L11n188

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_16,6,17,5 X_22,13,7,14 X_18,15,19,16 X_14,21,15,22 X_9,20,10,21 X_4,18,5,17 X_19,10,20,11 X₂₇₃₈ X_6,11,1,12
Gauss code	{1, -10, 2, -8, 3, -11}, {10, -1, -7, 9, 11, -2, 4, -6, 5, -3, 8, -5, -9, 7, 6, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{4}-t(2)^{4}+2t(1)^{2}t(2)^{3}-5t(1)t(2)^{3}+2t(2)^{3}-3t(1)^{2}t(2)^{2}+7t(1)t(2)^{2}-3t(2)^{2}+2t(1)^{2}t(2)-5t(1)t(2)+2t(2)-t(1)^{2}+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {9}{q^{5/2}}}-{\frac {12}{q^{7/2}}}+{\frac {12}{q^{9/2}}}-{\frac {11}{q^{11/2}}}+{\frac {8}{q^{13/2}}}-{\frac {5}{q^{15/2}}}+{\frac {2}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-2a^{7}z^{3}-3a^{7}z-a^{7}z^{-1}+2a^{5}z^{5}+6a^{5}z^{3}+7a^{5}z+3a^{5}z^{-1}+a^{3}z^{5}-4a^{3}z-2a^{3}z^{-1}-az^{3}-az$ (db)
Kauffman polynomial	$3a^{10}z^{4}-4a^{10}z^{2}+a^{9}z^{7}+3a^{9}z^{5}-6a^{9}z^{3}+a^{9}z+2a^{8}z^{8}-2a^{8}z^{4}+2a^{8}z^{2}-a^{8}+a^{7}z^{9}+5a^{7}z^{7}-12a^{7}z^{5}+12a^{7}z^{3}-5a^{7}z+a^{7}z^{-1}+6a^{6}z^{8}-7a^{6}z^{6}-3a^{6}z^{4}+10a^{6}z^{2}-3a^{6}+a^{5}z^{9}+9a^{5}z^{7}-26a^{5}z^{5}+28a^{5}z^{3}-14a^{5}z+3a^{5}z^{-1}+4a^{4}z^{8}-4a^{4}z^{6}-3a^{4}z^{4}+5a^{4}z^{2}-3a^{4}+5a^{3}z^{7}-10a^{3}z^{5}+8a^{3}z^{3}-7a^{3}z+2a^{3}z^{-1}+3a^{2}z^{6}-5a^{2}z^{4}+a^{2}z^{2}+az^{5}-2az^{3}+az$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

2

-2

5

1

4

-4

5

3

-2

-6

7

4

3

-8

6

0

-10

5

6

-1

-12

3

6

3

-14

2

5

-3

-16

3

-18

2

-2

Integral Khovanov Homology

(db, data source)

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