L11a44

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-4t(2)^{3}+8t(2)^{2}-4t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+10q^{7/2}-17q^{5/2}+20q^{3/2}-24{\sqrt {q}}+{\frac {23}{\sqrt {q}}}-{\frac {19}{q^{3/2}}}+{\frac {14}{q^{5/2}}}-{\frac {8}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+8az^{3}-9z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-5a^{3}z+9az-8za^{-1}+3za^{-3}+a^{5}z^{-1}-3a^{3}z^{-1}+4az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-4a^{3}z^{9}-13az^{9}-9z^{9}a^{-1}-3a^{4}z^{8}-9a^{2}z^{8}-17z^{8}a^{-2}-23z^{8}-a^{5}z^{7}+7a^{3}z^{7}+20az^{7}-5z^{7}a^{-1}-17z^{7}a^{-3}+10a^{4}z^{6}+41a^{2}z^{6}+26z^{6}a^{-2}-10z^{6}a^{-4}+67z^{6}+4a^{5}z^{5}+8a^{3}z^{5}+24az^{5}+50z^{5}a^{-1}+26z^{5}a^{-3}-4z^{5}a^{-5}-12a^{4}z^{4}-42a^{2}z^{4}-8z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}-46z^{4}-6a^{5}z^{3}-24a^{3}z^{3}-49az^{3}-49z^{3}a^{-1}-18z^{3}a^{-3}+6a^{4}z^{2}+14a^{2}z^{2}-z^{2}a^{-2}-3z^{2}a^{-4}+10z^{2}+4a^{5}z+16a^{3}z+26az+21za^{-1}+7za^{-3}-a^{4}-3a^{2}-a^{-2}-2-a^{5}z^{-1}-3a^{3}z^{-1}-4az^{-1}-2a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

3

8

7

1

-6

6

10

3

7

4

10

7

-3

2

14

10

4

0

11

12

1

-2

8

12

-4

6

11

5

-6

2

8

-6

-8

1

6

5

-10

2

-2

-12

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{14}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$