L11n369

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,14,8,15 X_20,16,21,15 X_18,11,19,12 X_12,17,13,18 X_22,20,17,19 X_16,22,5,21 X_13,8,14,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {6, -5, 7, -4, 8, -7}, {10, -1, -3, 9, 11, -2, 5, -6, -9, 3, 4, -8}

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

Polynomial invariants

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

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

5

1

-1

3

1

4

1

-3

-1

6

3

-3

6

0

-5

6

4

2

-7

3

6

3

-9

4

6

-2

-11

1

5

4

-13

2

-2

-15

1

Integral Khovanov Homology

(db, data source)

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