L10a82

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2t(2)^{2}t(1)^{2}-5t(2)t(1)^{2}+3t(1)^{2}-5t(2)^{2}t(1)+9t(2)t(1)-5t(1)+3t(2)^{2}-5t(2)+2}{t(1)t(2)}}$ (db)
Jones polynomial	$-7q^{9/2}+10q^{7/2}-{\frac {1}{q^{7/2}}}-13q^{5/2}+{\frac {2}{q^{5/2}}}+13q^{3/2}-{\frac {6}{q^{3/2}}}-q^{13/2}+4q^{11/2}-12{\sqrt {q}}+{\frac {9}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}+z^{5}a^{-3}+z^{3}a^{-3}+a^{3}z+za^{-3}+a^{3}z^{-1}+z^{5}a^{-1}-2az^{3}-2az-za^{-1}-az^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-z^{3}a^{-7}+4z^{6}a^{-6}-7z^{4}a^{-6}+2z^{2}a^{-6}+6z^{7}a^{-5}-11z^{5}a^{-5}+5z^{3}a^{-5}-za^{-5}+4z^{8}a^{-4}-11z^{4}a^{-4}+6z^{2}a^{-4}+z^{9}a^{-3}+11z^{7}a^{-3}+a^{3}z^{5}-27z^{5}a^{-3}-3a^{3}z^{3}+21z^{3}a^{-3}+3a^{3}z-5za^{-3}-a^{3}z^{-1}+7z^{8}a^{-2}+2a^{2}z^{6}-8z^{6}a^{-2}-3a^{2}z^{4}+2z^{2}a^{-2}+a^{2}+z^{9}a^{-1}+3az^{7}+8z^{7}a^{-1}-3az^{5}-19z^{5}a^{-1}-az^{3}+17z^{3}a^{-1}+2az-5za^{-1}-az^{-1}+3z^{8}-2z^{6}+z^{4}-2z^{2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

3

-3

10

4

1

3

8

6

3

-3

6

7

4

3

4

6

0

2

6

7

-1

0

4

7

3

-2

2

5

-3

-4

1

5

4

-6

1

-1

-8

1

Integral Khovanov Homology

(db, data source)

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