L10a152

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(3)-1)\left(-t(2)t(3)^{3}+t(1)t(2)^{2}t(3)^{2}-2t(2)^{2}t(3)^{2}-2t(1)t(2)t(3)^{2}+2t(2)t(3)^{2}-t(3)^{2}-t(1)t(2)^{2}t(3)-2t(1)t(3)+2t(1)t(2)t(3)-2t(2)t(3)+t(3)-t(1)t(2)\right)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$q^{7}-3q^{6}+7q^{5}-9q^{4}+12q^{3}+q^{-3}-12q^{2}-2q^{-2}+11q+6q^{-1}-8$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{2}a^{-6}+a^{-6}-2z^{4}a^{-4}-3z^{2}a^{-4}+a^{-4}z^{-2}+a^{-4}+z^{6}a^{-2}+2z^{4}a^{-2}+a^{2}z^{2}-z^{2}a^{-2}-2a^{-2}z^{-2}+2a^{2}-4a^{-2}-2z^{4}-4z^{2}+z^{-2}$ (db)
Kauffman polynomial	$z^{4}a^{-8}-z^{2}a^{-8}+3z^{5}a^{-7}-2z^{3}a^{-7}+6z^{6}a^{-6}-9z^{4}a^{-6}+8z^{2}a^{-6}-3a^{-6}+6z^{7}a^{-5}-6z^{5}a^{-5}+z^{3}a^{-5}+za^{-5}+4z^{8}a^{-4}-7z^{4}a^{-4}+3z^{2}a^{-4}-a^{-4}z^{-2}+2a^{-4}+z^{9}a^{-3}+10z^{7}a^{-3}-27z^{5}a^{-3}+24z^{3}a^{-3}-10za^{-3}+2a^{-3}z^{-1}+7z^{8}a^{-2}+a^{2}z^{6}-15z^{6}a^{-2}-4a^{2}z^{4}+15z^{4}a^{-2}+5a^{2}z^{2}-18z^{2}a^{-2}-2a^{-2}z^{-2}-2a^{2}+10a^{-2}+z^{9}a^{-1}+2az^{7}+6z^{7}a^{-1}-5az^{5}-23z^{5}a^{-1}+2az^{3}+23z^{3}a^{-1}+az-10za^{-1}+2a^{-1}z^{-1}+3z^{8}-8z^{6}+8z^{4}-7z^{2}-z^{-2}+4$ (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

3

1

-2

11

4

9

5

3

-2

7

4

3

5

6

0

3

5

6

-1

1

4

7

3

-1

2

4

-2

-3

4

-5

1

2

-1

-7

1

Integral Khovanov Homology

(db, data source)

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