L11n281

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₈₄₉₃ X_2,14,3,13 X_14,7,15,8 X_9,18,10,19 X_22,17,11,18 X_20,11,21,12 X_16,21,17,22 X_4,15,1,16 X_19,10,20,5
Gauss code	{1, -4, 3, -10}, {-2, -1, 5, -3, -6, 11}, {8, 2, 4, -5, 10, -9, 7, 6, -11, -8, 9, -7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-3

2

-5

3

2

-1

-7

5

-9

2

3

1

-11

7

5

2

-13

3

4

1

-15

3

5

-2

-17

1

3

2

-19

3

-3

-21

1

Integral Khovanov Homology

(db, data source)

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