L11n149

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_9,19,10,18 X₆₇₁₈ X_22,19,7,20 X_12,5,13,6 X_3,10,4,11 X_4,15,5,16 X_16,12,17,11 X_20,13,21,14 X_14,21,15,22 X_17,2,18,3
Gauss code	{1, 11, -6, -7, 5, -3}, {3, -1, -2, 6, 8, -5, 9, -10, 7, -8, -11, 2, 4, -9, 10, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{4}-t(2)^{4}-3t(1)t(2)^{3}+2t(2)^{3}-2t(1)^{2}t(2)^{2}+3t(1)t(2)^{2}-2t(2)^{2}+2t(1)^{2}t(2)-3t(1)t(2)-t(1)^{2}+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	${\frac {6}{q^{9/2}}}-{\frac {5}{q^{7/2}}}+{\frac {2}{q^{5/2}}}-{\frac {1}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {3}{q^{19/2}}}+{\frac {4}{q^{17/2}}}-{\frac {6}{q^{15/2}}}+{\frac {7}{q^{13/2}}}-{\frac {7}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{3}\right)-a^{9}z+a^{7}z^{5}+3a^{7}z^{3}+4a^{7}z+a^{7}z^{-1}-3a^{5}z^{3}-5a^{5}z-a^{5}z^{-1}-a^{3}z^{3}-a^{3}z$ (db)
Kauffman polynomial	$-z^{6}a^{12}+3z^{4}a^{12}-z^{2}a^{12}-3z^{7}a^{11}+11z^{5}a^{11}-9z^{3}a^{11}+za^{11}-3z^{8}a^{10}+10z^{6}a^{10}-7z^{4}a^{10}+z^{2}a^{10}-z^{9}a^{9}-z^{7}a^{9}+12z^{5}a^{9}-13z^{3}a^{9}+4za^{9}-4z^{8}a^{8}+13z^{6}a^{8}-14z^{4}a^{8}+6z^{2}a^{8}-z^{9}a^{7}+2z^{7}a^{7}-3z^{5}a^{7}+3z^{3}a^{7}-3za^{7}+a^{7}z^{-1}-z^{8}a^{6}+2z^{6}a^{6}-6z^{4}a^{6}+5z^{2}a^{6}-a^{6}-4z^{5}a^{5}+6z^{3}a^{5}-5za^{5}+a^{5}z^{-1}-2z^{4}a^{4}+z^{2}a^{4}-z^{3}a^{3}+za^{3}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

1

-4

2

1

-1

-6

3

-8

3

2

-1

-10

4

3

1

-12

3

0

-14

3

4

-1

-16

2

4

2

-18

1

2

-1

-20

2

-22

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{3}$
$r=-5$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$