L11a485

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_12,6,13,5 X₈₄₉₃ X_22,11,19,12 X_18,22,5,21 X_20,10,21,9 X_10,17,11,18 X_16,19,17,20 X_2,14,3,13
Gauss code	{1, -11, 5, -3}, {10, -8, 7, -6}, {4, -1, 2, -5, 8, -9, 6, -4, 11, -2, 3, -10, 9, -7}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)(w-1)^{3}\left(w^{2}-w+1\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$q^{-6}-q^{5}-6q^{-5}+5q^{4}+13q^{-4}-11q^{3}-20q^{-3}+19q^{2}+28q^{-2}-26q-30q^{-1}+32$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{4}-2a^{4}-2a^{2}z^{6}-z^{6}a^{-2}-4a^{2}z^{4}-2z^{4}a^{-2}+2a^{2}z^{2}+a^{2}z^{-2}+a^{-2}z^{-2}+6a^{2}+2a^{-2}+z^{8}+4z^{6}+5z^{4}-2z^{2}-2z^{-2}-6$ (db)
Kauffman polynomial	$a^{6}z^{6}+6a^{5}z^{7}-8a^{5}z^{5}+z^{5}a^{-5}+2a^{5}z+13a^{4}z^{8}-26a^{4}z^{6}+5z^{6}a^{-4}+12a^{4}z^{4}-4z^{4}a^{-4}+2a^{4}z^{2}-4a^{4}+12a^{3}z^{9}-14a^{3}z^{7}+11z^{7}a^{-3}-8a^{3}z^{5}-13z^{5}a^{-3}+2a^{3}z^{3}+4z^{3}a^{-3}+6a^{3}z+4a^{2}z^{10}+24a^{2}z^{8}+15z^{8}a^{-2}-69a^{2}z^{6}-20z^{6}a^{-2}+38a^{2}z^{4}+8z^{4}a^{-2}+10a^{2}z^{2}+2z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-12a^{2}-4a^{-2}+24az^{9}+12z^{9}a^{-1}-36az^{7}-5z^{7}a^{-1}-3az^{5}-17z^{5}a^{-1}+10az^{3}+12z^{3}a^{-1}+6az+2za^{-1}-2az^{-1}-2a^{-1}z^{-1}+4z^{10}+26z^{8}-67z^{6}+38z^{4}+10z^{2}+2z^{-2}-11$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

4

7

1

-6

5

12

4

8

3

14

7

-7

1

18

12

6

-1

16

18

2

-3

12

14

-2

-5

8

16

8

-7

5

12

-7

-9

1

8

7

-11

5

-5

-13

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-2$	${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=-1$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{16}$	${\mathbb {Z} }^{16}$
$r=0$	${\mathbb {Z} }^{18}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{18}$
$r=1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$