L11a155

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_14,6,15,5 X_20,11,21,12 X_22,18,7,17 X_18,22,19,21 X_16,13,17,14 X_12,19,13,20 X_4,16,5,15 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -10, 2, -9, 3, -11}, {10, -1, 11, -2, 4, -8, 7, -3, 9, -7, 5, -6, 8, -4, 6, -5}

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

Polynomial invariants

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

2

-2

4

5

1

4

2

6

2

-4

0

9

5

4

-2

9

7

-2

-4

8

0

-6

7

10

3

-8

4

7

-3

-10

2

7

5

-12

1

4

-3

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

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