L11a219

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(2)^{4}-3t(1)t(2)^{4}+2t(2)^{4}-3t(1)^{2}t(2)^{3}+7t(1)t(2)^{3}-3t(2)^{3}+3t(1)^{2}t(2)^{2}-7t(1)t(2)^{2}+3t(2)^{2}-3t(1)^{2}t(2)+7t(1)t(2)-3t(2)+2t(1)^{2}-3t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{17/2}-3q^{15/2}+7q^{13/2}-11q^{11/2}+14q^{9/2}-17q^{7/2}+15q^{5/2}-14q^{3/2}+10{\sqrt {q}}-{\frac {6}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+2za^{-7}+a^{-7}z^{-1}-2z^{5}a^{-5}-6z^{3}a^{-5}-6za^{-5}-3a^{-5}z^{-1}+z^{7}a^{-3}+4z^{5}a^{-3}+7z^{3}a^{-3}+7za^{-3}+2a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-6z^{3}a^{-1}+2az-4za^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-3z^{9}a^{-1}-7z^{9}a^{-3}-4z^{9}a^{-5}-7z^{8}a^{-2}-11z^{8}a^{-4}-7z^{8}a^{-6}-3z^{8}-az^{7}+7z^{7}a^{-1}+13z^{7}a^{-3}-3z^{7}a^{-5}-8z^{7}a^{-7}+32z^{6}a^{-2}+33z^{6}a^{-4}+7z^{6}a^{-6}-6z^{6}a^{-8}+12z^{6}+4az^{5}+2z^{5}a^{-1}+5z^{5}a^{-3}+22z^{5}a^{-5}+12z^{5}a^{-7}-3z^{5}a^{-9}-34z^{4}a^{-2}-24z^{4}a^{-4}+3z^{4}a^{-6}+7z^{4}a^{-8}-z^{4}a^{-10}-15z^{4}-5az^{3}-9z^{3}a^{-1}-12z^{3}a^{-3}-21z^{3}a^{-5}-11z^{3}a^{-7}+2z^{3}a^{-9}+11z^{2}a^{-2}+4z^{2}a^{-4}-7z^{2}a^{-6}-5z^{2}a^{-8}+z^{2}a^{-10}+6z^{2}+2az+2za^{-1}+5za^{-3}+10za^{-5}+5za^{-7}+3a^{-4}+3a^{-6}+a^{-8}-2a^{-3}z^{-1}-3a^{-5}z^{-1}-a^{-7}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

18

1

-1

16

2

14

5

1

-4

12

6

2

4

10

8

5

-3

8

9

6

3

6

7

9

2

4

7

8

-1

2

4

8

4

0

2

6

-4

-2

1

4

3

-4

2

-2

-6

1

Integral Khovanov Homology

(db, data source)

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