L11a20

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_18,16,19,15 X_16,9,17,10 X_8,17,9,18 X_22,19,5,20 X_20,13,21,14 X_14,21,15,22 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 5, -2, 11, -3, 8, -9, 4, -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	$-q^{7/2}+4q^{5/2}-9q^{3/2}+14{\sqrt {q}}-{\frac {19}{\sqrt {q}}}+{\frac {20}{q^{3/2}}}-{\frac {21}{q^{5/2}}}+{\frac {17}{q^{7/2}}}-{\frac {12}{q^{9/2}}}+{\frac {7}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+3z^{3}a^{5}+6za^{5}+3a^{5}z^{-1}-3z^{5}a^{3}-9z^{3}a^{3}-10za^{3}-3a^{3}z^{-1}+z^{7}a+4z^{5}a+8z^{3}a+7za+2az^{-1}-z^{5}a^{-1}-2z^{3}a^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{10}-a^{2}z^{10}-3a^{5}z^{9}-8a^{3}z^{9}-5az^{9}-4a^{6}z^{8}-12a^{4}z^{8}-17a^{2}z^{8}-9z^{8}-3a^{7}z^{7}-4a^{5}z^{7}+a^{3}z^{7}-6az^{7}-8z^{7}a^{-1}-a^{8}z^{6}+6a^{6}z^{6}+30a^{4}z^{6}+40a^{2}z^{6}-4z^{6}a^{-2}+13z^{6}+8a^{7}z^{5}+24a^{5}z^{5}+32a^{3}z^{5}+30az^{5}+13z^{5}a^{-1}-z^{5}a^{-3}+3a^{8}z^{4}+2a^{6}z^{4}-22a^{4}z^{4}-32a^{2}z^{4}+5z^{4}a^{-2}-6z^{4}-7a^{7}z^{3}-27a^{5}z^{3}-44a^{3}z^{3}-33az^{3}-8z^{3}a^{-1}+z^{3}a^{-3}-3a^{8}z^{2}-5a^{6}z^{2}+5a^{4}z^{2}+10a^{2}z^{2}-z^{2}a^{-2}+2z^{2}+3a^{7}z+14a^{5}z+22a^{3}z+15az+4za^{-1}+a^{8}+2a^{6}-2a^{2}-a^{7}z^{-1}-3a^{5}z^{-1}-3a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

3

-3

4

6

1

5

2

8

3

-5

0

11

6

5

-2

11

10

-1

-4

10

9

1

-6

7

11

4

-8

5

10

-5

-10

2

7

5

-12

1

5

-4

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$