L11a512

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_16,5,17,6 X_22,15,13,16 X_14,4,15,3 X_4,14,5,13 X_12,17,7,18 X_10,19,11,20 X_18,9,19,10 X_20,11,21,12 X₂₇₃₈ X_6,21,1,22
Gauss code	{1, -10, 4, -5, 2, -11}, {10, -1, 8, -7, 9, -6}, {5, -4, 3, -2, 6, -8, 7, -9, 11, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)^{2}t(3)^{3}-t(2)^{2}t(3)^{3}+t(1)^{2}t(2)t(3)^{3}-2t(1)t(2)t(3)^{3}+t(2)t(3)^{3}+t(1)^{2}t(3)^{2}+t(1)^{2}t(2)^{2}t(3)^{2}-4t(1)t(2)^{2}t(3)^{2}+2t(2)^{2}t(3)^{2}-2t(1)t(3)^{2}-3t(1)^{2}t(2)t(3)^{2}+7t(1)t(2)t(3)^{2}-3t(2)t(3)^{2}+t(3)^{2}-2t(1)^{2}t(3)-t(1)^{2}t(2)^{2}t(3)+2t(1)t(2)^{2}t(3)-t(2)^{2}t(3)+4t(1)t(3)+3t(1)^{2}t(2)t(3)-7t(1)t(2)t(3)+3t(2)t(3)-t(3)+t(1)^{2}-t(1)-t(1)^{2}t(2)+2t(1)t(2)-t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-11}+4q^{-10}-8q^{-9}+13q^{-8}-17q^{-7}+20q^{-6}-18q^{-5}+17q^{-4}-11q^{-3}+7q^{-2}-3q^{-1}+1$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{10}\left(-z^{2}\right)-a^{10}+3a^{8}z^{4}+7a^{8}z^{2}+a^{8}z^{-2}+4a^{8}-2a^{6}z^{6}-7a^{6}z^{4}-10a^{6}z^{2}-2a^{6}z^{-2}-8a^{6}-a^{4}z^{6}-a^{4}z^{4}+4a^{4}z^{2}+a^{4}z^{-2}+5a^{4}+a^{2}z^{4}+2a^{2}z^{2}$ (db)
Kauffman polynomial	$a^{13}z^{5}-a^{13}z^{3}+4a^{12}z^{6}-6a^{12}z^{4}+2a^{12}z^{2}+7a^{11}z^{7}-11a^{11}z^{5}+5a^{11}z^{3}-a^{11}z+7a^{10}z^{8}-7a^{10}z^{6}-a^{10}z^{4}+a^{10}+4a^{9}z^{9}+6a^{9}z^{7}-23a^{9}z^{5}+18a^{9}z^{3}-3a^{9}z+a^{8}z^{10}+14a^{8}z^{8}-36a^{8}z^{6}+39a^{8}z^{4}-24a^{8}z^{2}-a^{8}z^{-2}+8a^{8}+8a^{7}z^{9}-9a^{7}z^{7}-6a^{7}z^{5}+14a^{7}z^{3}-8a^{7}z+2a^{7}z^{-1}+a^{6}z^{10}+12a^{6}z^{8}-40a^{6}z^{6}+53a^{6}z^{4}-37a^{6}z^{2}-2a^{6}z^{-2}+12a^{6}+4a^{5}z^{9}-5a^{5}z^{7}-3a^{5}z^{5}+7a^{5}z^{3}-6a^{5}z+2a^{5}z^{-1}+5a^{4}z^{8}-14a^{4}z^{6}+16a^{4}z^{4}-13a^{4}z^{2}-a^{4}z^{-2}+6a^{4}+3a^{3}z^{7}-8a^{3}z^{5}+5a^{3}z^{3}+a^{2}z^{6}-3a^{2}z^{4}+2a^{2}z^{2}$ (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

1

-1

2

-2

-3

5

1

4

-5

7

3

-4

-7

10

4

6

-9

9

8

-1

-11

11

9

2

-13

7

10

3

-15

6

10

-4

-17

3

8

5

-19

1

5

-4

-21

3

-23

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-5$	$i=-3$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-6$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=-5$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=-3$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=0$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{5}$
$r=1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$