L11n61

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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_18,11,19,12 X_19,5,20,22 X_15,21,16,20 X_21,17,22,16 X_12,17,13,18 X_8,14,9,13 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 4, -8, 9, -3, -6, 7, 8, -4, -5, 6, -7, 5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)^{5}+t(1)t(2)^{4}-3t(2)^{4}-2t(1)t(2)^{3}+2t(2)^{3}+2t(1)t(2)^{2}-2t(2)^{2}-3t(1)t(2)+t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-2q^{5/2}+4q^{3/2}-5{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {5}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+2a^{5}z^{-1}-3a^{3}z^{3}-8a^{3}z-4a^{3}z^{-1}+2az^{5}+8az^{3}-2z^{3}a^{-1}+9az+3az^{-1}-4za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{5}z^{7}-5a^{5}z^{5}+9a^{5}z^{3}-7a^{5}z+2a^{5}z^{-1}+2a^{4}z^{8}-8a^{4}z^{6}+7a^{4}z^{4}+a^{4}z^{2}-2a^{4}+a^{3}z^{9}+a^{3}z^{7}-19a^{3}z^{5}+29a^{3}z^{3}-16a^{3}z+za^{-3}+4a^{3}z^{-1}+5a^{2}z^{8}-19a^{2}z^{6}+15a^{2}z^{4}+z^{4}a^{-2}+a^{2}z^{2}+z^{2}a^{-2}-3a^{2}-a^{-2}+az^{9}+2az^{7}+2z^{7}a^{-1}-21az^{5}-7z^{5}a^{-1}+28az^{3}+8z^{3}a^{-1}-13az-3za^{-1}+3az^{-1}+a^{-1}z^{-1}+3z^{8}-11z^{6}+9z^{4}+z^{2}-3$ (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

χ

6

2

4

2

-2

2

3

2

1

0

4

3

-1

-2

2

3

1

0

-4

3

4

1

-6

2

0

-8

1

4

3

-10

1

-1

-12

1

Integral Khovanov Homology

(db, data source)

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