L11a84

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_18,8,19,7 X_22,20,5,19 X_20,16,21,15 X_16,22,17,21 X_14,10,15,9 X_10,14,11,13 X_8,18,9,17 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 7, -8, 11, -2, 8, -7, 5, -6, 9, -3, 4, -5, 6, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {3t(1)t(2)^{3}-5t(2)^{3}-9t(1)t(2)^{2}+10t(2)^{2}+10t(1)t(2)-9t(2)-5t(1)+3}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{15/2}-4q^{13/2}+8q^{11/2}-12q^{9/2}+16q^{7/2}-17q^{5/2}+17q^{3/2}-15{\sqrt {q}}+{\frac {9}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {2}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-1}+2z^{5}a^{-3}-2az^{3}-z^{3}a^{-1}+4z^{3}a^{-3}-3z^{3}a^{-5}+a^{3}z-2az-3za^{-1}+4za^{-3}-3za^{-5}+za^{-7}+a^{3}z^{-1}-2a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-2z^{4}a^{-8}+z^{2}a^{-8}+4z^{7}a^{-7}-10z^{5}a^{-7}+6z^{3}a^{-7}-za^{-7}+6z^{8}a^{-6}-14z^{6}a^{-6}+7z^{4}a^{-6}-2z^{2}a^{-6}+4z^{9}a^{-5}-z^{7}a^{-5}-17z^{5}a^{-5}+14z^{3}a^{-5}-2za^{-5}+z^{10}a^{-4}+11z^{8}a^{-4}-33z^{6}a^{-4}+28z^{4}a^{-4}-7z^{2}a^{-4}-2a^{-4}+7z^{9}a^{-3}-9z^{7}a^{-3}+a^{3}z^{5}-3z^{5}a^{-3}-3a^{3}z^{3}+7z^{3}a^{-3}+3a^{3}z-za^{-3}-a^{3}z^{-1}+a^{-3}z^{-1}+z^{10}a^{-2}+8z^{8}a^{-2}+2a^{2}z^{6}-19z^{6}a^{-2}-3a^{2}z^{4}+14z^{4}a^{-2}+4z^{2}a^{-2}+a^{2}-5a^{-2}+3z^{9}a^{-1}+3az^{7}-z^{7}a^{-1}-3az^{5}+2z^{3}a^{-1}-3za^{-1}+2a^{-1}z^{-1}+3z^{8}+z^{6}-8z^{4}+8z^{2}-3$ (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

16

1

-1

14

3

12

5

1

-4

10

7

3

4

8

9

5

-4

6

8

7

1

4

9

0

2

6

8

-2

0

4

10

6

-2

2

5

-3

-4

4

-6

1

2

-1

-8

1

Integral Khovanov Homology

(db, data source)

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