L11a38

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)^{5}}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+8q^{5/2}-14q^{3/2}+18{\sqrt {q}}-{\frac {21}{\sqrt {q}}}+{\frac {20}{q^{3/2}}}-{\frac {18}{q^{5/2}}}+{\frac {13}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+2a^{5}z+a^{5}z^{-1}-2a^{3}z^{5}-6a^{3}z^{3}+z^{3}a^{-3}-8a^{3}z+za^{-3}-3a^{3}z^{-1}+az^{7}+4az^{5}-2z^{5}a^{-1}+9az^{3}-5z^{3}a^{-1}+10az-5za^{-1}+4az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-2a^{7}z^{3}+a^{7}z+3a^{6}z^{6}-5a^{6}z^{4}+3a^{6}z^{2}-a^{6}+5a^{5}z^{7}-6a^{5}z^{5}+3a^{5}z^{3}-2a^{5}z+a^{5}z^{-1}+6a^{4}z^{8}-5a^{4}z^{6}+z^{6}a^{-4}-2a^{4}z^{4}-2z^{4}a^{-4}+7a^{4}z^{2}+z^{2}a^{-4}-3a^{4}+4a^{3}z^{9}+6a^{3}z^{7}+4z^{7}a^{-3}-24a^{3}z^{5}-10z^{5}a^{-3}+28a^{3}z^{3}+8z^{3}a^{-3}-15a^{3}z-2za^{-3}+3a^{3}z^{-1}+a^{2}z^{10}+15a^{2}z^{8}+6z^{8}a^{-2}-32a^{2}z^{6}-12z^{6}a^{-2}+19a^{2}z^{4}+5z^{4}a^{-2}+2a^{2}z^{2}-3a^{2}+8az^{9}+4z^{9}a^{-1}+az^{7}+4z^{7}a^{-1}-36az^{5}-29z^{5}a^{-1}+43az^{3}+28z^{3}a^{-1}-21az-11za^{-1}+4az^{-1}+2a^{-1}z^{-1}+z^{10}+15z^{8}-37z^{6}+23z^{4}-3z^{2}-2$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

3

6

5

1

-4

4

9

3

6

2

9

5

-4

0

12

9

3

-2

10

11

1

-4

8

10

-2

-6

5

10

5

-8

2

8

-6

-10

1

5

4

-12

2

-2

-14

1

Integral Khovanov Homology

(db, data source)

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