L11n4

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_9,14,10,15 X₃₈₄₉ X_5,11,6,10 X_11,19,12,18 X_13,20,14,21 X_19,5,20,22 X_21,12,22,13 X_15,2,16,3
Gauss code	{1, 11, -5, -3}, {-6, -1, 2, 5, -4, 6, -7, 10, -8, 4, -11, -2, 3, 7, -9, 8, -10, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)^{5}-4t(1)t(2)^{4}+6t(1)t(2)^{3}-2t(2)^{3}-2t(1)t(2)^{2}+6t(2)^{2}-4t(2)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{3/2}-3{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {9}{q^{7/2}}}+{\frac {8}{q^{9/2}}}-{\frac {6}{q^{11/2}}}+{\frac {4}{q^{13/2}}}-{\frac {2}{q^{15/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}-a^{7}z^{3}-4a^{7}z-3a^{7}z^{-1}+2a^{5}z^{5}+8a^{5}z^{3}+9a^{5}z+4a^{5}z^{-1}-a^{3}z^{7}-5a^{3}z^{5}-9a^{3}z^{3}-8a^{3}z-2a^{3}z^{-1}+az^{5}+3az^{3}+az$ (db)
Kauffman polynomial	$-3z^{3}a^{9}+4za^{9}-a^{9}z^{-1}-z^{6}a^{8}-2z^{4}a^{8}+4z^{2}a^{8}-a^{8}-3z^{7}a^{7}+9z^{5}a^{7}-19z^{3}a^{7}+15za^{7}-3a^{7}z^{-1}-3z^{8}a^{6}+9z^{6}a^{6}-16z^{4}a^{6}+11z^{2}a^{6}-3a^{6}-z^{9}a^{5}-4z^{7}a^{5}+23z^{5}a^{5}-39z^{3}a^{5}+24za^{5}-4a^{5}z^{-1}-6z^{8}a^{4}+20z^{6}a^{4}-22z^{4}a^{4}+9z^{2}a^{4}-2a^{4}-z^{9}a^{3}-4z^{7}a^{3}+25z^{5}a^{3}-33z^{3}a^{3}+16za^{3}-2a^{3}z^{-1}-3z^{8}a^{2}+9z^{6}a^{2}-5z^{4}a^{2}+z^{2}a^{2}-a^{2}-3z^{7}a+11z^{5}a-10z^{3}a+3za-z^{6}+3z^{4}-z^{2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

0

2

1

-1

-2

5

2

3

-4

4

3

-1

-6

5

4

1

-8

3

4

1

-10

3

5

-2

-12

2

4

2

-14

2

-2

-16

2

Integral Khovanov Homology

(db, data source)

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