L10a162

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_16,6,17,5 X_10,20,11,19 X_18,10,19,9 X_20,12,13,11 X_2,13,3,14 X_14,3,15,4 X₄₇₅₈ X_12,16,7,15 X_6,18,1,17
Gauss code	{1, -6, 7, -8, 2, -10}, {8, -1, 4, -3, 5, -9}, {6, -7, 9, -2, 10, -4, 3, -5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(2)-1)(t(3)t(1)-t(1)+1)(t(1)t(3)-t(3)+1)(t(2)t(3)+1)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{7}-3q^{6}+6q^{5}-9q^{4}+12q^{3}-11q^{2}+12q-8+6q^{-1}-3q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-4}+4z^{4}a^{-4}+5z^{2}a^{-4}+a^{-4}z^{-2}+2a^{-4}-z^{8}a^{-2}-6z^{6}a^{-2}-13z^{4}a^{-2}-12z^{2}a^{-2}-2a^{-2}z^{-2}-5a^{-2}+z^{6}+4z^{4}+5z^{2}+z^{-2}+3$ (db)
Kauffman polynomial	$z^{4}a^{-8}-z^{2}a^{-8}+3z^{5}a^{-7}-3z^{3}a^{-7}+5z^{6}a^{-6}-6z^{4}a^{-6}+3z^{2}a^{-6}-a^{-6}+6z^{7}a^{-5}-9z^{5}a^{-5}+6z^{3}a^{-5}+5z^{8}a^{-4}-8z^{6}a^{-4}+7z^{4}a^{-4}-3z^{2}a^{-4}-a^{-4}z^{-2}+2a^{-4}+2z^{9}a^{-3}+3z^{7}a^{-3}-13z^{5}a^{-3}+12z^{3}a^{-3}-5za^{-3}+2a^{-3}z^{-1}+9z^{8}a^{-2}+a^{2}z^{6}-26z^{6}a^{-2}-3a^{2}z^{4}+29z^{4}a^{-2}+2a^{2}z^{2}-18z^{2}a^{-2}-2a^{-2}z^{-2}+6a^{-2}+2z^{9}a^{-1}+3az^{7}-9az^{5}-10z^{5}a^{-1}+6az^{3}+9z^{3}a^{-1}-5za^{-1}+2a^{-1}z^{-1}+4z^{8}-12z^{6}+12z^{4}-9z^{2}-z^{-2}+4$ (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

15

1

13

2

-2

11

4

1

3

9

6

3

-3

7

6

3

5

6

7

1

3

6

5

1

3

7

4

-1

3

5

-2

-3

1

4

3

-5

2

-2

-7

1

Integral Khovanov Homology

(db, data source)

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