L10a83

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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_14,17,15,18 X_16,5,17,6 X_4,15,5,16 X_18,13,19,14 X_10,20,11,19 X₂₇₃₈ X_6,11,1,12
Gauss code	{1, -9, 2, -6, 5, -10}, {9, -1, 3, -8, 10, -2, 7, -4, 6, -5, 4, -7, 8, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2t(2)^{2}t(1)^{2}-5t(2)t(1)^{2}+2t(1)^{2}-5t(2)^{2}t(1)+9t(2)t(1)-5t(1)+2t(2)^{2}-5t(2)+2}{t(1)t(2)}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {9}{q^{5/2}}}-{\frac {12}{q^{7/2}}}+{\frac {12}{q^{9/2}}}-{\frac {12}{q^{11/2}}}+{\frac {9}{q^{13/2}}}-{\frac {6}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$za^{9}-2z^{3}a^{7}-za^{7}+a^{7}z^{-1}+z^{5}a^{5}-2za^{5}-a^{5}z^{-1}+z^{5}a^{3}+z^{3}a^{3}-z^{3}a-za$ (db)
Kauffman polynomial	$a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{6}-6a^{10}z^{4}+3a^{10}z^{2}+4a^{9}z^{7}-6a^{9}z^{5}+2a^{9}z^{3}-a^{9}z+3a^{8}z^{8}-6a^{8}z^{4}+3a^{8}z^{2}+a^{7}z^{9}+7a^{7}z^{7}-14a^{7}z^{5}+7a^{7}z^{3}+a^{7}z-a^{7}z^{-1}+6a^{6}z^{8}-6a^{6}z^{6}+a^{6}+a^{5}z^{9}+7a^{5}z^{7}-14a^{5}z^{5}+7a^{5}z^{3}+a^{5}z-a^{5}z^{-1}+3a^{4}z^{8}-6a^{4}z^{4}+3a^{4}z^{2}+4a^{3}z^{7}-6a^{3}z^{5}+2a^{3}z^{3}-a^{3}z+3a^{2}z^{6}-6a^{2}z^{4}+3a^{2}z^{2}+az^{5}-2az^{3}+az$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

2

-2

4

1

3

-4

6

3

-3

-6

6

3

-8

6

0

-10

6

0

-12

3

6

3

-14

3

6

-3

-16

1

4

3

-18

2

-2

-20

1

Integral Khovanov Homology

(db, data source)

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