L11a191

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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_20,11,21,12 X_22,13,7,14 X_12,21,13,22 X_18,15,19,16 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, 5, -8, 7, -3, 9, -7, 8, -4, 6, -5}

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

Polynomial invariants

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

1

-1

-2

4

1

3

-4

5

2

-3

-6

7

3

4

-8

7

6

-1

-10

7

6

1

-12

5

7

2

-14

4

7

-3

-16

2

5

3

-18

1

4

-3

-20

2

-22

1

-1

Integral Khovanov Homology

(db, data source)

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