L11n399

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_22,16,19,15 X_7,20,8,21 X_19,8,20,9 X_13,18,14,5 X_11,14,12,15 X_17,12,18,13 X_16,22,17,21 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, 9, -3}, {10, -1, -4, 5, 11, -2, -7, 8, -6, 7, 3, -9, -8, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(2)-1)(t(3)-1)\left(t(1)t(3)^{2}-1\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$1-2q^{-1}+4q^{-2}-3q^{-3}+4q^{-4}-2q^{-5}+3q^{-6}-q^{-8}+q^{-9}-q^{-10}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10}z^{-2}-a^{10}+2a^{8}z^{2}+4a^{8}z^{-2}+5a^{8}-2a^{6}z^{2}-5a^{6}z^{-2}-7a^{6}-a^{4}z^{6}-4a^{4}z^{4}-3a^{4}z^{2}+2a^{4}z^{-2}+a^{4}+a^{2}z^{4}+3a^{2}z^{2}+2a^{2}$ (db)
Kauffman polynomial	$a^{11}z^{7}-6a^{11}z^{5}+10a^{11}z^{3}-6a^{11}z+a^{11}z^{-1}+a^{10}z^{8}-6a^{10}z^{6}+9a^{10}z^{4}-6a^{10}z^{2}-a^{10}z^{-2}+4a^{10}+2a^{9}z^{7}-16a^{9}z^{5}+33a^{9}z^{3}-22a^{9}z+5a^{9}z^{-1}+a^{8}z^{8}-9a^{8}z^{6}+22a^{8}z^{4}-22a^{8}z^{2}-4a^{8}z^{-2}+15a^{8}+2a^{7}z^{7}-15a^{7}z^{5}+34a^{7}z^{3}-30a^{7}z+9a^{7}z^{-1}+a^{6}z^{8}-5a^{6}z^{6}+11a^{6}z^{4}-19a^{6}z^{2}-5a^{6}z^{-2}+16a^{6}+3a^{5}z^{7}-12a^{5}z^{5}+15a^{5}z^{3}-13a^{5}z+5a^{5}z^{-1}+a^{4}z^{8}-a^{4}z^{6}-6a^{4}z^{4}+2a^{4}z^{2}-2a^{4}z^{-2}+4a^{4}+2a^{3}z^{7}-7a^{3}z^{5}+4a^{3}z^{3}+a^{3}z+a^{2}z^{6}-4a^{2}z^{4}+5a^{2}z^{2}-2a^{2}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

1

-1

1

-1

-3

3

1

2

-5

2

3

1

-7

1

3

1

-9

1

2

-11

1

5

3

1

-13

1

2

3

-15

1

2

-1

-17

1

0

-19

0

-21

1

-1

Integral Khovanov Homology

(db, data source)

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