L11a175

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_20,12,21,11 X_14,6,15,5 X_4,21,5,22 X_16,7,17,8 X_22,15,7,16 X_18,14,19,13 X_6,18,1,17 X_12,20,13,19
Gauss code	{1, -2, 3, -6, 5, -10}, {7, -1, 2, -3, 4, -11, 9, -5, 8, -7, 10, -9, 11, -4, 6, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}-4t(1)^{2}t(2)^{3}+5t(1)t(2)^{3}-2t(2)^{3}+4t(1)^{2}t(2)^{2}-5t(1)t(2)^{2}+4t(2)^{2}-2t(1)^{2}t(2)+5t(1)t(2)-4t(2)-2t(1)+2}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	${\frac {9}{q^{9/2}}}+q^{7/2}-{\frac {12}{q^{7/2}}}-3q^{5/2}+{\frac {13}{q^{5/2}}}+5q^{3/2}-{\frac {13}{q^{3/2}}}-{\frac {1}{q^{15/2}}}+{\frac {3}{q^{13/2}}}-{\frac {6}{q^{11/2}}}-9{\sqrt {q}}+{\frac {11}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{3}z^{7}-az^{7}+a^{5}z^{5}-4a^{3}z^{5}-4az^{5}+z^{5}a^{-1}+3a^{5}z^{3}-5a^{3}z^{3}-4az^{3}+3z^{3}a^{-1}+2a^{5}z-4a^{3}z+za^{-1}+a^{5}z^{-1}-2a^{3}z^{-1}+2az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-6a^{3}z^{9}-9az^{9}-3z^{9}a^{-1}-10a^{4}z^{8}-4a^{2}z^{8}-z^{8}a^{-2}+5z^{8}-11a^{5}z^{7}+11a^{3}z^{7}+38az^{7}+16z^{7}a^{-1}-9a^{6}z^{6}+25a^{4}z^{6}+36a^{2}z^{6}+5z^{6}a^{-2}+7z^{6}-6a^{7}z^{5}+23a^{5}z^{5}+12a^{3}z^{5}-45az^{5}-28z^{5}a^{-1}-3a^{8}z^{4}+12a^{6}z^{4}-14a^{4}z^{4}-43a^{2}z^{4}-8z^{4}a^{-2}-22z^{4}-a^{9}z^{3}+4a^{7}z^{3}-12a^{5}z^{3}-22a^{3}z^{3}+12az^{3}+17z^{3}a^{-1}-3a^{6}z^{2}+4a^{4}z^{2}+13a^{2}z^{2}+4z^{2}a^{-2}+10z^{2}-a^{7}z+4a^{5}z+10a^{3}z+4az-za^{-1}-a^{2}-a^{5}z^{-1}-2a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

8

1

-1

6

2

4

3

1

-2

2

6

2

4

0

5

3

-2

8

6

2

-4

6

0

-6

6

7

-1

-8

4

7

3

-10

2

5

-3

-12

1

4

3

-14

2

-2

-16

1

Integral Khovanov Homology

(db, data source)

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