L11a31

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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_20,14,21,13 X_12,20,13,19 X_14,22,15,21 X_8,16,9,15 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 5, -7, 6, -8, 9, -3, 4, -5, 7, -6, 8, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{5}-3t(2)^{5}-4t(1)t(2)^{4}+5t(2)^{4}+5t(1)t(2)^{3}-5t(2)^{3}-5t(1)t(2)^{2}+5t(2)^{2}+5t(1)t(2)-4t(2)-3t(1)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{17/2}-3q^{15/2}+7q^{13/2}-10q^{11/2}+13q^{9/2}-15q^{7/2}+14q^{5/2}-12q^{3/2}+8{\sqrt {q}}-{\frac {6}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+2za^{-7}-2z^{5}a^{-5}-5z^{3}a^{-5}+2a^{-5}z^{-1}+z^{7}a^{-3}+3z^{5}a^{-3}-5za^{-3}-4a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-6z^{3}a^{-1}+3az-2za^{-1}+az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-3z^{9}a^{-1}-7z^{9}a^{-3}-4z^{9}a^{-5}-5z^{8}a^{-2}-9z^{8}a^{-4}-6z^{8}a^{-6}-2z^{8}-az^{7}+11z^{7}a^{-1}+22z^{7}a^{-3}+3z^{7}a^{-5}-7z^{7}a^{-7}+30z^{6}a^{-2}+35z^{6}a^{-4}+6z^{6}a^{-6}-6z^{6}a^{-8}+7z^{6}+5az^{5}-13z^{5}a^{-1}-27z^{5}a^{-3}+3z^{5}a^{-5}+9z^{5}a^{-7}-3z^{5}a^{-9}-42z^{4}a^{-2}-44z^{4}a^{-4}+2z^{4}a^{-6}+8z^{4}a^{-8}-z^{4}a^{-10}-5z^{4}-8az^{3}+10z^{3}a^{-1}+27z^{3}a^{-3}+2z^{3}a^{-5}-5z^{3}a^{-7}+2z^{3}a^{-9}+23z^{2}a^{-2}+27z^{2}a^{-4}-4z^{2}a^{-6}-6z^{2}a^{-8}+z^{2}a^{-10}-z^{2}+5az-6za^{-1}-17za^{-3}-5za^{-5}+za^{-7}-5a^{-2}-6a^{-4}+a^{-6}+2a^{-8}+1-az^{-1}+a^{-1}z^{-1}+4a^{-3}z^{-1}+2a^{-5}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

18

1

-1

16

2

14

5

1

-4

12

5

2

3

10

8

5

-3

8

7

5

2

6

7

8

1

4

5

7

-2

2

4

8

4

0

2

4

-2

4

-4

1

2

-1

-6

1

Integral Khovanov Homology

(db, data source)

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