L10a124

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uvw^{3}-3uvw^{2}+3uvw-uv-uw^{3}+4uw^{2}-4uw+2u-2vw^{3}+4vw^{2}-4vw+v+w^{3}-3w^{2}+3w-1}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{2}+4q-7+11q^{-1}-11q^{-2}+14q^{-3}-11q^{-4}+9q^{-5}-5q^{-6}+2q^{-7}-q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{8}z^{-2}-a^{8}+3a^{6}z^{2}+4a^{6}z^{-2}+6a^{6}-3a^{4}z^{4}-8a^{4}z^{2}-5a^{4}z^{-2}-10a^{4}+a^{2}z^{6}+3a^{2}z^{4}+5a^{2}z^{2}+2a^{2}z^{-2}+5a^{2}-z^{4}-z^{2}$ (db)
Kauffman polynomial	$z^{5}a^{9}-3z^{3}a^{9}+3za^{9}-a^{9}z^{-1}+2z^{6}a^{8}-4z^{4}a^{8}+3z^{2}a^{8}+a^{8}z^{-2}-2a^{8}+2z^{7}a^{7}+2z^{5}a^{7}-12z^{3}a^{7}+13za^{7}-5a^{7}z^{-1}+2z^{8}a^{6}+3z^{6}a^{6}-11z^{4}a^{6}+14z^{2}a^{6}+4a^{6}z^{-2}-10a^{6}+z^{9}a^{5}+5z^{7}a^{5}-4z^{5}a^{5}-12z^{3}a^{5}+21za^{5}-9a^{5}z^{-1}+6z^{8}a^{4}-3z^{6}a^{4}-12z^{4}a^{4}+19z^{2}a^{4}+5a^{4}z^{-2}-14a^{4}+z^{9}a^{3}+9z^{7}a^{3}-16z^{5}a^{3}+11za^{3}-5a^{3}z^{-1}+4z^{8}a^{2}-12z^{4}a^{2}+11z^{2}a^{2}+2a^{2}z^{-2}-7a^{2}+6z^{7}a-10z^{5}a+2z^{3}a+4z^{6}-7z^{4}+3z^{2}+z^{5}a^{-1}-z^{3}a^{-1}$ (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

0

-5

7

4

3

-7

4

7

3

-9

5

7

-2

-11

1

5

4

-13

1

4

-3

-15

1

-17

1

-1

Integral Khovanov Homology

(db, data source)

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