L11n104

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_7,16,8,17 X_22,18,5,17 X_18,14,19,13 X_21,14,22,15 X_9,20,10,21 X_15,8,16,9 X_19,10,20,11 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 8, -7, 9, 11, -2, 5, 6, -8, 3, 4, -5, -9, 7, -6, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{3}-2t(2)^{3}-6t(1)t(2)^{2}+6t(2)^{2}+6t(1)t(2)-6t(2)-2t(1)+1}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	${\frac {10}{q^{9/2}}}-{\frac {11}{q^{7/2}}}+{\frac {8}{q^{5/2}}}-{\frac {7}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {3}{q^{15/2}}}+{\frac {6}{q^{13/2}}}-{\frac {9}{q^{11/2}}}-{\sqrt {q}}+{\frac {4}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{9}z^{-1}+4za^{7}+3a^{7}z^{-1}-4z^{3}a^{5}-6za^{5}-2a^{5}z^{-1}+z^{5}a^{3}+z^{3}a^{3}-z^{3}a$ (db)
Kauffman polynomial	$a^{10}z^{4}-2a^{10}z^{2}+a^{10}+3a^{9}z^{5}-4a^{9}z^{3}+a^{9}z-a^{9}z^{-1}+a^{8}z^{8}-2a^{8}z^{6}+9a^{8}z^{4}-10a^{8}z^{2}+3a^{8}+a^{7}z^{9}-a^{7}z^{7}+5a^{7}z^{5}-6a^{7}z^{3}+7a^{7}z-3a^{7}z^{-1}+5a^{6}z^{8}-12a^{6}z^{6}+17a^{6}z^{4}-10a^{6}z^{2}+3a^{6}+a^{5}z^{9}+5a^{5}z^{7}-12a^{5}z^{5}+4a^{5}z^{3}+6a^{5}z-2a^{5}z^{-1}+4a^{4}z^{8}-6a^{4}z^{6}+2a^{4}z^{4}-2a^{4}z^{2}+6a^{3}z^{7}-13a^{3}z^{5}+5a^{3}z^{3}+4a^{2}z^{6}-7a^{2}z^{4}+az^{5}-az^{3}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

3

-3

-2

4

1

3

-4

5

4

-1

-6

6

3

-8

4

5

1

-10

5

6

-1

-12

2

5

3

-14

1

4

-3

-16

2

-18

1

-1

Integral Khovanov Homology

(db, data source)

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