L11a217

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2u^{2}v^{3}-7u^{2}v^{2}+6u^{2}v-u^{2}+uv^{4}-9uv^{3}+17uv^{2}-9uv+u-v^{4}+6v^{3}-7v^{2}+2v}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {15}{q^{9/2}}}-q^{7/2}+{\frac {19}{q^{7/2}}}+3q^{5/2}-{\frac {23}{q^{5/2}}}-8q^{3/2}+{\frac {22}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {9}{q^{11/2}}}+14{\sqrt {q}}-{\frac {19}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)+3a^{5}z^{3}+2a^{5}z+a^{5}z^{-1}-2a^{3}z^{5}-3a^{3}z^{3}-5a^{3}z-2a^{3}z^{-1}-za^{-3}-az^{5}+3az^{3}+2z^{3}a^{-1}+5az+2az^{-1}-za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{4}z^{10}-2a^{2}z^{10}-6a^{5}z^{9}-12a^{3}z^{9}-6az^{9}-7a^{6}z^{8}-13a^{4}z^{8}-14a^{2}z^{8}-8z^{8}-4a^{7}z^{7}+5a^{5}z^{7}+16a^{3}z^{7}+az^{7}-6z^{7}a^{-1}-a^{8}z^{6}+15a^{6}z^{6}+39a^{4}z^{6}+39a^{2}z^{6}-3z^{6}a^{-2}+13z^{6}+9a^{7}z^{5}+13a^{5}z^{5}+9a^{3}z^{5}+15az^{5}+9z^{5}a^{-1}-z^{5}a^{-3}+2a^{8}z^{4}-8a^{6}z^{4}-30a^{4}z^{4}-37a^{2}z^{4}+4z^{4}a^{-2}-13z^{4}-6a^{7}z^{3}-14a^{5}z^{3}-22a^{3}z^{3}-23az^{3}-7z^{3}a^{-1}+2z^{3}a^{-3}-a^{8}z^{2}+2a^{6}z^{2}+8a^{4}z^{2}+13a^{2}z^{2}-z^{2}a^{-2}+7z^{2}+2a^{7}z+5a^{5}z+10a^{3}z+12az+4za^{-1}-za^{-3}-a^{2}-a^{5}z^{-1}-2a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

2

-2

4

6

1

5

2

8

2

-6

0

11

6

5

-2

12

9

-3

-4

11

10

1

-6

9

13

4

-8

6

10

-4

-10

3

9

6

-12

1

6

-5

-14

3

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-2$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$