L11a174

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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_16,8,17,7 X_22,16,7,15 X_6,21,1,22 X_20,11,21,12 X_14,6,15,5 X_18,13,19,14 X_12,19,13,20 X_4,18,5,17
Gauss code	{1, -2, 3, -11, 8, -6}, {4, -1, 2, -3, 7, -10, 9, -8, 5, -4, 11, -9, 10, -7, 6, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2u^{2}v^{4}-4u^{2}v^{3}+4u^{2}v^{2}-2u^{2}v-2uv^{4}+7uv^{3}-11uv^{2}+7uv-2u-2v^{3}+4v^{2}-4v+2}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {11}{q^{9/2}}}-q^{7/2}+{\frac {14}{q^{7/2}}}+3q^{5/2}-{\frac {17}{q^{5/2}}}-7q^{3/2}+{\frac {17}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {6}{q^{11/2}}}+11{\sqrt {q}}-{\frac {15}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+4a^{3}z^{5}+4az^{5}-z^{5}a^{-1}-3a^{5}z^{3}+5a^{3}z^{3}+6az^{3}-3z^{3}a^{-1}-2a^{5}z+4az-3za^{-1}+a^{5}z^{-1}-2a^{3}z^{-1}+2az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-3a^{8}z^{4}+2a^{8}z^{2}+3a^{7}z^{7}-9a^{7}z^{5}+7a^{7}z^{3}-a^{7}z+4a^{6}z^{8}-9a^{6}z^{6}+3a^{6}z^{4}+a^{6}z^{2}+4a^{5}z^{9}-9a^{5}z^{7}+9a^{5}z^{5}-8a^{5}z^{3}+a^{5}z^{-1}+2a^{4}z^{10}-4a^{4}z^{6}+2a^{4}z^{4}-2a^{4}z^{2}+9a^{3}z^{9}-26a^{3}z^{7}+35a^{3}z^{5}+z^{5}a^{-3}-18a^{3}z^{3}-2z^{3}a^{-3}-2a^{3}z+2a^{3}z^{-1}+2a^{2}z^{10}+3a^{2}z^{8}-14a^{2}z^{6}+3z^{6}a^{-2}+19a^{2}z^{4}-5z^{4}a^{-2}-7a^{2}z^{2}+a^{2}+5az^{9}-8az^{7}+6z^{7}a^{-1}+2az^{5}-14z^{5}a^{-1}+11az^{3}+12z^{3}a^{-1}-8az-5za^{-1}+2az^{-1}+a^{-1}z^{-1}+7z^{8}-17z^{6}+18z^{4}-6z^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

2

-2

4

5

1

4

2

6

2

-4

0

9

5

4

-2

9

7

-2

-4

8

0

-6

7

10

3

-8

4

7

-3

-10

2

7

5

-12

1

4

-3

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

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