L11a189

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2t(1)t(2)^{4}-2t(2)^{4}+2t(1)^{2}t(2)^{3}-7t(1)t(2)^{3}+3t(2)^{3}-3t(1)^{2}t(2)^{2}+9t(1)t(2)^{2}-3t(2)^{2}+3t(1)^{2}t(2)-7t(1)t(2)+2t(2)-2t(1)^{2}+2t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	${\frac {15}{q^{9/2}}}-{\frac {14}{q^{7/2}}}+{\frac {10}{q^{5/2}}}-{\frac {7}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {2}{q^{19/2}}}+{\frac {5}{q^{17/2}}}-{\frac {9}{q^{15/2}}}+{\frac {12}{q^{13/2}}}-{\frac {15}{q^{11/2}}}-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{9}-2za^{9}-a^{9}z^{-1}+z^{5}a^{7}+2z^{3}a^{7}+3za^{7}+2a^{7}z^{-1}+2z^{5}a^{5}+4z^{3}a^{5}+2za^{5}+z^{5}a^{3}-3za^{3}-a^{3}z^{-1}-z^{3}a-za$ (db)
Kauffman polynomial	$-z^{6}a^{12}+4z^{4}a^{12}-4z^{2}a^{12}-2z^{7}a^{11}+6z^{5}a^{11}-4z^{3}a^{11}-3z^{8}a^{10}+9z^{6}a^{10}-11z^{4}a^{10}+9z^{2}a^{10}-2a^{10}-3z^{9}a^{9}+9z^{7}a^{9}-17z^{5}a^{9}+20z^{3}a^{9}-7za^{9}+a^{9}z^{-1}-z^{10}a^{8}-4z^{8}a^{8}+17z^{6}a^{8}-28z^{4}a^{8}+22z^{2}a^{8}-5a^{8}-6z^{9}a^{7}+15z^{7}a^{7}-22z^{5}a^{7}+19z^{3}a^{7}-10za^{7}+2a^{7}z^{-1}-z^{10}a^{6}-6z^{8}a^{6}+17z^{6}a^{6}-20z^{4}a^{6}+9z^{2}a^{6}-3a^{6}-3z^{9}a^{5}-z^{7}a^{5}+11z^{5}a^{5}-13z^{3}a^{5}+3za^{5}-5z^{8}a^{4}+7z^{6}a^{4}-2z^{4}a^{4}-z^{2}a^{4}+a^{4}-5z^{7}a^{3}+9z^{5}a^{3}-6z^{3}a^{3}+5za^{3}-a^{3}z^{-1}-3z^{6}a^{2}+5z^{4}a^{2}-z^{2}a^{2}-z^{5}a+2z^{3}a-za$ (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

6

3

-3

-6

8

4

-8

8

7

-1

-10

7

0

-12

5

8

3

-14

4

7

-3

-16

1

5

4

-18

1

4

-3

-20

1

-22

1

-1

Integral Khovanov Homology

(db, data source)

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