L11a395

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_16,7,17,8 X_8,15,5,16 X_20,11,21,12 X_22,19,9,20 X_18,14,19,13 X_14,18,15,17 X_12,21,13,22 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, 5, -9, 7, -8, 4, -3, 8, -7, 6, -5, 9, -6}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uvw^{3}-4uvw^{2}+4uvw-2uv-2uw^{3}+5uw^{2}-5uw+2u-2vw^{3}+5vw^{2}-5vw+2v+2w^{3}-4w^{2}+4w-1}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$1-4q^{-1}+8q^{-2}-11q^{-3}+15q^{-4}-15q^{-5}+17q^{-6}-12q^{-7}+9q^{-8}-5q^{-9}+2q^{-10}-q^{-11}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{12}z^{-2}+4a^{10}z^{-2}+4a^{10}-6z^{2}a^{8}-5a^{8}z^{-2}-11a^{8}+4z^{4}a^{6}+9z^{2}a^{6}+2a^{6}z^{-2}+7a^{6}-z^{6}a^{4}-2z^{4}a^{4}-z^{2}a^{4}+z^{4}a^{2}+z^{2}a^{2}$ (db)
Kauffman polynomial	$z^{5}a^{13}-3z^{3}a^{13}+3za^{13}-a^{13}z^{-1}+2z^{6}a^{12}-4z^{4}a^{12}+3z^{2}a^{12}+a^{12}z^{-2}-2a^{12}+2z^{7}a^{11}+2z^{5}a^{11}-12z^{3}a^{11}+13za^{11}-5a^{11}z^{-1}+2z^{8}a^{10}+4z^{6}a^{10}-13z^{4}a^{10}+14z^{2}a^{10}+4a^{10}z^{-2}-10a^{10}+2z^{9}a^{9}+z^{7}a^{9}+3z^{5}a^{9}-16z^{3}a^{9}+21za^{9}-9a^{9}z^{-1}+z^{10}a^{8}+4z^{8}a^{8}-4z^{6}a^{8}-8z^{4}a^{8}+20z^{2}a^{8}+5a^{8}z^{-2}-14a^{8}+6z^{9}a^{7}-9z^{7}a^{7}+4z^{5}a^{7}-6z^{3}a^{7}+11za^{7}-5a^{7}z^{-1}+z^{10}a^{6}+8z^{8}a^{6}-22z^{6}a^{6}+11z^{4}a^{6}+6z^{2}a^{6}+2a^{6}z^{-2}-7a^{6}+4z^{9}a^{5}-4z^{7}a^{5}-8z^{5}a^{5}+6z^{3}a^{5}+6z^{8}a^{4}-15z^{6}a^{4}+8z^{4}a^{4}-2z^{2}a^{4}+4z^{7}a^{3}-10z^{5}a^{3}+5z^{3}a^{3}+z^{6}a^{2}-2z^{4}a^{2}+z^{2}a^{2}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

1

-1

3

-3

5

1

4

-5

7

4

-3

-7

8

4

-9

7

0

-11

10

8

2

-13

5

10

5

-15

4

7

-3

-17

1

5

4

-19

1

4

-3

-21

1

-23

1

-1

Integral Khovanov Homology

(db, data source)

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