L11a167

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Planar diagram presentation	X₈₁₉₂ X_16,9,17,10 X₆₇₁₈ X_20,13,21,14 X_10,4,11,3 X_14,6,15,5 X_4,12,5,11 X_22,17,7,18 X_18,21,19,22 X_12,19,13,20 X_2,16,3,15
Gauss code	{1, -11, 5, -7, 6, -3}, {3, -1, 2, -5, 7, -10, 4, -6, 11, -2, 8, -9, 10, -4, 9, -8}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(2)t(1)-t(1)+1)(t(1)t(2)-t(2)+1)\left(2t(2)^{2}-3t(2)+2\right)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {13}{q^{9/2}}}-q^{7/2}+{\frac {18}{q^{7/2}}}+3q^{5/2}-{\frac {21}{q^{5/2}}}-7q^{3/2}+{\frac {19}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {9}{q^{11/2}}}+12{\sqrt {q}}-{\frac {18}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{5}z^{5}-2a^{5}z^{3}-a^{5}z-a^{5}z^{-1}+a^{3}z^{7}+3a^{3}z^{5}+2a^{3}z^{3}+2a^{3}z^{-1}+az^{7}+4az^{5}-z^{5}a^{-1}+7az^{3}-3z^{3}a^{-1}+5az-3za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{4}z^{10}-2a^{2}z^{10}-6a^{5}z^{9}-11a^{3}z^{9}-5az^{9}-7a^{6}z^{8}-9a^{4}z^{8}-8a^{2}z^{8}-6z^{8}-4a^{7}z^{7}+10a^{5}z^{7}+22a^{3}z^{7}+3az^{7}-5z^{7}a^{-1}-a^{8}z^{6}+17a^{6}z^{6}+30a^{4}z^{6}+22a^{2}z^{6}-3z^{6}a^{-2}+7z^{6}+9a^{7}z^{5}-2a^{5}z^{5}-15a^{3}z^{5}+4az^{5}+7z^{5}a^{-1}-z^{5}a^{-3}+2a^{8}z^{4}-11a^{6}z^{4}-26a^{4}z^{4}-20a^{2}z^{4}+5z^{4}a^{-2}-2z^{4}-4a^{7}z^{3}+2a^{5}z^{3}+3a^{3}z^{3}-9az^{3}-4z^{3}a^{-1}+2z^{3}a^{-3}-a^{8}z^{2}+4a^{6}z^{2}+12a^{4}z^{2}+7a^{2}z^{2}-2z^{2}a^{-2}-2z^{2}-a^{5}z+5az+3za^{-1}-za^{-3}-2a^{6}-5a^{4}-3a^{2}+1+a^{5}z^{-1}+2a^{3}z^{-1}-a^{-1}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

2

-2

4

5

1

4

2

8

3

-5

0

10

4

6

-2

10

9

-1

-4

11

9

2

-6

7

10

3

-8

6

11

-5

-10

3

7

4

-12

1

6

-5

-14

3

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{5}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$