L11n300

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_18,13,19,14 X_17,11,18,22 X_7,17,8,16 X_21,8,22,9 X_14,10,15,9 X_20,16,21,15 X_10,19,5,20 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -5, 6, 7, -9}, {11, -2, 3, -7, 8, 5, -4, -3, 9, -8, -6, 4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)+t(3)-1)(t(3)t(2)-t(2)-t(3))}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{3}-3q^{2}+4q-5+7q^{-1}-5q^{-2}+6q^{-3}-3q^{-4}+2q^{-5}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{4}z^{2}-2a^{4}z^{-2}-2a^{4}-a^{2}z^{4}+a^{2}z^{-2}+z^{2}a^{-2}+2a^{2}-z^{4}-z^{2}$ (db)
Kauffman polynomial	$a^{3}z^{9}+az^{9}+a^{4}z^{8}+4a^{2}z^{8}+3z^{8}-3a^{3}z^{7}+3z^{7}a^{-1}-4a^{4}z^{6}-16a^{2}z^{6}+z^{6}a^{-2}-11z^{6}+a^{5}z^{5}+3a^{3}z^{5}-9az^{5}-11z^{5}a^{-1}+8a^{4}z^{4}+21a^{2}z^{4}-3z^{4}a^{-2}+10z^{4}+8az^{3}+8z^{3}a^{-1}+2a^{6}z^{2}-2a^{4}z^{2}-9a^{2}z^{2}+z^{2}a^{-2}-4z^{2}+2a^{5}z+2a^{3}z-2a^{6}-3a^{4}-2a^{2}-2a^{5}z^{-1}-2a^{3}z^{-1}+a^{6}z^{-2}+2a^{4}z^{-2}+a^{2}z^{-2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

2

-2

3

2

1

3

2

-1

4

2

-3

2

4

2

-5

4

3

1

-7

1

4

3

-9

1

2

-1

-11

2

Integral Khovanov Homology

(db, data source)

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