L11a154

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(1)t(2)^{4}+2t(1)^{2}t(2)^{3}-6t(1)t(2)^{3}+3t(2)^{3}-4t(1)^{2}t(2)^{2}+9t(1)t(2)^{2}-4t(2)^{2}+3t(1)^{2}t(2)-6t(1)t(2)+2t(2)+2t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {5}{q^{9/2}}}-3q^{7/2}+{\frac {8}{q^{7/2}}}+6q^{5/2}-{\frac {12}{q^{5/2}}}-9q^{3/2}+{\frac {13}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}+12{\sqrt {q}}-{\frac {14}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{5}+2za^{5}+a^{5}z^{-1}-z^{5}a^{3}-2z^{3}a^{3}-2za^{3}-2z^{5}a-5z^{3}a-5za-2az^{-1}-z^{5}a^{-1}-z^{3}a^{-1}+za^{-1}+a^{-1}z^{-1}+z^{3}a^{-3}+za^{-3}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-3a^{3}z^{9}-6az^{9}-3z^{9}a^{-1}-3a^{4}z^{8}-4a^{2}z^{8}-4z^{8}a^{-2}-5z^{8}-3a^{5}z^{7}+7a^{3}z^{7}+18az^{7}+5z^{7}a^{-1}-3z^{7}a^{-3}-2a^{6}z^{6}+4a^{4}z^{6}+17a^{2}z^{6}+11z^{6}a^{-2}-z^{6}a^{-4}+23z^{6}-a^{7}z^{5}+6a^{5}z^{5}-12a^{3}z^{5}-27az^{5}+z^{5}a^{-1}+9z^{5}a^{-3}+4a^{6}z^{4}-a^{4}z^{4}-28a^{2}z^{4}-8z^{4}a^{-2}+3z^{4}a^{-4}-34z^{4}+3a^{7}z^{3}-5a^{5}z^{3}+7a^{3}z^{3}+21az^{3}-6z^{3}a^{-3}-a^{6}z^{2}-2a^{4}z^{2}+15a^{2}z^{2}+5z^{2}a^{-2}-2z^{2}a^{-4}+23z^{2}-2a^{7}z+4a^{5}z-a^{3}z-10az-2za^{-1}+za^{-3}+a^{4}-3a^{2}-2a^{-2}-5-a^{5}z^{-1}+2az^{-1}+a^{-1}z^{-1}$ (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

2

6

4

1

-3

4

5

2

3

2

7

4

-3

0

7

5

2

-2

7

8

1

-4

5

6

-1

-6

3

7

4

-8

2

5

-3

-10

3

-12

1

2

-1

-14

1

Integral Khovanov Homology

(db, data source)

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