L11a162

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X₆₇₁₈ X_16,11,17,12 X_14,6,15,5 X_4,16,5,15 X_20,13,21,14 X_22,17,7,18 X_18,21,19,22 X_12,19,13,20
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 5, -11, 8, -6, 7, -5, 9, -10, 11, -8, 10, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}-4t(1)^{2}t(2)^{3}+4t(1)t(2)^{3}-2t(2)^{3}+4t(1)^{2}t(2)^{2}-5t(1)t(2)^{2}+4t(2)^{2}-2t(1)^{2}t(2)+4t(1)t(2)-4t(2)-2t(1)+2}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {11}{q^{9/2}}}+{\frac {7}{q^{7/2}}}-{\frac {5}{q^{5/2}}}+{\frac {2}{q^{3/2}}}+{\frac {1}{q^{23/2}}}-{\frac {3}{q^{21/2}}}+{\frac {6}{q^{19/2}}}-{\frac {9}{q^{17/2}}}+{\frac {12}{q^{15/2}}}-{\frac {13}{q^{13/2}}}+{\frac {12}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-z^{5}a^{9}-3z^{3}a^{9}-2za^{9}-a^{9}z^{-1}+z^{7}a^{7}+4z^{5}a^{7}+5z^{3}a^{7}+4za^{7}+2a^{7}z^{-1}+z^{7}a^{5}+4z^{5}a^{5}+4z^{3}a^{5}+za^{5}-z^{5}a^{3}-4z^{3}a^{3}-4za^{3}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{14}z^{4}-a^{14}z^{2}+3a^{13}z^{5}-3a^{13}z^{3}+5a^{12}z^{6}-6a^{12}z^{4}+2a^{12}z^{2}+6a^{11}z^{7}-9a^{11}z^{5}+6a^{11}z^{3}-a^{11}z+6a^{10}z^{8}-13a^{10}z^{6}+15a^{10}z^{4}-8a^{10}z^{2}+2a^{10}+4a^{9}z^{9}-8a^{9}z^{7}+8a^{9}z^{5}-6a^{9}z^{3}+3a^{9}z-a^{9}z^{-1}+a^{8}z^{10}+6a^{8}z^{8}-27a^{8}z^{6}+36a^{8}z^{4}-22a^{8}z^{2}+5a^{8}+6a^{7}z^{9}-20a^{7}z^{7}+23a^{7}z^{5}-16a^{7}z^{3}+7a^{7}z-2a^{7}z^{-1}+a^{6}z^{10}+2a^{6}z^{8}-17a^{6}z^{6}+22a^{6}z^{4}-12a^{6}z^{2}+3a^{6}+2a^{5}z^{9}-5a^{5}z^{7}-2a^{5}z^{5}+7a^{5}z^{3}-2a^{5}z+2a^{4}z^{8}-8a^{4}z^{6}+8a^{4}z^{4}-a^{4}z^{2}-a^{4}+a^{3}z^{7}-5a^{3}z^{5}+8a^{3}z^{3}-5a^{3}z+a^{3}z^{-1}$ (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

1

2

χ

0

1

-2

1

-1

-4

4

1

3

-6

4

2

-2

-8

7

3

4

-10

6

5

-1

-12

7

6

1

-14

5

6

1

-16

4

7

-3

-18

2

5

3

-20

1

4

-3

-22

2

-24

1

-1

Integral Khovanov Homology

(db, data source)

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