L11n10

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

A Braid Representative

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Polynomial invariants

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

2

1

0

-14

4

2

-16

1

2

1

-18

3

0

-20

2

0

-22

1

3

-2

-24

1

-26

1

-1

Integral Khovanov Homology

(db, data source)

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