L10a117

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Planar diagram presentation	X_12,1,13,2 X_2,13,3,14 X_14,3,15,4 X_4,11,5,12 X_20,9,11,10 X_16,5,17,6 X_18,7,19,8 X_6,17,7,18 X_8,19,9,20 X_10,15,1,16
Gauss code	{1, -2, 3, -4, 6, -8, 7, -9, 5, -10}, {4, -1, 2, -3, 10, -6, 8, -7, 9, -5}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{4}\left(-v^{3}\right)-u^{3}v^{4}+2u^{3}v^{3}-2u^{3}v^{2}-2u^{2}v^{3}+3u^{2}v^{2}-2u^{2}v-2uv^{2}+2uv-u-v}{u^{2}v^{2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{9/2}}}-{\frac {3}{q^{11/2}}}+{\frac {4}{q^{13/2}}}-{\frac {5}{q^{15/2}}}+{\frac {6}{q^{17/2}}}-{\frac {6}{q^{19/2}}}+{\frac {5}{q^{21/2}}}-{\frac {4}{q^{23/2}}}+{\frac {2}{q^{25/2}}}-{\frac {1}{q^{27/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$z^{5}a^{11}+4z^{3}a^{11}+4za^{11}+a^{11}z^{-1}-z^{7}a^{9}-5z^{5}a^{9}-7z^{3}a^{9}-3za^{9}-a^{9}z^{-1}-z^{7}a^{7}-6z^{5}a^{7}-11z^{3}a^{7}-6za^{7}$ (db)
Kauffman polynomial	$-z^{3}a^{17}+za^{17}-2z^{4}a^{16}+z^{2}a^{16}-3z^{5}a^{15}+3z^{3}a^{15}-2za^{15}-3z^{6}a^{14}+3z^{4}a^{14}-z^{2}a^{14}-3z^{7}a^{13}+6z^{5}a^{13}-5z^{3}a^{13}+2za^{13}-2z^{8}a^{12}+4z^{6}a^{12}-2z^{4}a^{12}+2z^{2}a^{12}-z^{9}a^{11}+2z^{7}a^{11}-2z^{5}a^{11}+7z^{3}a^{11}-5za^{11}+a^{11}z^{-1}-3z^{8}a^{10}+11z^{6}a^{10}-10z^{4}a^{10}+3z^{2}a^{10}-a^{10}-z^{9}a^{9}+4z^{7}a^{9}-5z^{5}a^{9}+5z^{3}a^{9}-4za^{9}+a^{9}z^{-1}-z^{8}a^{8}+4z^{6}a^{8}-3z^{4}a^{8}-z^{2}a^{8}-z^{7}a^{7}+6z^{5}a^{7}-11z^{3}a^{7}+6za^{7}$ (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		\

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-6

1

-8

1

0

-10

2

-12

2

1

-1

-14

3

2

1

-16

3

2

-1

-18

3

0

-20

2

3

1

-22

2

3

-1

-24

2

-26

1

2

-1

-28

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-8$	$i=-6$
$r=-10$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-9$	${\mathbb {Z} }^{2}$
$r=-8$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-7$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-1$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$