L11n374

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_2,9,3,10 X_12,3,13,4 X_10,5,11,6 X_16,11,5,12 X_4,15,1,16 X_13,20,14,21 X_7,19,8,18 X_17,9,18,8 X_19,22,20,17 X_21,14,22,15
Gauss code	{1, -2, 3, -6}, {-9, 8, -10, 7, -11, 10}, {4, -1, -8, 9, 2, -4, 5, -3, -7, 11, 6, -5}

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

Polynomial invariants

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

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

1

-1

1

-1

-3

5

1

4

-5

3

0

-7

5

3

2

-9

1

4

3

0

-11

5

0

-13

2

5

3

-15

1

3

-2

-17

2

-19

1

-1

Integral Khovanov Homology

(db, data source)

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