L11n388

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_5,12,6,13 X₈₄₉₃ X_22,11,19,12 X_18,22,5,21 X_20,10,21,9 X_10,17,11,18 X_16,19,17,20 X_2,14,3,13
Gauss code	{1, -11, 5, -3}, {10, -8, 7, -6}, {-4, -1, 2, -5, 8, -9, 6, 4, 11, -2, 3, -10, 9, -7}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)(w-1)^{3}}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-3q^{-6}+6q^{-5}-9q^{-4}+12q^{-3}-q^{2}-10q^{-2}+5q+11q^{-1}-7$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{6}z^{-2}-2a^{6}-a^{4}z^{4}+2a^{4}z^{2}+4a^{4}z^{-2}+7a^{4}+a^{2}z^{6}+2a^{2}z^{4}-2a^{2}z^{2}-5a^{2}z^{-2}-8a^{2}-z^{4}+2z^{-2}+3$ (db)
Kauffman polynomial	$3a^{4}z^{8}+3a^{2}z^{8}+6a^{5}z^{7}+13a^{3}z^{7}+7az^{7}+3a^{6}z^{6}+3a^{4}z^{6}+5a^{2}z^{6}+5z^{6}-13a^{5}z^{5}-27a^{3}z^{5}-13az^{5}+z^{5}a^{-1}-8a^{4}z^{4}-16a^{2}z^{4}-8z^{4}+6a^{7}z^{3}+24a^{5}z^{3}+24a^{3}z^{3}+6az^{3}-3a^{4}z^{2}-3a^{2}z^{2}-6a^{7}z-19a^{5}z-21a^{3}z-8az+3a^{6}+10a^{4}+11a^{2}+5+a^{7}z^{-1}+5a^{5}z^{-1}+9a^{3}z^{-1}+5az^{-1}-a^{6}z^{-2}-4a^{4}z^{-2}-5a^{2}z^{-2}-2z^{-2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

χ

5

1

-1

3

4

1

3

1

-2

-1

8

4

-3

6

7

1

-5

6

4

2

-7

3

6

3

-9

3

6

-3

-11

3

-13

3

-3

Integral Khovanov Homology

(db, data source)

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