L11a141

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(2)^{6}-t(1)t(2)^{6}-3t(1)^{2}t(2)^{5}+4t(1)t(2)^{5}-t(2)^{5}+4t(1)^{2}t(2)^{4}-8t(1)t(2)^{4}+3t(2)^{4}-4t(1)^{2}t(2)^{3}+9t(1)t(2)^{3}-4t(2)^{3}+3t(1)^{2}t(2)^{2}-8t(1)t(2)^{2}+4t(2)^{2}-t(1)^{2}t(2)+4t(1)t(2)-3t(2)-t(1)+1}{t(1)t(2)^{3}}}$ (db)
Jones polynomial	$-q^{13/2}+4q^{11/2}-8q^{9/2}+14q^{7/2}-19q^{5/2}+21q^{3/2}-22{\sqrt {q}}+{\frac {18}{\sqrt {q}}}-{\frac {14}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{9}a^{-1}-az^{7}+6z^{7}a^{-1}-z^{7}a^{-3}-4az^{5}+12z^{5}a^{-1}-4z^{5}a^{-3}-4az^{3}+7z^{3}a^{-1}-4z^{3}a^{-3}+az-3za^{-1}+za^{-3}+2az^{-1}-3a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-3z^{10}a^{-2}-3z^{10}-7az^{9}-14z^{9}a^{-1}-7z^{9}a^{-3}-7a^{2}z^{8}-5z^{8}a^{-2}-8z^{8}a^{-4}-4z^{8}-4a^{3}z^{7}+16az^{7}+36z^{7}a^{-1}+9z^{7}a^{-3}-7z^{7}a^{-5}-a^{4}z^{6}+18a^{2}z^{6}+16z^{6}a^{-2}+9z^{6}a^{-4}-4z^{6}a^{-6}+22z^{6}+10a^{3}z^{5}-11az^{5}-38z^{5}a^{-1}-6z^{5}a^{-3}+10z^{5}a^{-5}-z^{5}a^{-7}+2a^{4}z^{4}-12a^{2}z^{4}-10z^{4}a^{-2}+2z^{4}a^{-4}+6z^{4}a^{-6}-20z^{4}-5a^{3}z^{3}+2az^{3}+14z^{3}a^{-1}+3z^{3}a^{-3}-3z^{3}a^{-5}+z^{3}a^{-7}+2a^{2}z^{2}-3z^{2}a^{-2}-5z^{2}a^{-4}-2z^{2}a^{-6}+2z^{2}+3az+4za^{-1}+za^{-3}+3a^{-2}+a^{-4}+3-2az^{-1}-3a^{-1}z^{-1}-a^{-3}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

3

-3

10

5

1

4

8

9

3

-6

6

10

5

4

11

9

-2

2

11

10

1

0

8

12

4

-2

6

10

-4

3

9

6

-6

1

5

-4

-8

3

-10

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$