L11a172

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_16,8,17,7 X_22,16,7,15 X_6,21,1,22 X_18,11,19,12 X_14,6,15,5 X_20,13,21,14 X_12,19,13,20 X_4,18,5,17
Gauss code	{1, -2, 3, -11, 8, -6}, {4, -1, 2, -3, 7, -10, 9, -8, 5, -4, 11, -7, 10, -9, 6, -5}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(2)^{6}-t(1)t(2)^{6}-2t(1)^{2}t(2)^{5}+3t(1)t(2)^{5}-t(2)^{5}+3t(1)^{2}t(2)^{4}-5t(1)t(2)^{4}+2t(2)^{4}-3t(1)^{2}t(2)^{3}+5t(1)t(2)^{3}-3t(2)^{3}+2t(1)^{2}t(2)^{2}-5t(1)t(2)^{2}+3t(2)^{2}-t(1)^{2}t(2)+3t(1)t(2)-2t(2)-t(1)+1}{t(1)t(2)^{3}}}$ (db)
Jones polynomial	${\frac {13}{q^{9/2}}}-{\frac {15}{q^{7/2}}}-q^{5/2}+{\frac {14}{q^{5/2}}}+3q^{3/2}-{\frac {13}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {3}{q^{15/2}}}+{\frac {6}{q^{13/2}}}-{\frac {10}{q^{11/2}}}-6{\sqrt {q}}+{\frac {9}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{5}z^{7}-5a^{5}z^{5}-8a^{5}z^{3}-4a^{5}z+a^{3}z^{9}+7a^{3}z^{7}+18a^{3}z^{5}+20a^{3}z^{3}+8a^{3}z+a^{3}z^{-1}-az^{7}-5az^{5}-8az^{3}-5az-az^{-1}$ (db)
Kauffman polynomial	$a^{10}z^{4}-a^{10}z^{2}+3a^{9}z^{5}-3a^{9}z^{3}+5a^{8}z^{6}-5a^{8}z^{4}+a^{8}z^{2}+7a^{7}z^{7}-12a^{7}z^{5}+10a^{7}z^{3}-2a^{7}z+7a^{6}z^{8}-14a^{6}z^{6}+12a^{6}z^{4}-2a^{6}z^{2}+5a^{5}z^{9}-9a^{5}z^{7}+5a^{5}z^{5}-2a^{5}z^{3}+2a^{5}z+2a^{4}z^{10}+2a^{4}z^{8}-16a^{4}z^{6}+16a^{4}z^{4}-6a^{4}z^{2}+9a^{3}z^{9}-33a^{3}z^{7}+46a^{3}z^{5}-36a^{3}z^{3}+11a^{3}z-a^{3}z^{-1}+2a^{2}z^{10}-2a^{2}z^{8}-9a^{2}z^{6}+11a^{2}z^{4}-5a^{2}z^{2}+a^{2}+4az^{9}-16az^{7}+z^{7}a^{-1}+22az^{5}-4z^{5}a^{-1}-17az^{3}+4z^{3}a^{-1}+7az-az^{-1}+3z^{8}-12z^{6}+13z^{4}-3z^{2}$ (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

6

1

4

2

-2

2

4

1

3

0

5

2

-3

-2

8

4

-4

7

6

-1

-6

8

7

1

-8

6

8

2

-10

4

7

-3

-12

2

6

4

-14

1

4

-3

-16

2

-18

1

-1

Integral Khovanov Homology

(db, data source)

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