L10a142

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Link Presentations

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{3}w^{2}+uv^{2}w^{3}-2uv^{2}w^{2}+uv^{2}w+uvw^{2}-2uvw+uv+uw-u+v^{3}w^{3}-v^{3}w^{2}-v^{2}w^{3}+2v^{2}w^{2}-v^{2}w-vw^{2}+2vw-v-w}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{-3}-2q^{-4}+4q^{-5}-4q^{-6}+7q^{-7}-7q^{-8}+7q^{-9}-5q^{-10}+4q^{-11}-2q^{-12}+q^{-13}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a^{12}z^{2}+a^{12}z^{-2}+3a^{12}-3a^{10}z^{4}-12a^{10}z^{2}-2a^{10}z^{-2}-11a^{10}+2a^{8}z^{6}+10a^{8}z^{4}+15a^{8}z^{2}+a^{8}z^{-2}+8a^{8}+a^{6}z^{6}+4a^{6}z^{4}+3a^{6}z^{2}$ (db)
Kauffman polynomial	$a^{16}z^{4}-2a^{16}z^{2}+a^{16}+2a^{15}z^{5}-3a^{15}z^{3}+2a^{14}z^{6}-a^{14}z^{4}-2a^{14}z^{2}+2a^{13}z^{7}-2a^{13}z^{5}+a^{13}z^{3}+2a^{12}z^{8}-5a^{12}z^{6}+9a^{12}z^{4}-8a^{12}z^{2}-a^{12}z^{-2}+5a^{12}+a^{11}z^{9}-8a^{11}z^{5}+18a^{11}z^{3}-11a^{11}z+2a^{11}z^{-1}+5a^{10}z^{8}-22a^{10}z^{6}+38a^{10}z^{4}-31a^{10}z^{2}-2a^{10}z^{-2}+13a^{10}+a^{9}z^{9}-11a^{9}z^{5}+18a^{9}z^{3}-11a^{9}z+2a^{9}z^{-1}+3a^{8}z^{8}-14a^{8}z^{6}+23a^{8}z^{4}-20a^{8}z^{2}-a^{8}z^{-2}+8a^{8}+2a^{7}z^{7}-7a^{7}z^{5}+4a^{7}z^{3}+a^{6}z^{6}-4a^{6}z^{4}+3a^{6}z^{2}$ (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		\

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-5

1

-7

2

1

-1

-9

2

-11

2

0

-13

5

2

3

-15

3

0

-17

4

0

-19

1

3

2

-21

3

4

-1

-23

1

3

2

-25

1

-1

-27

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-7$	$i=-5$
$r=-10$	${\mathbb {Z} }$
$r=-9$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{3}$
$r=-7$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-5$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-1$	${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$