L11a185

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2u^{2}v^{3}-6u^{2}v^{2}+4u^{2}v-u^{2}+2uv^{4}-8uv^{3}+11uv^{2}-8uv+2u-v^{4}+4v^{3}-6v^{2}+2v}{uv^{2}}}$ (db)
Jones polynomial	${\frac {18}{q^{9/2}}}-{\frac {16}{q^{7/2}}}+{\frac {11}{q^{5/2}}}-{\frac {7}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {4}{q^{19/2}}}+{\frac {7}{q^{17/2}}}-{\frac {12}{q^{15/2}}}+{\frac {16}{q^{13/2}}}-{\frac {18}{q^{11/2}}}-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{3}\right)+a^{9}z^{-1}+a^{7}z^{5}-2a^{7}z-2a^{7}z^{-1}+2a^{5}z^{5}+3a^{5}z^{3}+2a^{5}z+2a^{5}z^{-1}+a^{3}z^{5}-2a^{3}z-a^{3}z^{-1}-az^{3}-az$ (db)
Kauffman polynomial	$-z^{6}a^{12}+2z^{4}a^{12}-4z^{7}a^{11}+11z^{5}a^{11}-7z^{3}a^{11}-6z^{8}a^{10}+16z^{6}a^{10}-12z^{4}a^{10}+3z^{2}a^{10}-4z^{9}a^{9}+2z^{7}a^{9}+15z^{5}a^{9}-18z^{3}a^{9}+7za^{9}-a^{9}z^{-1}-z^{10}a^{8}-11z^{8}a^{8}+32z^{6}a^{8}-28z^{4}a^{8}+9z^{2}a^{8}-7z^{9}a^{7}+5z^{7}a^{7}+16z^{5}a^{7}-25z^{3}a^{7}+12za^{7}-2a^{7}z^{-1}-z^{10}a^{6}-10z^{8}a^{6}+22z^{6}a^{6}-18z^{4}a^{6}+7z^{2}a^{6}-a^{6}-3z^{9}a^{5}-6z^{7}a^{5}+21z^{5}a^{5}-22z^{3}a^{5}+10za^{5}-2a^{5}z^{-1}-5z^{8}a^{4}+4z^{6}a^{4}+z^{4}a^{4}-z^{2}a^{4}-5z^{7}a^{3}+8z^{5}a^{3}-6z^{3}a^{3}+4za^{3}-a^{3}z^{-1}-3z^{6}a^{2}+5z^{4}a^{2}-2z^{2}a^{2}-z^{5}a+2z^{3}a-za$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

2

-2

5

1

4

-4

7

3

-4

-6

9

4

5

-8

9

7

-2

-10

9

0

-12

7

9

2

-14

5

9

-4

-16

3

8

5

-18

1

4

-3

-20

3

-22

1

-1

Integral Khovanov Homology

(db, data source)

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