L11n424

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_7,16,8,17 X_5,14,6,15 X_3,10,4,11 X_13,4,14,5 X_2,18,3,17 X_18,9,19,10 X_21,7,22,12 X_11,13,12,22 X_15,20,16,21 X_19,1,20,6
Gauss code	{1, -6, -4, 5, -3, 11}, {-2, -1, 7, 4, -9, 8}, {-5, 3, -10, 2, 6, -7, -11, 10, -8, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(w-1)(uw-v-w+1)(uvw-uv-uw+v)}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{-8}+3q^{-7}-6q^{-6}+9q^{-5}-10q^{-4}+12q^{-3}-9q^{-2}+2q+8q^{-1}-4$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{6}-2z^{2}a^{6}-2a^{6}+z^{6}a^{4}+4z^{4}a^{4}+8z^{2}a^{4}+a^{4}z^{-2}+6a^{4}-3z^{4}a^{2}-8z^{2}a^{2}-2a^{2}z^{-2}-7a^{2}+2z^{2}+z^{-2}+3$ (db)
Kauffman polynomial	$z^{5}a^{9}-2z^{3}a^{9}+3z^{6}a^{8}-6z^{4}a^{8}+z^{2}a^{8}+5z^{7}a^{7}-12z^{5}a^{7}+8z^{3}a^{7}-2za^{7}+5z^{8}a^{6}-14z^{6}a^{6}+18z^{4}a^{6}-12z^{2}a^{6}+4a^{6}+2z^{9}a^{5}-8z^{5}a^{5}+14z^{3}a^{5}-6za^{5}+8z^{8}a^{4}-28z^{6}a^{4}+49z^{4}a^{4}-38z^{2}a^{4}-a^{4}z^{-2}+13a^{4}+2z^{9}a^{3}-4z^{7}a^{3}+6z^{5}a^{3}+2z^{3}a^{3}-7za^{3}+2a^{3}z^{-1}+3z^{8}a^{2}-11z^{6}a^{2}+28z^{4}a^{2}-32z^{2}a^{2}-2a^{2}z^{-2}+13a^{2}+z^{7}a+z^{5}a-2z^{3}a-3za+2az^{-1}+3z^{4}-7z^{2}-z^{-2}+5$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

3

2

1

2

-2

-1

6

2

4

-3

5

4

-1

-5

7

4

3

-7

5

7

2

-9

4

5

-1

-11

2

5

3

-13

1

4

-3

-15

2

-17

1

-1

Integral Khovanov Homology

(db, data source)

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