L10a173

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)t(4)^{2}t(3)^{2}+t(1)t(3)^{2}-t(1)t(2)t(3)^{2}+t(2)t(3)^{2}-t(1)t(4)t(3)^{2}+t(1)t(2)t(4)t(3)^{2}-2t(2)t(4)t(3)^{2}+2t(4)t(3)^{2}-t(3)^{2}-t(1)t(4)^{2}t(3)+t(1)t(2)t(4)^{2}t(3)-2t(2)t(4)^{2}t(3)+2t(4)^{2}t(3)-2t(1)t(3)+2t(1)t(2)t(3)-t(2)t(3)+3t(1)t(4)t(3)-4t(1)t(2)t(4)t(3)+3t(2)t(4)t(3)-4t(4)t(3)+t(3)+t(1)t(4)^{2}-t(1)t(2)t(4)^{2}+t(2)t(4)^{2}-t(4)^{2}+t(1)-2t(1)t(4)+2t(1)t(2)t(4)-t(2)t(4)+t(4)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	${\frac {15}{q^{9/2}}}-{\frac {17}{q^{7/2}}}+{\frac {11}{q^{5/2}}}-{\frac {9}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {8}{q^{15/2}}}+{\frac {11}{q^{13/2}}}-{\frac {16}{q^{11/2}}}-{\sqrt {q}}+{\frac {4}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z^{-3}+a^{9}z+a^{9}z^{-1}-3a^{7}z^{3}-3a^{7}z^{-3}-5a^{7}z-6a^{7}z^{-1}+2a^{5}z^{5}+5a^{5}z^{3}+3a^{5}z^{-3}+9a^{5}z+9a^{5}z^{-1}+a^{3}z^{5}-a^{3}z^{3}-a^{3}z^{-3}-5a^{3}z-4a^{3}z^{-1}-az^{3}$ (db)
Kauffman polynomial	$a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{6}-4a^{10}z^{4}+a^{10}z^{2}+6a^{9}z^{7}-12a^{9}z^{5}+14a^{9}z^{3}-a^{9}z^{-3}-12a^{9}z+5a^{9}z^{-1}+5a^{8}z^{8}-a^{8}z^{6}-13a^{8}z^{4}+17a^{8}z^{2}+3a^{8}z^{-2}-10a^{8}+2a^{7}z^{9}+12a^{7}z^{7}-36a^{7}z^{5}+42a^{7}z^{3}-3a^{7}z^{-3}-29a^{7}z+12a^{7}z^{-1}+12a^{6}z^{8}-17a^{6}z^{6}-4a^{6}z^{4}+23a^{6}z^{2}+6a^{6}z^{-2}-19a^{6}+2a^{5}z^{9}+14a^{5}z^{7}-40a^{5}z^{5}+39a^{5}z^{3}-3a^{5}z^{-3}-23a^{5}z+12a^{5}z^{-1}+7a^{4}z^{8}-9a^{4}z^{6}+7a^{4}z^{2}+3a^{4}z^{-2}-10a^{4}+8a^{3}z^{7}-16a^{3}z^{5}+12a^{3}z^{3}-a^{3}z^{-3}-7a^{3}z+5a^{3}z^{-1}+4a^{2}z^{6}-5a^{2}z^{4}+az^{5}-az^{3}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

3

-3

-2

6

1

5

-4

6

4

-2

-6

11

5

6

-8

8

10

2

-10

8

7

1

-12

5

10

5

-14

3

6

-3

-16

5

-18

1

3

-2

-20

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-8$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{3}$
$r=-6$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-5$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{8}$
$r=-3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{11}$
$r=-1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=0$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$