L11n88

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Planar diagram presentation	X₆₁₇₂ X_3,13,4,12 X_7,16,8,17 X_17,22,18,5 X_9,15,10,14 X_19,10,20,11 X_21,9,22,8 X_13,18,14,19 X_15,21,16,20 X₂₅₃₆ X_11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 7, -5, 6, -11, 2, -8, 5, -9, 3, -4, 8, -6, 9, -7, 4}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)^{3}\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {3}{q^{9/2}}}-4q^{7/2}+{\frac {8}{q^{7/2}}}+8q^{5/2}-{\frac {12}{q^{5/2}}}-13q^{3/2}+{\frac {15}{q^{3/2}}}+15{\sqrt {q}}-{\frac {17}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-a^{3}z^{5}+4az^{5}-2z^{5}a^{-1}-3a^{3}z^{3}+7az^{3}-5z^{3}a^{-1}+z^{3}a^{-3}+a^{5}z-5a^{3}z+7az-4za^{-1}+za^{-3}+a^{5}z^{-1}-3a^{3}z^{-1}+4az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$6a^{5}z^{3}-4a^{5}z+a^{5}z^{-1}+3a^{4}z^{6}+z^{6}a^{-4}+6a^{4}z^{4}-2z^{4}a^{-4}-7a^{4}z^{2}+z^{2}a^{-4}+a^{4}+10a^{3}z^{7}+4z^{7}a^{-3}-21a^{3}z^{5}-10z^{5}a^{-3}+28a^{3}z^{3}+8z^{3}a^{-3}-16a^{3}z-3za^{-3}+3a^{3}z^{-1}+10a^{2}z^{8}+6z^{8}a^{-2}-22a^{2}z^{6}-13z^{6}a^{-2}+27a^{2}z^{4}+5z^{4}a^{-2}-17a^{2}z^{2}+3a^{2}+a^{-2}+3az^{9}+3z^{9}a^{-1}+13az^{7}+7z^{7}a^{-1}-48az^{5}-37z^{5}a^{-1}+49az^{3}+35z^{3}a^{-1}-22az-13za^{-1}+4az^{-1}+2a^{-1}z^{-1}+16z^{8}-39z^{6}+28z^{4}-11z^{2}+2$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

3

6

5

1

-4

4

8

3

5

2

7

5

-2

0

10

8

2

-2

7

9

2

-4

5

8

-3

-6

3

7

4

-8

5

-5

-10

3

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-4$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$