L11n85

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_14,8,15,7 X_9,18,10,19 X_22,19,5,20 X_20,15,21,16 X_16,21,17,22 X_17,8,18,9 X_10,14,11,13 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, 8, -4, -9, 11, -2, 9, -3, 6, -7, -8, 4, 5, -6, 7, -5}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{3}-5uv^{2}+6uv-3u-3v^{3}+6v^{2}-5v+1}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	${\frac {9}{q^{9/2}}}-{\frac {8}{q^{7/2}}}+{\frac {5}{q^{5/2}}}-{\frac {2}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {2}{q^{19/2}}}+{\frac {5}{q^{17/2}}}-{\frac {8}{q^{15/2}}}+{\frac {9}{q^{13/2}}}-{\frac {11}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+3za^{9}+2a^{9}z^{-1}-3z^{3}a^{7}-3za^{7}+z^{5}a^{5}+z^{3}a^{5}-a^{5}z^{-1}-2z^{3}a^{3}-2za^{3}$ (db)
Kauffman polynomial	$-z^{6}a^{12}+4z^{4}a^{12}-5z^{2}a^{12}+2a^{12}-2z^{7}a^{11}+6z^{5}a^{11}-5z^{3}a^{11}+2za^{11}-a^{11}z^{-1}-2z^{8}a^{10}+2z^{6}a^{10}+8z^{4}a^{10}-12z^{2}a^{10}+5a^{10}-z^{9}a^{9}-3z^{7}a^{9}+12z^{5}a^{9}-8z^{3}a^{9}+4za^{9}-2a^{9}z^{-1}-5z^{8}a^{8}+8z^{6}a^{8}+z^{4}a^{8}-4z^{2}a^{8}+3a^{8}-z^{9}a^{7}-4z^{7}a^{7}+8z^{5}a^{7}-z^{3}a^{7}-2za^{7}-3z^{8}a^{6}+4z^{6}a^{6}-7z^{4}a^{6}+6z^{2}a^{6}-a^{6}-3z^{7}a^{5}+2z^{5}a^{5}-z^{3}a^{5}-2za^{5}+a^{5}z^{-1}-z^{6}a^{4}-4z^{4}a^{4}+3z^{2}a^{4}-3z^{3}a^{3}+2za^{3}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

2

-4

4

1

-3

-6

4

1

3

-8

5

4

-1

-10

6

4

2

-12

4

6

2

-14

4

5

-1

-16

1

4

3

-18

1

4

-3

-20

1

-22

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=0$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$