L11n370

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_14,8,15,7 X_15,20,16,21 X_11,19,12,18 X_17,13,18,12 X_19,22,20,17 X_21,16,22,5 X_8,14,9,13 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	$-q^{-6}+5q^{-5}-7q^{-4}+q^{3}+11q^{-3}-3q^{2}-10q^{-2}+6q+11q^{-1}-9$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}-a^{4}z^{4}-a^{4}z^{2}-2a^{4}z^{-2}-2a^{4}+a^{2}z^{6}+3a^{2}z^{4}+4a^{2}z^{2}+a^{2}z^{-2}+z^{2}a^{-2}+3a^{2}+a^{-2}-2z^{4}-4z^{2}-2$ (db)
Kauffman polynomial	$a^{7}z^{3}+5a^{6}z^{4}-a^{6}z^{2}+a^{6}z^{-2}-2a^{6}+2a^{5}z^{7}+2a^{5}z^{3}+2a^{5}z-2a^{5}z^{-1}+3a^{4}z^{8}-4a^{4}z^{6}+6a^{4}z^{4}-3a^{4}z^{2}+2a^{4}z^{-2}-2a^{4}+a^{3}z^{9}+6a^{3}z^{7}-16a^{3}z^{5}+12a^{3}z^{3}-2a^{3}z-2a^{3}z^{-1}+6a^{2}z^{8}-9a^{2}z^{6}+z^{6}a^{-2}-3a^{2}z^{4}-3z^{4}a^{-2}+3a^{2}z^{2}+3z^{2}a^{-2}+a^{2}z^{-2}-a^{2}-a^{-2}+az^{9}+7az^{7}+3z^{7}a^{-1}-25az^{5}-9z^{5}a^{-1}+19az^{3}+8z^{3}a^{-1}-6az-2za^{-1}+3z^{8}-4z^{6}-7z^{4}+8z^{2}-1$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

2

-2

3

4

1

3

1

5

2

-3

-1

6

4

2

-3

6

7

1

-5

5

4

1

-7

2

6

4

-9

3

5

-2

-11

4

-13

1

-1

Integral Khovanov Homology

(db, data source)

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