L11n98

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_13,21,14,20 X_7,17,8,16 X_9,19,10,18 X_17,9,18,8 X_19,11,20,10 X_15,5,16,22 X_21,15,22,14 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 6, -5, 7, 11, -2, -3, 9, -8, 4, -6, 5, -7, 3, -9, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(2)^{7}-t(1)t(2)^{5}+t(2)^{5}+2t(1)t(2)^{4}-2t(2)^{4}-2t(1)t(2)^{3}+2t(2)^{3}+t(1)t(2)^{2}-t(2)^{2}-t(1)}{{\sqrt {t(1)}}t(2)^{7/2}}}$ (db)
Jones polynomial	$q^{17/2}-2q^{15/2}+4q^{13/2}-4q^{11/2}+4q^{9/2}-4q^{7/2}+2q^{5/2}-q^{3/2}-{\sqrt {q}}-{\frac {1}{q^{3/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-3}+z^{5}a^{-1}-8z^{5}a^{-3}+6z^{3}a^{-1}-20z^{3}a^{-3}+3z^{3}a^{-5}+z^{3}a^{-7}+10za^{-1}-20za^{-3}+7za^{-5}+za^{-7}+4a^{-1}z^{-1}-7a^{-3}z^{-1}+3a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-z^{8}a^{-2}-z^{8}a^{-4}+9z^{7}a^{-1}+10z^{7}a^{-3}-z^{7}a^{-7}+9z^{6}a^{-2}+9z^{6}a^{-4}-2z^{6}a^{-6}-2z^{6}a^{-8}-28z^{5}a^{-1}-33z^{5}a^{-3}-3z^{5}a^{-5}-2z^{5}a^{-9}-24z^{4}a^{-2}-26z^{4}a^{-4}+z^{4}a^{-6}+2z^{4}a^{-8}-z^{4}a^{-10}+37z^{3}a^{-1}+46z^{3}a^{-3}+6z^{3}a^{-5}+3z^{3}a^{-9}+23z^{2}a^{-2}+24z^{2}a^{-4}+z^{2}a^{-8}+2z^{2}a^{-10}-20za^{-1}-29za^{-3}-8za^{-5}+za^{-7}-7a^{-2}-7a^{-4}-a^{-10}+4a^{-1}z^{-1}+7a^{-3}z^{-1}+3a^{-5}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

18

1

-1

16

1

14

3

1

-2

12

1

0

10

3

0

8

1

2

1

0

6

1

3

2

4

1

2

-1

2

0

1

-2

1

-4

1

Integral Khovanov Homology

(db, data source)

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