L10a150

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(w-1)\left(2vw^{2}-2vw+v-w^{2}+2w-2\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$q^{10}-4q^{9}+8q^{8}-11q^{7}+13q^{6}-13q^{5}+13q^{4}-8q^{3}+6q^{2}-2q+1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-4}-z^{6}a^{-6}+z^{4}a^{-2}-3z^{4}a^{-4}-2z^{4}a^{-6}+z^{4}a^{-8}+3z^{2}a^{-2}-4z^{2}a^{-4}+z^{2}a^{-8}+3a^{-2}-4a^{-4}+a^{-6}+a^{-2}z^{-2}-2a^{-4}z^{-2}+a^{-6}z^{-2}$ (db)
Kauffman polynomial	$z^{4}a^{-12}+4z^{5}a^{-11}-2z^{3}a^{-11}+8z^{6}a^{-10}-9z^{4}a^{-10}+3z^{2}a^{-10}-a^{-10}+9z^{7}a^{-9}-12z^{5}a^{-9}+5z^{3}a^{-9}-za^{-9}+5z^{8}a^{-8}+2z^{6}a^{-8}-15z^{4}a^{-8}+11z^{2}a^{-8}-3a^{-8}+z^{9}a^{-7}+13z^{7}a^{-7}-29z^{5}a^{-7}+16z^{3}a^{-7}-3za^{-7}+7z^{8}a^{-6}-7z^{6}a^{-6}-13z^{4}a^{-6}+18z^{2}a^{-6}+a^{-6}z^{-2}-7a^{-6}+z^{9}a^{-5}+6z^{7}a^{-5}-18z^{5}a^{-5}+11z^{3}a^{-5}+za^{-5}-2a^{-5}z^{-1}+2z^{8}a^{-4}-12z^{4}a^{-4}+16z^{2}a^{-4}+2a^{-4}z^{-2}-8a^{-4}+2z^{7}a^{-3}-5z^{5}a^{-3}+2z^{3}a^{-3}+3za^{-3}-2a^{-3}z^{-1}+z^{6}a^{-2}-4z^{4}a^{-2}+6z^{2}a^{-2}+a^{-2}z^{-2}-4a^{-2}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

χ

21

1

19

3

-3

17

5

1

4

15

6

3

-3

13

7

5

2

11

8

0

9

5

0

7

3

8

5

3

5

-2

3

1

5

4

1

-1

1

Integral Khovanov Homology

(db, data source)

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