L10a132

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}-2t(1)t(3)t(2)^{2}+2t(3)t(2)^{2}-2t(1)t(3)^{2}t(2)+2t(3)^{2}t(2)-2t(1)t(2)+4t(1)t(3)t(2)-4t(3)t(2)+2t(2)+2t(1)-2t(1)t(3)+2t(3)-1}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{-10}-2q^{-9}+5q^{-8}-7q^{-7}+10q^{-6}-9q^{-5}+10q^{-4}-7q^{-3}+5q^{-2}-3q^{-1}+1$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{10}z^{-2}+a^{10}-3a^{8}z^{2}-2a^{8}z^{-2}-6a^{8}+3a^{6}z^{4}+8a^{6}z^{2}+a^{6}z^{-2}+5a^{6}-a^{4}z^{6}-3a^{4}z^{4}-2a^{4}z^{2}+a^{2}z^{4}+2a^{2}z^{2}$ (db)
Kauffman polynomial	$z^{4}a^{12}-2z^{2}a^{12}+a^{12}+2z^{5}a^{11}-2z^{3}a^{11}+3z^{6}a^{10}-3z^{4}a^{10}+3z^{2}a^{10}+a^{10}z^{-2}-3a^{10}+3z^{7}a^{9}-z^{5}a^{9}-3z^{3}a^{9}+6za^{9}-2a^{9}z^{-1}+2z^{8}a^{8}+3z^{6}a^{8}-12z^{4}a^{8}+15z^{2}a^{8}+2a^{8}z^{-2}-8a^{8}+z^{9}a^{7}+3z^{7}a^{7}-5z^{5}a^{7}-3z^{3}a^{7}+6za^{7}-2a^{7}z^{-1}+5z^{8}a^{6}-9z^{6}a^{6}+6z^{2}a^{6}+a^{6}z^{-2}-5a^{6}+z^{9}a^{5}+3z^{7}a^{5}-12z^{5}a^{5}+6z^{3}a^{5}+3z^{8}a^{4}-8z^{6}a^{4}+5z^{4}a^{4}-2z^{2}a^{4}+3z^{7}a^{3}-10z^{5}a^{3}+8z^{3}a^{3}+z^{6}a^{2}-3z^{4}a^{2}+2z^{2}a^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

1

-1

2

-2

-3

3

1

2

-5

5

3

-2

-7

5

2

3

-9

4

5

1

-11

6

5

1

-13

2

5

3

-15

3

5

-2

-17

1

4

3

-19

1

-1

-21

1

Integral Khovanov Homology

(db, data source)

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