L10a129

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Link Presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}+t(1)t(2)^{2}-2t(1)t(3)t(2)^{2}+3t(3)t(2)^{2}-t(2)^{2}-2t(1)t(3)^{2}t(2)+3t(3)^{2}t(2)-3t(1)t(2)+6t(1)t(3)t(2)-6t(3)t(2)+2t(2)+t(1)t(3)^{2}-t(3)^{2}+2t(1)-3t(1)t(3)+2t(3)-1}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{2}+4q-7+11q^{-1}-13q^{-2}+15q^{-3}-12q^{-4}+11q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{8}+3z^{2}a^{6}+a^{6}z^{-2}+3a^{6}-3z^{4}a^{4}-6z^{2}a^{4}-2a^{4}z^{-2}-5a^{4}+z^{6}a^{2}+3z^{4}a^{2}+5z^{2}a^{2}+a^{2}z^{-2}+3a^{2}-z^{4}-z^{2}$ (db)
Kauffman polynomial	$z^{5}a^{9}-2z^{3}a^{9}+za^{9}+3z^{6}a^{8}-6z^{4}a^{8}+5z^{2}a^{8}-2a^{8}+4z^{7}a^{7}-4z^{5}a^{7}-2z^{3}a^{7}+3za^{7}+3z^{8}a^{6}+4z^{6}a^{6}-17z^{4}a^{6}+17z^{2}a^{6}+a^{6}z^{-2}-8a^{6}+z^{9}a^{5}+10z^{7}a^{5}-18z^{5}a^{5}+5z^{3}a^{5}+5za^{5}-2a^{5}z^{-1}+7z^{8}a^{4}-z^{6}a^{4}-19z^{4}a^{4}+20z^{2}a^{4}+2a^{4}z^{-2}-9a^{4}+z^{9}a^{3}+12z^{7}a^{3}-24z^{5}a^{3}+10z^{3}a^{3}+3za^{3}-2a^{3}z^{-1}+4z^{8}a^{2}+2z^{6}a^{2}-15z^{4}a^{2}+11z^{2}a^{2}+a^{2}z^{-2}-4a^{2}+6z^{7}a-10z^{5}a+4z^{3}a+4z^{6}-7z^{4}+3z^{2}+z^{5}a^{-1}-z^{3}a^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

5

1

-1

3

1

4

1

-3

-1

7

3

4

-3

7

5

-2

-5

8

6

2

-7

6

9

3

-9

5

6

-1

-11

2

7

5

-13

1

4

-3

-15

2

-17

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{8}$
$r=-1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=0$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$