L10a172

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(1)t(4)^{2}t(3)^{2}+t(1)t(2)t(4)^{2}t(3)^{2}-t(2)t(4)^{2}t(3)^{2}+t(4)^{2}t(3)^{2}-t(2)t(3)^{2}-t(1)t(2)t(4)t(3)^{2}+2t(2)t(4)t(3)^{2}-t(4)t(3)^{2}+2t(1)t(4)^{2}t(3)-t(1)t(2)t(4)^{2}t(3)-t(4)^{2}t(3)-t(1)t(2)t(3)+2t(2)t(3)-t(1)t(4)t(3)+2t(1)t(2)t(4)t(3)-t(2)t(4)t(3)+2t(4)t(3)-t(3)-t(1)t(4)^{2}-t(1)+t(1)t(2)-t(2)+2t(1)t(4)-t(1)t(2)t(4)-t(4)+1}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$8q^{9/2}-11q^{7/2}+9q^{5/2}-{\frac {1}{q^{5/2}}}-11q^{3/2}+{\frac {2}{q^{3/2}}}-q^{15/2}+3q^{13/2}-6q^{11/2}+6{\sqrt {q}}-{\frac {6}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-5}-3z^{3}a^{-5}-a^{-5}z^{-3}-3za^{-5}-2a^{-5}z^{-1}+z^{7}a^{-3}+5z^{5}a^{-3}+10z^{3}a^{-3}+3a^{-3}z^{-3}+11za^{-3}+7a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-8z^{3}a^{-1}+az^{-3}-3a^{-1}z^{-3}+3az-11za^{-1}+3az^{-1}-8a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-9}+3z^{4}a^{-8}+6z^{5}a^{-7}-5z^{3}a^{-7}+2za^{-7}+8z^{6}a^{-6}-10z^{4}a^{-6}+2z^{2}a^{-6}+9z^{7}a^{-5}-20z^{5}a^{-5}+17z^{3}a^{-5}-a^{-5}z^{-3}-12za^{-5}+5a^{-5}z^{-1}+5z^{8}a^{-4}-4z^{6}a^{-4}-17z^{4}a^{-4}+20z^{2}a^{-4}+3a^{-4}z^{-2}-10a^{-4}+z^{9}a^{-3}+12z^{7}a^{-3}-51z^{5}a^{-3}+61z^{3}a^{-3}-3a^{-3}z^{-3}-35za^{-3}+12a^{-3}z^{-1}+7z^{8}a^{-2}-19z^{6}a^{-2}+26z^{2}a^{-2}+6a^{-2}z^{-2}-19a^{-2}+z^{9}a^{-1}+az^{7}+4z^{7}a^{-1}-5az^{5}-30z^{5}a^{-1}+10az^{3}+48z^{3}a^{-1}-az^{-3}-3a^{-1}z^{-3}-10az-31za^{-1}+5az^{-1}+12a^{-1}z^{-1}+2z^{8}-7z^{6}+4z^{4}+8z^{2}+3z^{-2}-10$ (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

5

3

-2

8

6

3

6

5

7

2

4

6

4

2

4

9

5

0

2

0

-2

1

5

4

-4

1

-1

-6

1

Integral Khovanov Homology

(db, data source)

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