L11a42

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(v^{4}-4v^{3}+4v^{2}-4v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {7}{q^{9/2}}}-3q^{7/2}+{\frac {12}{q^{7/2}}}+6q^{5/2}-{\frac {16}{q^{5/2}}}-12q^{3/2}+{\frac {18}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {3}{q^{11/2}}}+15{\sqrt {q}}-{\frac {18}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+4az^{5}-2z^{5}a^{-1}+a^{5}z^{3}-6a^{3}z^{3}+8az^{3}-6z^{3}a^{-1}+z^{3}a^{-3}+2a^{5}z-7a^{3}z+9az-6za^{-1}+2za^{-3}+a^{5}z^{-1}-3a^{3}z^{-1}+4az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-3a^{3}z^{9}-6az^{9}-3z^{9}a^{-1}-5a^{4}z^{8}-9a^{2}z^{8}-4z^{8}a^{-2}-8z^{8}-5a^{5}z^{7}-7a^{3}z^{7}-z^{7}a^{-1}-3z^{7}a^{-3}-3a^{6}z^{6}+2a^{4}z^{6}+14a^{2}z^{6}+8z^{6}a^{-2}-z^{6}a^{-4}+18z^{6}-a^{7}z^{5}+7a^{5}z^{5}+23a^{3}z^{5}+24az^{5}+18z^{5}a^{-1}+9z^{5}a^{-3}+5a^{6}z^{4}+7a^{4}z^{4}-2z^{4}a^{-2}+3z^{4}a^{-4}-7z^{4}+2a^{7}z^{3}-4a^{5}z^{3}-24a^{3}z^{3}-33az^{3}-24z^{3}a^{-1}-9z^{3}a^{-3}-3a^{6}z^{2}-8a^{4}z^{2}-8a^{2}z^{2}-2z^{2}a^{-2}-2z^{2}a^{-4}-3z^{2}-a^{7}z+2a^{5}z+13a^{3}z+19az+13za^{-1}+4za^{-3}+a^{6}+3a^{4}+3a^{2}+2-a^{5}z^{-1}-3a^{3}z^{-1}-4az^{-1}-2a^{-1}z^{-1}$ (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

10

1

-1

8

2

6

4

1

-3

4

8

2

6

2

7

4

-3

0

11

8

3

-2

9

0

-4

7

9

-2

-6

5

9

4

-8

2

7

-5

-10

1

5

4

-12

2

-2

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$