L11n31

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_5,12,6,13 X₈₄₉₃ X_9,16,10,17 X_13,22,14,5 X_15,10,16,11 X_21,14,22,15 X_11,20,12,21 X_2,18,3,17
Gauss code	{1, -11, 5, -3}, {-4, -1, 2, -5, -6, 8, -10, 4, -7, 9, -8, 6, 11, -2, 3, 10, -9, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}-3t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {7}{q^{5/2}}}-{\frac {10}{q^{7/2}}}+{\frac {9}{q^{9/2}}}-{\frac {9}{q^{11/2}}}+{\frac {6}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {2}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{9}z^{-1}-z^{3}a^{7}+a^{7}z^{-1}+z^{5}a^{5}+2z^{3}a^{5}+3za^{5}+2a^{5}z^{-1}+z^{5}a^{3}+z^{3}a^{3}-2za^{3}-2a^{3}z^{-1}-z^{3}a-za$ (db)
Kauffman polynomial	$3a^{10}z^{4}-8a^{10}z^{2}+4a^{10}+a^{9}z^{7}-a^{9}z^{5}+a^{9}z^{3}-2a^{9}z-a^{9}z^{-1}+2a^{8}z^{8}-7a^{8}z^{6}+18a^{8}z^{4}-21a^{8}z^{2}+9a^{8}+a^{7}z^{9}+a^{7}z^{7}-8a^{7}z^{5}+17a^{7}z^{3}-7a^{7}z-a^{7}z^{-1}+5a^{6}z^{8}-14a^{6}z^{6}+20a^{6}z^{4}-11a^{6}z^{2}+4a^{6}+a^{5}z^{9}+4a^{5}z^{7}-16a^{5}z^{5}+22a^{5}z^{3}-11a^{5}z+2a^{5}z^{-1}+3a^{4}z^{8}-4a^{4}z^{6}-a^{4}z^{4}+3a^{4}z^{2}-2a^{4}+4a^{3}z^{7}-8a^{3}z^{5}+4a^{3}z^{3}-5a^{3}z+2a^{3}z^{-1}+3a^{2}z^{6}-6a^{2}z^{4}+a^{2}z^{2}+az^{5}-2az^{3}+az$ (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

2

-2

4

1

3

-4

5

4

-1

-6

5

2

3

-8

4

5

1

-10

5

0

-12

1

4

3

-14

2

5

-3

-16

1

-18

2

-2

Integral Khovanov Homology

(db, data source)

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