L11a35

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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_18,13,19,14 X_14,17,15,18 X_22,20,5,19 X_20,12,21,11 X_12,22,13,21 X_8,16,9,15 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 7, -8, 4, -5, 9, -3, 5, -4, 6, -7, 8, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)^{5}+2t(1)t(2)^{4}-6t(2)^{4}-7t(1)t(2)^{3}+9t(2)^{3}+9t(1)t(2)^{2}-7t(2)^{2}-6t(1)t(2)+2t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-3q^{9/2}+{\frac {3}{q^{9/2}}}+6q^{7/2}-{\frac {7}{q^{7/2}}}-10q^{5/2}+{\frac {10}{q^{5/2}}}+14q^{3/2}-{\frac {14}{q^{3/2}}}+q^{11/2}-{\frac {1}{q^{11/2}}}-16{\sqrt {q}}+{\frac {15}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}+za^{-5}-3a^{3}z^{3}-2z^{3}a^{-3}-5a^{3}z-2a^{3}z^{-1}-za^{-3}+a^{-3}z^{-1}+2az^{5}+z^{5}a^{-1}+5az^{3}-z^{3}a^{-1}+6az+3az^{-1}-4za^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}-z^{2}a^{-6}+a^{5}z^{7}-4a^{5}z^{5}+3z^{5}a^{-5}+6a^{5}z^{3}-3z^{3}a^{-5}-4a^{5}z+za^{-5}+a^{5}z^{-1}+3a^{4}z^{8}-11a^{4}z^{6}+5z^{6}a^{-4}+12a^{4}z^{4}-5z^{4}a^{-4}-4a^{4}z^{2}+3z^{2}a^{-4}-a^{-4}+3a^{3}z^{9}-5a^{3}z^{7}+6z^{7}a^{-3}-11a^{3}z^{5}-5z^{5}a^{-3}+22a^{3}z^{3}+z^{3}a^{-3}-11a^{3}z-za^{-3}+2a^{3}z^{-1}+a^{-3}z^{-1}+a^{2}z^{10}+8a^{2}z^{8}+6z^{8}a^{-2}-36a^{2}z^{6}-5z^{6}a^{-2}+38a^{2}z^{4}-5z^{4}a^{-2}-14a^{2}z^{2}+8z^{2}a^{-2}+2a^{2}-2a^{-2}+7az^{9}+4z^{9}a^{-1}-11az^{7}+z^{7}a^{-1}-18az^{5}-19z^{5}a^{-1}+35az^{3}+23z^{3}a^{-1}-18az-13za^{-1}+3az^{-1}+3a^{-1}z^{-1}+z^{10}+11z^{8}-35z^{6}+27z^{4}-6z^{2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

2

8

4

1

-3

6

2

4

8

4

-4

2

8

6

2

0

8

9

1

-2

6

7

-1

-4

4

8

4

-6

3

6

-3

-8

1

5

4

-10

2

-2

-12

1

Integral Khovanov Homology

(db, data source)

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