L10a133

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Link Presentations

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

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

Polynomial invariants

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

2

-2

17

3

1

2

15

5

3

-2

13

4

2

11

4

5

1

9

4

0

7

2

5

3

5

2

3

-1

3

1

4

3

1

0

-1

1

Integral Khovanov Homology

(db, data source)

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