{2}t(2)+6t(1)t(2)-3t(2)-2t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}-q^{7/2}+{\frac {11}{q^{7/2}}}+4q^{5/2}-{\frac {15}{q^{5/2}}}-8q^{3/2}+{\frac {16}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {3}{q^{11/2}}}+12{\sqrt {q}}-{\frac {15}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+4az^{5}-z^{5}a^{-1}+a^{5}z^{3}-6a^{3}z^{3}+6az^{3}-2z^{3}a^{-1}+2a^{5}z-7a^{3}z+3az-za^{-1}+2a^{5}z^{-1}-3a^{3}z^{-1}+az^{-1}$ (db)
Kauffman polynomial	$-2a^{3}z^{9}-2az^{9}-5a^{4}z^{8}-11a^{2}z^{8}-6z^{8}-6a^{5}z^{7}-10a^{3}z^{7}-11az^{7}-7z^{7}a^{-1}-3a^{6}z^{6}+2a^{4}z^{6}+14a^{2}z^{6}-4z^{6}a^{-2}+5z^{6}-a^{7}z^{5}+12a^{5}z^{5}+26a^{3}z^{5}+26az^{5}+12z^{5}a^{-1}-z^{5}a^{-3}+4a^{6}z^{4}+9a^{4}z^{4}+3a^{2}z^{4}+6z^{4}a^{-2}+4z^{4}+2a^{7}z^{3}-13a^{5}z^{3}-23a^{3}z^{3}-15az^{3}-6z^{3}a^{-1}+z^{3}a^{-3}-a^{6}z^{2}-10a^{4}z^{2}-10a^{2}z^{2}-2z^{2}a^{-2}-3z^{2}-a^{7}z+9a^{5}z+12a^{3}z+3az+za^{-1}+3a^{4}+3a^{2}+1-2a^{5}z^{-1}-3a^{3}z^{-1}-az^{-1}$ (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

5

1

4

2

7

3

-4

0

8

5

3

-2

9

8

-1

-4

6

7

-1

-6

5

9

4

-8

3

6

-3

-10

5

-12

1

3

-2

-14

1

Integral Khovanov Homology

(db, data source)

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