L10a81

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2u^{2}v^{3}-5u^{2}v^{2}+2u^{2}v+uv^{4}-6uv^{3}+9uv^{2}-6uv+u+2v^{3}-5v^{2}+2v}{uv^{2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {13}{q^{7/2}}}+{\frac {14}{q^{9/2}}}-{\frac {13}{q^{11/2}}}+{\frac {10}{q^{13/2}}}-{\frac {7}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$za^{9}+a^{9}z^{-1}-3z^{3}a^{7}-5za^{7}-2a^{7}z^{-1}+2z^{5}a^{5}+5z^{3}a^{5}+5za^{5}+2a^{5}z^{-1}+z^{5}a^{3}-3za^{3}-a^{3}z^{-1}-z^{3}a-za$ (db)
Kauffman polynomial	$a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{6}-5a^{10}z^{4}+a^{10}z^{2}+5a^{9}z^{7}-9a^{9}z^{5}+6a^{9}z^{3}-5a^{9}z+a^{9}z^{-1}+5a^{8}z^{8}-8a^{8}z^{6}+5a^{8}z^{4}-2a^{8}z^{2}+2a^{7}z^{9}+6a^{7}z^{7}-22a^{7}z^{5}+26a^{7}z^{3}-12a^{7}z+2a^{7}z^{-1}+10a^{6}z^{8}-22a^{6}z^{6}+20a^{6}z^{4}-6a^{6}z^{2}+a^{6}+2a^{5}z^{9}+6a^{5}z^{7}-22a^{5}z^{5}+26a^{5}z^{3}-12a^{5}z+2a^{5}z^{-1}+5a^{4}z^{8}-8a^{4}z^{6}+5a^{4}z^{4}-2a^{4}z^{2}+5a^{3}z^{7}-9a^{3}z^{5}+6a^{3}z^{3}-5a^{3}z+a^{3}z^{-1}+3a^{2}z^{6}-5a^{2}z^{4}+a^{2}z^{2}+az^{5}-2az^{3}+az$ (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

5

1

4

-4

6

3

-3

-6

7

4

3

-8

7

6

-1

-10

6

7

-1

-12

4

7

3

-14

3

6

-3

-16

1

5

4

-18

2

-2

-20

1

Integral Khovanov Homology

(db, data source)

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