L11a216

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {4u^{2}v^{3}-6u^{2}v^{2}+4u^{2}v-u^{2}+2uv^{4}-8uv^{3}+13uv^{2}-8uv+2u-v^{4}+4v^{3}-6v^{2}+4v}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {4}{q^{25/2}}}+{\frac {8}{q^{23/2}}}-{\frac {13}{q^{21/2}}}+{\frac {18}{q^{19/2}}}-{\frac {20}{q^{17/2}}}+{\frac {20}{q^{15/2}}}-{\frac {18}{q^{13/2}}}+{\frac {12}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}(-z)+3a^{11}z^{3}+3a^{11}z-a^{11}z^{-1}-2a^{9}z^{5}-2a^{9}z^{3}+3a^{9}z+3a^{9}z^{-1}-3a^{7}z^{5}-8a^{7}z^{3}-7a^{7}z-2a^{7}z^{-1}-a^{5}z^{5}-2a^{5}z^{3}-a^{5}z$ (db)
Kauffman polynomial	$a^{16}z^{6}-2a^{16}z^{4}+a^{16}z^{2}+4a^{15}z^{7}-10a^{15}z^{5}+7a^{15}z^{3}-a^{15}z+6a^{14}z^{8}-13a^{14}z^{6}+7a^{14}z^{4}-a^{14}z^{2}+4a^{13}z^{9}+2a^{13}z^{7}-22a^{13}z^{5}+18a^{13}z^{3}-3a^{13}z+a^{12}z^{10}+14a^{12}z^{8}-36a^{12}z^{6}+27a^{12}z^{4}-8a^{12}z^{2}-a^{12}+8a^{11}z^{9}-3a^{11}z^{7}-19a^{11}z^{5}+16a^{11}z^{3}-3a^{11}z+a^{11}z^{-1}+a^{10}z^{10}+14a^{10}z^{8}-29a^{10}z^{6}+17a^{10}z^{4}+a^{10}z^{2}-3a^{10}+4a^{9}z^{9}+5a^{9}z^{7}-19a^{9}z^{5}+19a^{9}z^{3}-10a^{9}z+3a^{9}z^{-1}+6a^{8}z^{8}-4a^{8}z^{6}-5a^{8}z^{4}+8a^{8}z^{2}-3a^{8}+6a^{7}z^{7}-11a^{7}z^{5}+12a^{7}z^{3}-8a^{7}z+2a^{7}z^{-1}+3a^{6}z^{6}-4a^{6}z^{4}+a^{6}z^{2}+a^{5}z^{5}-2a^{5}z^{3}+a^{5}z$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

3

1

-2

-8

5

-10

7

3

-4

-12

11

5

6

-14

9

7

-2

-16

11

0

-18

8

10

2

-20

5

10

-5

-22

3

8

5

-24

1

5

-4

-26

3

-28

1

-1

Integral Khovanov Homology

(db, data source)

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