L11a21

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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_18,10,19,9 X_20,14,21,13 X_22,16,5,15 X_14,22,15,21 X_16,20,17,19 X_8,18,9,17 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 4, -2, 11, -3, 5, -7, 6, -8, 9, -4, 8, -5, 7, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}+1\right)\left(2t(2)^{2}-3t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{23/2}-4q^{21/2}+9q^{19/2}-13q^{17/2}+17q^{15/2}-18q^{13/2}+17q^{11/2}-15q^{9/2}+9q^{7/2}-6q^{5/2}+2q^{3/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-5}-z^{7}a^{-7}+z^{5}a^{-3}-4z^{5}a^{-5}-3z^{5}a^{-7}+z^{5}a^{-9}+4z^{3}a^{-3}-6z^{3}a^{-5}-2z^{3}a^{-7}+2z^{3}a^{-9}+5za^{-3}-5za^{-5}-za^{-7}+za^{-9}+2a^{-3}z^{-1}-2a^{-5}z^{-1}-a^{-7}z^{-1}+a^{-9}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-14}+4z^{5}a^{-13}-z^{3}a^{-13}+9z^{6}a^{-12}-9z^{4}a^{-12}+4z^{2}a^{-12}-a^{-12}+12z^{7}a^{-11}-16z^{5}a^{-11}+7z^{3}a^{-11}-za^{-11}+10z^{8}a^{-10}-10z^{6}a^{-10}-4z^{4}a^{-10}+4z^{2}a^{-10}+5z^{9}a^{-9}+4z^{7}a^{-9}-22z^{5}a^{-9}+12z^{3}a^{-9}-2za^{-9}-a^{-9}z^{-1}+z^{10}a^{-8}+13z^{8}a^{-8}-33z^{6}a^{-8}+19z^{4}a^{-8}-6z^{2}a^{-8}+3a^{-8}+7z^{9}a^{-7}-11z^{7}a^{-7}-8z^{5}a^{-7}+15z^{3}a^{-7}-4za^{-7}-a^{-7}z^{-1}+z^{10}a^{-6}+5z^{8}a^{-6}-21z^{6}a^{-6}+19z^{4}a^{-6}-4z^{2}a^{-6}+2z^{9}a^{-5}-2z^{7}a^{-5}-11z^{5}a^{-5}+20z^{3}a^{-5}-10za^{-5}+2a^{-5}z^{-1}+2z^{8}a^{-4}-7z^{6}a^{-4}+6z^{4}a^{-4}+2z^{2}a^{-4}-3a^{-4}+z^{7}a^{-3}-5z^{5}a^{-3}+9z^{3}a^{-3}-7za^{-3}+2a^{-3}z^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

9

χ

24

1

-1

22

3

20

6

1

-5

18

7

3

4

16

10

6

-4

14

8

7

1

12

9

10

1

10

6

8

-2

8

3

9

6

3

6

-3

4

1

5

4

2

1

-1

0

1

Integral Khovanov Homology

(db, data source)

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