L10a144

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(3)^{2}t(2)^{3}-t(3)^{2}t(2)^{3}+t(3)t(2)^{3}+t(1)t(3)^{3}t(2)^{2}-t(3)^{3}t(2)^{2}-2t(1)t(3)^{2}t(2)^{2}+2t(3)^{2}t(2)^{2}+t(1)t(3)t(2)^{2}-2t(3)t(2)^{2}+t(2)^{2}-t(1)t(3)^{3}t(2)+2t(1)t(3)^{2}t(2)-t(3)^{2}t(2)+t(1)t(2)-2t(1)t(3)t(2)+2t(3)t(2)-t(2)-t(1)t(3)^{2}+t(1)t(3)-t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{10}-2q^{9}+4q^{8}-6q^{7}+9q^{6}-8q^{5}+8q^{4}-6q^{3}+5q^{2}-2q+1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-4}-z^{6}a^{-6}+z^{4}a^{-2}-3z^{4}a^{-4}-4z^{4}a^{-6}+z^{4}a^{-8}+3z^{2}a^{-2}-z^{2}a^{-4}-6z^{2}a^{-6}+3z^{2}a^{-8}+2a^{-2}+a^{-4}-5a^{-6}+2a^{-8}+a^{-4}z^{-2}-2a^{-6}z^{-2}+a^{-8}z^{-2}$ (db)
Kauffman polynomial	$z^{9}a^{-5}+z^{9}a^{-7}+2z^{8}a^{-4}+5z^{8}a^{-6}+3z^{8}a^{-8}+2z^{7}a^{-3}+z^{7}a^{-5}+2z^{7}a^{-7}+3z^{7}a^{-9}+z^{6}a^{-2}-4z^{6}a^{-4}-16z^{6}a^{-6}-8z^{6}a^{-8}+3z^{6}a^{-10}-6z^{5}a^{-3}-9z^{5}a^{-5}-10z^{5}a^{-7}-5z^{5}a^{-9}+2z^{5}a^{-11}-4z^{4}a^{-2}-z^{4}a^{-4}+22z^{4}a^{-6}+12z^{4}a^{-8}-6z^{4}a^{-10}+z^{4}a^{-12}+3z^{3}a^{-3}+10z^{3}a^{-5}+14z^{3}a^{-7}+4z^{3}a^{-9}-3z^{3}a^{-11}+5z^{2}a^{-2}-z^{2}a^{-4}-19z^{2}a^{-6}-5z^{2}a^{-8}+6z^{2}a^{-10}-2z^{2}a^{-12}+za^{-3}-8za^{-5}-8za^{-7}+za^{-9}-2a^{-2}+3a^{-4}+9a^{-6}+3a^{-8}-2a^{-10}+2a^{-5}z^{-1}+2a^{-7}z^{-1}-a^{-4}z^{-2}-2a^{-6}z^{-2}-a^{-8}z^{-2}$ (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

1

-1

17

2

15

4

2

-2

13

5

2

3

11

4

5

1

9

4

0

7

2

4

2

5

3

4

-1

3

1

4

3

1

-1

1

Integral Khovanov Homology

(db, data source)

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