L11a40

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,10,13,9 X_18,15,19,16 X_16,7,17,8 X_8,17,9,18 X_22,20,5,19 X_20,14,21,13 X_14,22,15,21 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 5, -6, 3, -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 $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)^{5}}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{13/2}+4q^{11/2}-8q^{9/2}+14q^{7/2}-19q^{5/2}+20q^{3/2}-21{\sqrt {q}}+{\frac {17}{\sqrt {q}}}-{\frac {13}{q^{3/2}}}+{\frac {7}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-4z^{5}a^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+6az^{3}-9z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}-2a^{3}z+8az-10za^{-1}+5za^{-3}-za^{-5}-a^{3}z^{-1}+4az^{-1}-4a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-z^{3}a^{-7}+4z^{6}a^{-6}-6z^{4}a^{-6}+3z^{2}a^{-6}+7z^{7}a^{-5}-10z^{5}a^{-5}+6z^{3}a^{-5}-2za^{-5}+7z^{8}a^{-4}+a^{4}z^{6}-3z^{6}a^{-4}-3a^{4}z^{4}-10z^{4}a^{-4}+3a^{4}z^{2}+9z^{2}a^{-4}-a^{4}-a^{-4}+4z^{9}a^{-3}+3a^{3}z^{7}+10z^{7}a^{-3}-8a^{3}z^{5}-31z^{5}a^{-3}+8a^{3}z^{3}+26z^{3}a^{-3}-4a^{3}z-10za^{-3}+a^{3}z^{-1}+a^{-3}z^{-1}+z^{10}a^{-2}+4a^{2}z^{8}+14z^{8}a^{-2}-5a^{2}z^{6}-21z^{6}a^{-2}-5a^{2}z^{4}-3z^{4}a^{-2}+10a^{2}z^{2}+13z^{2}a^{-2}-4a^{2}-4a^{-2}+3az^{9}+7z^{9}a^{-1}+5az^{7}+5z^{7}a^{-1}-26az^{5}-38z^{5}a^{-1}+29az^{3}+40z^{3}a^{-1}-15az-19za^{-1}+4az^{-1}+4a^{-1}z^{-1}+z^{10}+11z^{8}-20z^{6}-z^{4}+14z^{2}-7$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

3

-3

10

5

1

4

8

9

3

-6

6

10

5

4

10

9

-1

2

11

10

1

0

8

12

4

-2

5

9

-4

2

8

6

-6

1

5

-4

-8

2

-10

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$