L11n43

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_9,18,10,19 X₈₄₉₃ X_5,14,6,15 X_15,22,16,5 X_17,20,18,21 X_21,16,22,17 X_19,10,20,11 X_2,12,3,11
Gauss code	{1, -11, 5, -3}, {-6, -1, 2, -5, -4, 10, 11, -2, 3, 6, -7, 9, -8, 4, -10, 8, -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	$-{\sqrt {q}}+{\frac {2}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {7}{q^{5/2}}}-{\frac {10}{q^{7/2}}}+{\frac {9}{q^{9/2}}}-{\frac {9}{q^{11/2}}}+{\frac {7}{q^{13/2}}}-{\frac {4}{q^{15/2}}}+{\frac {2}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{9}z^{-1}-z^{3}a^{7}+za^{7}+2a^{7}z^{-1}+z^{5}a^{5}+z^{3}a^{5}-za^{5}-a^{5}z^{-1}+z^{5}a^{3}+2z^{3}a^{3}+2za^{3}+a^{3}z^{-1}-z^{3}a-2za-az^{-1}$ (db)
Kauffman polynomial	$-3z^{4}a^{10}+6z^{2}a^{10}-3a^{10}-z^{7}a^{9}-z^{5}a^{9}+3z^{3}a^{9}-za^{9}+a^{9}z^{-1}-2z^{8}a^{8}+5z^{6}a^{8}-14z^{4}a^{8}+18z^{2}a^{8}-7a^{8}-z^{9}a^{7}-3z^{5}a^{7}+5z^{3}a^{7}-4za^{7}+2a^{7}z^{-1}-4z^{8}a^{6}+8z^{6}a^{6}-14z^{4}a^{6}+13z^{2}a^{6}-4a^{6}-z^{9}a^{5}-z^{7}a^{5}-z^{5}a^{5}+6z^{3}a^{5}-5za^{5}+a^{5}z^{-1}-2z^{8}a^{4}+z^{6}a^{4}+z^{4}a^{4}-2z^{7}a^{3}+7z^{3}a^{3}-5za^{3}+a^{3}z^{-1}-2z^{6}a^{2}+4z^{4}a^{2}-z^{2}a^{2}-a^{2}-z^{5}a+3z^{3}a-3za+az^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

1

-1

-2

4

1

3

-4

5

3

-2

-6

5

2

3

-8

4

5

1

-10

5

0

-12

2

4

2

-14

2

5

-3

-16

2

-18

2

-2

Integral Khovanov Homology

(db, data source)

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