L11a32

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2t(1)t(2)^{3}-6t(2)^{3}-9t(1)t(2)^{2}+12t(2)^{2}+12t(1)t(2)-9t(2)-6t(1)+2}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{15/2}-3q^{13/2}+7q^{11/2}-11q^{9/2}+16q^{7/2}-19q^{5/2}+18q^{3/2}-17{\sqrt {q}}+{\frac {12}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-1}+z^{5}a^{-3}-2az^{3}-2z^{3}a^{-1}-z^{3}a^{-3}-2z^{3}a^{-5}+a^{3}z+az-4za^{-1}-za^{-3}+za^{-7}+2az^{-1}-2a^{-1}z^{-1}-a^{-3}z^{-1}+a^{-5}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-3z^{4}a^{-8}+2z^{2}a^{-8}+3z^{7}a^{-7}-8z^{5}a^{-7}+5z^{3}a^{-7}-za^{-7}+5z^{8}a^{-6}-14z^{6}a^{-6}+15z^{4}a^{-6}-11z^{2}a^{-6}+4a^{-6}+4z^{9}a^{-5}-5z^{7}a^{-5}-4z^{5}a^{-5}+6z^{3}a^{-5}-za^{-5}-a^{-5}z^{-1}+z^{10}a^{-4}+12z^{8}a^{-4}-42z^{6}a^{-4}+56z^{4}a^{-4}-36z^{2}a^{-4}+9a^{-4}+8z^{9}a^{-3}-11z^{7}a^{-3}+a^{3}z^{5}-z^{5}a^{-3}-2a^{3}z^{3}+7z^{3}a^{-3}+a^{3}z-za^{-3}-a^{-3}z^{-1}+z^{10}a^{-2}+13z^{8}a^{-2}+3a^{2}z^{6}-35z^{6}a^{-2}-4a^{2}z^{4}+38z^{4}a^{-2}+a^{2}z^{2}-19z^{2}a^{-2}+4a^{-2}+4z^{9}a^{-1}+6az^{7}+3z^{7}a^{-1}-11az^{5}-17z^{5}a^{-1}+11az^{3}+19z^{3}a^{-1}-8az-10za^{-1}+2az^{-1}+2a^{-1}z^{-1}+6z^{8}-5z^{6}-4z^{4}+5z^{2}-2$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

16

1

-1

14

2

12

5

1

-4

10

6

2

4

8

10

5

-5

6

9

6

3

4

9

10

1

2

8

9

-1

0

5

10

5

-2

3

7

-4

5

-6

1

3

-2

-8

1

Integral Khovanov Homology

(db, data source)

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