L10a138

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,6,15,5 X₈₄₉₃ X_2,16,3,15 X_16,7,17,8 X_18,10,19,9 X_20,12,13,11 X_12,14,5,13 X_4,17,1,18 X_10,20,11,19
Gauss code	{1, -4, 3, -9}, {2, -1, 5, -3, 6, -10, 7, -8}, {8, -2, 4, -5, 9, -6, 10, -7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)(t(3)-1)\left(t(2)^{2}t(3)^{2}+t(2)t(3)+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{9}+3q^{8}-4q^{7}+6q^{6}-7q^{5}+8q^{4}-6q^{3}+6q^{2}-3q+3-q^{-1}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-6}-4z^{4}a^{-6}-3z^{2}a^{-6}+a^{-6}z^{-2}+a^{-6}+z^{8}a^{-4}+6z^{6}a^{-4}+11z^{4}a^{-4}+5z^{2}a^{-4}-2a^{-4}z^{-2}-4a^{-4}-z^{6}a^{-2}-4z^{4}a^{-2}-2z^{2}a^{-2}+a^{-2}z^{-2}+3a^{-2}$ (db)
Kauffman polynomial	$z^{3}a^{-11}+3z^{4}a^{-10}-2z^{2}a^{-10}+4z^{5}a^{-9}-3z^{3}a^{-9}+4z^{6}a^{-8}-3z^{4}a^{-8}-2z^{2}a^{-8}+a^{-8}+4z^{7}a^{-7}-6z^{5}a^{-7}+4z^{8}a^{-6}-12z^{6}a^{-6}+12z^{4}a^{-6}-6z^{2}a^{-6}+a^{-6}z^{-2}-a^{-6}+2z^{9}a^{-5}-4z^{7}a^{-5}-3z^{5}a^{-5}+3z^{3}a^{-5}+4za^{-5}-2a^{-5}z^{-1}+7z^{8}a^{-4}-31z^{6}a^{-4}+39z^{4}a^{-4}-13z^{2}a^{-4}+2a^{-4}z^{-2}-4a^{-4}+2z^{9}a^{-3}-7z^{7}a^{-3}+3z^{5}a^{-3}+2z^{3}a^{-3}+4za^{-3}-2a^{-3}z^{-1}+3z^{8}a^{-2}-15z^{6}a^{-2}+21z^{4}a^{-2}-7z^{2}a^{-2}+a^{-2}z^{-2}-3a^{-2}+z^{7}a^{-1}-4z^{5}a^{-1}+3z^{3}a^{-1}$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

7

χ

19

1

-1

17

2

15

2

1

-1

13

4

2

11

5

4

-1

9

3

2

1

7

3

5

2

5

3

0

3

2

5

3

1

0

-1

2

-3

1

-1

Integral Khovanov Homology

(db, data source)

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