L11n151

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Planar diagram presentation	X₈₁₉₂ X_18,9,19,10 X₆₇₁₈ X_22,19,7,20 X_12,5,13,6 X_3,10,4,11 X_4,15,5,16 X_11,16,12,17 X_20,13,21,14 X_14,21,15,22 X_17,2,18,3
Gauss code	{1, 11, -6, -7, 5, -3}, {3, -1, 2, 6, -8, -5, 9, -10, 7, 8, -11, -2, 4, -9, 10, -4}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2u^{2}v^{4}-2u^{2}v^{3}+u^{2}v^{2}+uv^{3}-uv^{2}+uv+v^{2}-2v+2}{uv^{2}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{25/2}}}-{\frac {2}{q^{23/2}}}+{\frac {3}{q^{21/2}}}-{\frac {4}{q^{19/2}}}+{\frac {4}{q^{17/2}}}-{\frac {4}{q^{15/2}}}+{\frac {3}{q^{13/2}}}-{\frac {3}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-za^{13}-a^{13}z^{-1}+z^{5}a^{11}+5z^{3}a^{11}+6za^{11}+2a^{11}z^{-1}-z^{7}a^{9}-5z^{5}a^{9}-6z^{3}a^{9}-za^{9}-z^{7}a^{7}-6z^{5}a^{7}-11z^{3}a^{7}-7za^{7}-a^{7}z^{-1}$ (db)
Kauffman polynomial	$-z^{2}a^{16}-2z^{3}a^{15}+za^{15}-3z^{4}a^{14}+4z^{2}a^{14}-2a^{14}-z^{7}a^{13}+2z^{5}a^{13}-z^{3}a^{13}-za^{13}+a^{13}z^{-1}-2z^{8}a^{12}+10z^{6}a^{12}-20z^{4}a^{12}+19z^{2}a^{12}-5a^{12}-z^{9}a^{11}+4z^{7}a^{11}-7z^{5}a^{11}+12z^{3}a^{11}-9za^{11}+2a^{11}z^{-1}-3z^{8}a^{10}+14z^{6}a^{10}-19z^{4}a^{10}+11z^{2}a^{10}-3a^{10}-z^{9}a^{9}+4z^{7}a^{9}-3z^{5}a^{9}-z^{8}a^{8}+4z^{6}a^{8}-2z^{4}a^{8}-3z^{2}a^{8}+a^{8}-z^{7}a^{7}+6z^{5}a^{7}-11z^{3}a^{7}+7za^{7}-a^{7}z^{-1}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-6

1

-8

1

0

-10

2

-12

1

0

-14

3

2

1

-16

1

0

-18

3

0

-20

1

2

1

-22

1

2

-1

-24

1

-26

1

-1

Integral Khovanov Homology

(db, data source)

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