L10a131

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}-t(1)t(3)t(2)^{2}+3t(3)t(2)^{2}-t(2)^{2}-3t(1)t(3)^{2}t(2)+t(3)^{2}t(2)-t(1)t(2)+4t(1)t(3)t(2)-4t(3)t(2)+3t(2)+t(1)t(3)^{2}+2t(1)-3t(1)t(3)+t(3)-2}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{10}-3q^{9}+6q^{8}-9q^{7}+11q^{6}-11q^{5}+11q^{4}-7q^{3}+6q^{2}-2q+1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-4}-z^{6}a^{-6}+z^{4}a^{-2}-3z^{4}a^{-4}-3z^{4}a^{-6}+z^{4}a^{-8}+3z^{2}a^{-2}-3z^{2}a^{-4}-3z^{2}a^{-6}+2z^{2}a^{-8}+3a^{-2}-3a^{-4}-a^{-6}+a^{-8}+a^{-2}z^{-2}-2a^{-4}z^{-2}+a^{-6}z^{-2}$ (db)
Kauffman polynomial	$z^{9}a^{-5}+z^{9}a^{-7}+2z^{8}a^{-4}+6z^{8}a^{-6}+4z^{8}a^{-8}+2z^{7}a^{-3}+4z^{7}a^{-5}+8z^{7}a^{-7}+6z^{7}a^{-9}+z^{6}a^{-2}-z^{6}a^{-4}-10z^{6}a^{-6}-3z^{6}a^{-8}+5z^{6}a^{-10}-5z^{5}a^{-3}-14z^{5}a^{-5}-22z^{5}a^{-7}-10z^{5}a^{-9}+3z^{5}a^{-11}-4z^{4}a^{-2}-10z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}a^{-8}-6z^{4}a^{-10}+z^{4}a^{-12}+z^{3}a^{-3}+8z^{3}a^{-5}+19z^{3}a^{-7}+9z^{3}a^{-9}-3z^{3}a^{-11}+6z^{2}a^{-2}+12z^{2}a^{-4}+6z^{2}a^{-6}+4z^{2}a^{-8}+3z^{2}a^{-10}-z^{2}a^{-12}+4za^{-3}-6za^{-7}-2za^{-9}-4a^{-2}-6a^{-4}-3a^{-6}-a^{-8}-a^{-10}-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-2}z^{-2}+2a^{-4}z^{-2}+a^{-6}z^{-2}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

χ

21

1

19

2

-2

17

4

1

3

15

6

3

-3

13

5

3

2

11

6

0

9

5

0

7

3

7

4

5

3

4

-1

3

1

5

4

1

-1

1

Integral Khovanov Homology

(db, data source)

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