L11n282

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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_18,10,19,9 X_17,11,18,22 X_11,21,12,20 X_21,17,22,16 X_4,15,1,16 X_10,20,5,19
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 $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)(t(3)-1)\left(t(3)^{2}-t(3)+1\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-2q^{6}+4q^{5}-7q^{4}+9q^{3}-7q^{2}-q^{-2}+9q+4q^{-1}-5$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+3z^{4}a^{-2}-z^{4}a^{-4}-z^{4}+z^{2}a^{-2}-z^{2}-5a^{-2}+4a^{-4}-a^{-6}+2-5a^{-2}z^{-2}+4a^{-4}z^{-2}-a^{-6}z^{-2}+2z^{-2}$ (db)
Kauffman polynomial	$z^{8}a^{-2}+z^{8}a^{-4}+4z^{7}a^{-1}+6z^{7}a^{-3}+2z^{7}a^{-5}+8z^{6}a^{-2}+5z^{6}a^{-4}+z^{6}a^{-6}+4z^{6}+az^{5}-6z^{5}a^{-1}-8z^{5}a^{-3}-z^{5}a^{-5}-20z^{4}a^{-2}-9z^{4}a^{-4}+2z^{4}a^{-6}-9z^{4}-az^{3}+5z^{3}a^{-3}+7z^{3}a^{-5}+3z^{3}a^{-7}+7z^{2}a^{-2}+3z^{2}a^{-4}+4z^{2}-5za^{-1}-12za^{-3}-10za^{-5}-3za^{-7}+5a^{-2}+4a^{-4}+a^{-6}+3+5a^{-1}z^{-1}+9a^{-3}z^{-1}+5a^{-5}z^{-1}+a^{-7}z^{-1}-5a^{-2}z^{-2}-4a^{-4}z^{-2}-a^{-6}z^{-2}-2z^{-2}$ (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		\

-3

-2

-1

0

1

2

3

4

5

χ

13

2

-2

11

2

9

5

2

-3

7

4

2

5

3

5

2

3

6

4

2

1

3

7

4

-1

1

2

-1

-3

3

-5

1

-1

Integral Khovanov Homology

(db, data source)

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