L11n284

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,6,13,5 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	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)(t(3)-1)\left(t(2)t(3)^{3}+1\right)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-2q^{7}+3q^{6}-4q^{5}+6q^{4}-4q^{3}+6q^{2}-3q-q^{-1}+3$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^{2}a^{-8}-a^{-8}z^{-2}-z^{6}a^{-6}-5z^{4}a^{-6}-5z^{2}a^{-6}+4a^{-6}z^{-2}+2a^{-6}+z^{8}a^{-4}+6z^{6}a^{-4}+11z^{4}a^{-4}+6z^{2}a^{-4}-5a^{-4}z^{-2}-5a^{-4}-z^{6}a^{-2}-4z^{4}a^{-2}-2z^{2}a^{-2}+2a^{-2}z^{-2}+3a^{-2}$ (db)
Kauffman polynomial	$2z^{9}a^{-3}+2z^{9}a^{-5}+3z^{8}a^{-2}+7z^{8}a^{-4}+4z^{8}a^{-6}+z^{7}a^{-1}-7z^{7}a^{-3}-6z^{7}a^{-5}+2z^{7}a^{-7}-15z^{6}a^{-2}-35z^{6}a^{-4}-20z^{6}a^{-6}-4z^{5}a^{-1}+z^{5}a^{-3}-3z^{5}a^{-5}-8z^{5}a^{-7}+21z^{4}a^{-2}+49z^{4}a^{-4}+29z^{4}a^{-6}+z^{4}a^{-8}+3z^{3}a^{-1}+4z^{3}a^{-3}+7z^{3}a^{-5}+6z^{3}a^{-7}-9z^{2}a^{-2}-21z^{2}a^{-4}-14z^{2}a^{-6}-2z^{2}a^{-8}+5za^{-3}+9za^{-5}+5za^{-7}+za^{-9}-3a^{-2}-4a^{-4}-2a^{-6}-5a^{-3}z^{-1}-9a^{-5}z^{-1}-5a^{-7}z^{-1}-a^{-9}z^{-1}+2a^{-2}z^{-2}+5a^{-4}z^{-2}+4a^{-6}z^{-2}+a^{-8}z^{-2}$ (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

15

2

-2

13

2

1

11

3

2

-1

9

3

2

1

2

7

3

5

2

5

3

2

1

2

3

2

5

3

1

0

-1

2

-3

1

-1

Integral Khovanov Homology

(db, data source)

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