L11a188

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Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_16,6,17,5 X_18,11,19,12 X_20,13,21,14 X_22,15,7,16 X_12,19,13,20 X_14,21,15,22 X_4,18,5,17 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -10, 2, -9, 3, -11}, {10, -1, 11, -2, 4, -7, 5, -8, 6, -3, 9, -4, 7, -5, 8, -6}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{5}-u^{2}v^{4}+u^{2}v^{3}-u^{2}v^{2}+u^{2}v-u^{2}+uv^{6}-3uv^{5}+3uv^{4}-3uv^{3}+3uv^{2}-3uv+u-v^{6}+v^{5}-v^{4}+v^{3}-v^{2}+v}{uv^{3}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}+{\frac {5}{q^{7/2}}}-{\frac {4}{q^{5/2}}}+{\frac {2}{q^{3/2}}}+{\frac {1}{q^{23/2}}}-{\frac {2}{q^{21/2}}}+{\frac {4}{q^{19/2}}}-{\frac {6}{q^{17/2}}}+{\frac {8}{q^{15/2}}}-{\frac {9}{q^{13/2}}}+{\frac {8}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{5}\right)-4a^{9}z^{3}-4a^{9}z-2a^{9}z^{-1}+a^{7}z^{7}+5a^{7}z^{5}+9a^{7}z^{3}+10a^{7}z+5a^{7}z^{-1}+a^{5}z^{7}+4a^{5}z^{5}+2a^{5}z^{3}-4a^{5}z-3a^{5}z^{-1}-a^{3}z^{5}-4a^{3}z^{3}-3a^{3}z$ (db)
Kauffman polynomial	$a^{14}z^{4}-2a^{14}z^{2}+2a^{13}z^{5}-3a^{13}z^{3}+3a^{12}z^{6}-6a^{12}z^{4}+5a^{12}z^{2}-a^{12}+3a^{11}z^{7}-6a^{11}z^{5}+6a^{11}z^{3}+3a^{10}z^{8}-8a^{10}z^{6}+10a^{10}z^{4}-2a^{10}z^{2}+3a^{9}z^{9}-13a^{9}z^{7}+28a^{9}z^{5}-28a^{9}z^{3}+12a^{9}z-2a^{9}z^{-1}+a^{8}z^{10}-11a^{8}z^{6}+25a^{8}z^{4}-22a^{8}z^{2}+5a^{8}+5a^{7}z^{9}-25a^{7}z^{7}+50a^{7}z^{5}-53a^{7}z^{3}+25a^{7}z-5a^{7}z^{-1}+a^{6}z^{10}-a^{6}z^{8}-9a^{6}z^{6}+18a^{6}z^{4}-15a^{6}z^{2}+5a^{6}+2a^{5}z^{9}-8a^{5}z^{7}+9a^{5}z^{5}-9a^{5}z^{3}+10a^{5}z-3a^{5}z^{-1}+2a^{4}z^{8}-9a^{4}z^{6}+10a^{4}z^{4}-2a^{4}z^{2}+a^{3}z^{7}-5a^{3}z^{5}+7a^{3}z^{3}-3a^{3}z$ (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

1

2

χ

0

1

-2

1

-1

-4

3

1

2

-6

3

2

-1

-8

5

2

3

-10

4

0

-12

5

4

1

-14

3

4

1

-16

3

5

-2

-18

1

3

2

-20

1

3

-2

-22

1

-24

1

-1

Integral Khovanov Homology

(db, data source)

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