L10n58

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X_7,17,8,16 X_5,1,6,10 X₃₇₄₆ X_9,5,10,4 X_17,11,18,20 X_13,19,14,18 X_19,15,20,14 X_2,11,3,12 X_15,9,16,8
Gauss code	{1, -9, -4, 5, -3, 4, -2, 10, -5, 3}, {9, -1, -7, 8, -10, 2, -6, 7, -8, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(u^{2}v+uv^{2}-2uv+u+v\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$8q^{9/2}-8q^{7/2}+4q^{5/2}-3q^{3/2}-q^{19/2}+3q^{17/2}-5q^{15/2}+8q^{13/2}-8q^{11/2}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-za^{-9}-a^{-9}z^{-1}+3z^{3}a^{-7}+7za^{-7}+5a^{-7}z^{-1}-2z^{5}a^{-5}-8z^{3}a^{-5}-13za^{-5}-8a^{-5}z^{-1}+3z^{3}a^{-3}+7za^{-3}+4a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-11}-2z^{3}a^{-11}+za^{-11}+3z^{6}a^{-10}-7z^{4}a^{-10}+5z^{2}a^{-10}-2a^{-10}+3z^{7}a^{-9}-3z^{5}a^{-9}-4z^{3}a^{-9}+za^{-9}+a^{-9}z^{-1}+z^{8}a^{-8}+8z^{6}a^{-8}-26z^{4}a^{-8}+24z^{2}a^{-8}-9a^{-8}+7z^{7}a^{-7}-14z^{5}a^{-7}+12z^{3}a^{-7}-9za^{-7}+5a^{-7}z^{-1}+z^{8}a^{-6}+8z^{6}a^{-6}-25z^{4}a^{-6}+31z^{2}a^{-6}-14a^{-6}+4z^{7}a^{-5}-10z^{5}a^{-5}+20z^{3}a^{-5}-19za^{-5}+8a^{-5}z^{-1}+3z^{6}a^{-4}-6z^{4}a^{-4}+12z^{2}a^{-4}-8a^{-4}+6z^{3}a^{-3}-10za^{-3}+4a^{-3}z^{-1}$ (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		\

0

1

2

3

4

5

6

7

8

χ

20

1

18

2

-2

16

3

1

2

14

5

2

-3

12

3

0

10

5

0

8

3

0

6

1

5

4

2

3

-1

2

3

Integral Khovanov Homology

(db, data source)

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