L11a60

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_14,8,15,7 X_20,18,21,17 X_18,9,19,10 X_8,19,9,20 X_22,16,5,15 X_16,22,17,21 X_10,14,11,13 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 5, -9, 11, -2, 9, -3, 7, -8, 4, -5, 6, -4, 8, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(1)t(2)^{3}-4t(2)^{3}-8t(1)t(2)^{2}+11t(2)^{2}+11t(1)t(2)-8t(2)-4t(1)+2}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {6}{q^{9/2}}}-4q^{7/2}+{\frac {9}{q^{7/2}}}+7q^{5/2}-{\frac {13}{q^{5/2}}}-11q^{3/2}+{\frac {16}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}+14{\sqrt {q}}-{\frac {16}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}-3a^{5}z-a^{5}z^{-1}+3a^{3}z^{3}+z^{3}a^{-3}+a^{3}z-a^{3}z^{-1}-az^{5}-z^{5}a^{-1}+az^{3}-z^{3}a^{-1}+2az+2az^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-2a^{3}z^{9}-6az^{9}-4z^{9}a^{-1}-3a^{4}z^{8}-5a^{2}z^{8}-6z^{8}a^{-2}-8z^{8}-3a^{5}z^{7}-4a^{3}z^{7}+8az^{7}+5z^{7}a^{-1}-4z^{7}a^{-3}-2a^{6}z^{6}+a^{4}z^{6}+10a^{2}z^{6}+17z^{6}a^{-2}-z^{6}a^{-4}+25z^{6}-a^{7}z^{5}+3a^{5}z^{5}+12a^{3}z^{5}+5az^{5}+8z^{5}a^{-1}+11z^{5}a^{-3}+3a^{6}z^{4}+a^{4}z^{4}-9a^{2}z^{4}-12z^{4}a^{-2}+2z^{4}a^{-4}-21z^{4}+3a^{7}z^{3}+a^{5}z^{3}-14a^{3}z^{3}-18az^{3}-12z^{3}a^{-1}-6z^{3}a^{-3}+2a^{4}z^{2}+6a^{2}z^{2}+2z^{2}a^{-2}+6z^{2}-3a^{7}z-2a^{5}z+8a^{3}z+12az+5za^{-1}-a^{6}-2a^{4}-3a^{2}-1+a^{7}z^{-1}+a^{5}z^{-1}-a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

3

6

4

1

-3

4

7

3

4

2

7

4

-3

0

9

7

2

-2

8

0

-4

5

8

-3

-6

4

8

4

-8

2

5

-3

-10

1

5

4

-12

1

-1

-14

1

Integral Khovanov Homology

(db, data source)

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