L11a30

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(2)^{5}+2t(1)t(2)^{4}-6t(2)^{4}-9t(1)t(2)^{3}+13t(2)^{3}+13t(1)t(2)^{2}-9t(2)^{2}-6t(1)t(2)+2t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {14}{q^{9/2}}}-q^{7/2}+{\frac {17}{q^{7/2}}}+3q^{5/2}-{\frac {20}{q^{5/2}}}-7q^{3/2}+{\frac {20}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {8}{q^{11/2}}}+12{\sqrt {q}}-{\frac {17}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)+3a^{5}z^{3}+3a^{5}z+2a^{5}z^{-1}-2a^{3}z^{5}-4a^{3}z^{3}-7a^{3}z-4a^{3}z^{-1}-za^{-3}-az^{5}+2az^{3}+2z^{3}a^{-1}+4az+3az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{10}-a^{2}z^{10}-4a^{5}z^{9}-8a^{3}z^{9}-4az^{9}-6a^{6}z^{8}-15a^{4}z^{8}-15a^{2}z^{8}-6z^{8}-4a^{7}z^{7}-4a^{5}z^{7}+a^{3}z^{7}-4az^{7}-5z^{7}a^{-1}-a^{8}z^{6}+12a^{6}z^{6}+42a^{4}z^{6}+40a^{2}z^{6}-3z^{6}a^{-2}+8z^{6}+10a^{7}z^{5}+33a^{5}z^{5}+37a^{3}z^{5}+22az^{5}+7z^{5}a^{-1}-z^{5}a^{-3}+2a^{8}z^{4}-4a^{6}z^{4}-35a^{4}z^{4}-41a^{2}z^{4}+5z^{4}a^{-2}-7z^{4}-8a^{7}z^{3}-36a^{5}z^{3}-52a^{3}z^{3}-30az^{3}-4z^{3}a^{-1}+2z^{3}a^{-3}-a^{8}z^{2}+a^{6}z^{2}+12a^{4}z^{2}+17a^{2}z^{2}-2z^{2}a^{-2}+5z^{2}+3a^{7}z+16a^{5}z+25a^{3}z+16az+3za^{-1}-za^{-3}-a^{6}-2a^{4}-3a^{2}-1-2a^{5}z^{-1}-4a^{3}z^{-1}-3az^{-1}-a^{-1}z^{-1}$ (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

2

-2

4

5

1

4

2

7

2

-5

0

10

5

-2

11

8

-3

-4

9

0

-6

8

11

3

-8

6

9

-3

-10

3

9

6

-12

1

5

-4

-14

3

-16

1

-1

Integral Khovanov Homology

(db, data source)

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