L11a156

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{4}-5u^{2}v^{3}+6u^{2}v^{2}-2u^{2}v-uv^{4}+6uv^{3}-9uv^{2}+6uv-u-2v^{3}+6v^{2}-5v+1}{uv^{2}}}$ (db)
Jones polynomial	$q^{17/2}-3q^{15/2}+6q^{13/2}-11q^{11/2}+14q^{9/2}-16q^{7/2}+16q^{5/2}-14q^{3/2}+10{\sqrt {q}}-{\frac {7}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+2za^{-7}-2z^{5}a^{-5}-6z^{3}a^{-5}-5za^{-5}-a^{-5}z^{-1}+z^{7}a^{-3}+4z^{5}a^{-3}+7z^{3}a^{-3}+7za^{-3}+2a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-6z^{3}a^{-1}+2az-5za^{-1}+az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-3z^{9}a^{-1}-7z^{9}a^{-3}-4z^{9}a^{-5}-8z^{8}a^{-2}-12z^{8}a^{-4}-7z^{8}a^{-6}-3z^{8}-az^{7}+5z^{7}a^{-1}+10z^{7}a^{-3}-3z^{7}a^{-5}-7z^{7}a^{-7}+35z^{6}a^{-2}+39z^{6}a^{-4}+10z^{6}a^{-6}-5z^{6}a^{-8}+11z^{6}+4az^{5}+11z^{5}a^{-1}+20z^{5}a^{-3}+25z^{5}a^{-5}+9z^{5}a^{-7}-3z^{5}a^{-9}-35z^{4}a^{-2}-34z^{4}a^{-4}-6z^{4}a^{-6}+4z^{4}a^{-8}-z^{4}a^{-10}-12z^{4}-6az^{3}-23z^{3}a^{-1}-34z^{3}a^{-3}-26z^{3}a^{-5}-6z^{3}a^{-7}+3z^{3}a^{-9}+11z^{2}a^{-2}+11z^{2}a^{-4}+2z^{2}a^{-6}-z^{2}a^{-8}+z^{2}a^{-10}+4z^{2}+4az+12za^{-1}+15za^{-3}+9za^{-5}+za^{-7}-za^{-9}-a^{-2}-az^{-1}-2a^{-1}z^{-1}-2a^{-3}z^{-1}-a^{-5}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

18

1

-1

16

2

14

4

1

-3

12

7

2

5

10

7

4

-3

8

9

7

2

6

8

0

4

6

8

-2

2

5

9

4

0

2

5

-3

-2

1

5

4

-4

2

-2

-6

1

Integral Khovanov Homology

(db, data source)

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