L10a167

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(1)t(4)^{2}t(3)^{2}+t(1)t(2)t(4)^{2}t(3)^{2}-2t(2)t(4)^{2}t(3)^{2}+t(4)^{2}t(3)^{2}+t(1)t(4)t(3)^{2}+t(2)t(4)t(3)^{2}-t(4)t(3)^{2}+t(1)t(4)^{2}t(3)-t(1)t(2)t(4)^{2}t(3)+t(2)t(4)^{2}t(3)+t(1)t(3)+t(2)t(3)-2t(1)t(4)t(3)+t(1)t(2)t(4)t(3)-2t(2)t(4)t(3)+t(4)t(3)-t(3)-2t(1)+t(1)t(2)-t(2)+t(1)t(4)-t(1)t(2)t(4)+t(2)t(4)+1}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$7q^{9/2}-10q^{7/2}+8q^{5/2}-{\frac {1}{q^{5/2}}}-9q^{3/2}+{\frac {1}{q^{3/2}}}-q^{15/2}+3q^{13/2}-6q^{11/2}+5{\sqrt {q}}-{\frac {5}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}-2z^{5}a^{-1}+5z^{5}a^{-3}-z^{5}a^{-5}+az^{3}-9z^{3}a^{-1}+10z^{3}a^{-3}-3z^{3}a^{-5}+4az-14za^{-1}+13za^{-3}-3za^{-5}+4az^{-1}-11a^{-1}z^{-1}+10a^{-3}z^{-1}-3a^{-5}z^{-1}+az^{-3}-3a^{-1}z^{-3}+3a^{-3}z^{-3}-a^{-5}z^{-3}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-5z^{8}a^{-2}-4z^{8}a^{-4}-z^{8}-az^{7}-6z^{7}a^{-3}-7z^{7}a^{-5}+13z^{6}a^{-2}+4z^{6}a^{-4}-7z^{6}a^{-6}+2z^{6}+6az^{5}+14z^{5}a^{-1}+27z^{5}a^{-3}+13z^{5}a^{-5}-6z^{5}a^{-7}+6z^{4}a^{-2}+11z^{4}a^{-4}+8z^{4}a^{-6}-3z^{4}a^{-8}+6z^{4}-13az^{3}-30z^{3}a^{-1}-30z^{3}a^{-3}-6z^{3}a^{-5}+6z^{3}a^{-7}-z^{3}a^{-9}-33z^{2}a^{-2}-16z^{2}a^{-4}-17z^{2}+13az+28za^{-1}+21za^{-3}+3za^{-5}-3za^{-7}+24a^{-2}+11a^{-4}-a^{-6}+13-6az^{-1}-14a^{-1}z^{-1}-12a^{-3}z^{-1}-3a^{-5}z^{-1}+a^{-7}z^{-1}-6a^{-2}z^{-2}-3a^{-4}z^{-2}-3z^{-2}+az^{-3}+3a^{-1}z^{-3}+3a^{-3}z^{-3}+a^{-5}z^{-3}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

16

1

14

3

1

-2

12

3

10

4

3

-1

8

6

3

6

5

7

2

4

3

1

2

4

8

4

0

1

0

-2

4

-4

1

0

-6

1

Integral Khovanov Homology

(db, data source)

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