L11n167

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_14,5,15,6 X_11,19,12,18 X_22,19,7,20 X_20,15,21,16 X_16,21,17,22 X_17,13,18,12 X₆₇₁₈ X_4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, -5, 9, 11, -4, 7, -8, -9, 5, 6, -7, 8, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(2)^{4}-t(1)t(2)^{4}-3t(1)^{2}t(2)^{3}+3t(1)t(2)^{3}-t(2)^{3}+2t(1)^{2}t(2)^{2}-5t(1)t(2)^{2}+2t(2)^{2}-t(1)^{2}t(2)+3t(1)t(2)-3t(2)-t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {8}{q^{11/2}}}-{\frac {9}{q^{13/2}}}+{\frac {9}{q^{15/2}}}-{\frac {8}{q^{17/2}}}+{\frac {5}{q^{19/2}}}-{\frac {3}{q^{21/2}}}+{\frac {1}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{5}\right)-3a^{9}z^{3}-2a^{9}z+a^{7}z^{7}+5a^{7}z^{5}+9a^{7}z^{3}+7a^{7}z+a^{7}z^{-1}-2a^{5}z^{5}-8a^{5}z^{3}-8a^{5}z-a^{5}z^{-1}$ (db)
Kauffman polynomial	$-z^{4}a^{14}+z^{2}a^{14}-3z^{5}a^{13}+4z^{3}a^{13}-za^{13}-4z^{6}a^{12}+4z^{4}a^{12}-4z^{7}a^{11}+4z^{5}a^{11}-2za^{11}-3z^{8}a^{10}+4z^{6}a^{10}-4z^{4}a^{10}+z^{2}a^{10}-z^{9}a^{9}-2z^{7}a^{9}+5z^{5}a^{9}-3z^{3}a^{9}-4z^{8}a^{8}+10z^{6}a^{8}-11z^{4}a^{8}+6z^{2}a^{8}-z^{9}a^{7}+2z^{7}a^{7}-5z^{5}a^{7}+11z^{3}a^{7}-7za^{7}+a^{7}z^{-1}-z^{8}a^{6}+2z^{6}a^{6}-2z^{4}a^{6}+4z^{2}a^{6}-a^{6}-3z^{5}a^{5}+10z^{3}a^{5}-8za^{5}+a^{5}z^{-1}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

2

-6

2

1

-1

-8

4

1

3

-10

4

2

-2

-12

5

4

1

-14

4

0

-16

4

5

-1

-18

2

5

3

-20

1

3

-2

-22

2

-24

1

-1

Integral Khovanov Homology

(db, data source)

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