L11a37

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_18,15,19,16 X_16,7,17,8 X_8,17,9,18 X_22,11,5,12 X_20,13,21,14 X_14,19,15,20 X_12,21,13,22 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 6, -9, 7, -8, 3, -4, 5, -3, 8, -7, 9, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {3t(2)^{3}+4t(1)t(2)^{2}-6t(2)^{2}-6t(1)t(2)+4t(2)+3t(1)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{3/2}}}+{\frac {2}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {5}{q^{9/2}}}-{\frac {7}{q^{11/2}}}+{\frac {7}{q^{13/2}}}-{\frac {8}{q^{15/2}}}+{\frac {6}{q^{17/2}}}-{\frac {5}{q^{19/2}}}+{\frac {4}{q^{21/2}}}-{\frac {2}{q^{23/2}}}+{\frac {1}{q^{25/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{13}z^{-1}+3za^{11}+2a^{11}z^{-1}-2z^{3}a^{9}-2za^{9}-a^{9}z^{-1}-2z^{3}a^{7}+a^{7}z^{-1}-2z^{3}a^{5}-2za^{5}-a^{5}z^{-1}-z^{3}a^{3}-za^{3}$ (db)
Kauffman polynomial	$-z^{8}a^{14}+6z^{6}a^{14}-12z^{4}a^{14}+10z^{2}a^{14}-3a^{14}-2z^{9}a^{13}+11z^{7}a^{13}-18z^{5}a^{13}+10z^{3}a^{13}-3za^{13}+a^{13}z^{-1}-z^{10}a^{12}+z^{8}a^{12}+16z^{6}a^{12}-39z^{4}a^{12}+28z^{2}a^{12}-7a^{12}-5z^{9}a^{11}+25z^{7}a^{11}-38z^{5}a^{11}+23z^{3}a^{11}-8za^{11}+2a^{11}z^{-1}-z^{10}a^{10}-z^{8}a^{10}+19z^{6}a^{10}-31z^{4}a^{10}+18z^{2}a^{10}-4a^{10}-3z^{9}a^{9}+11z^{7}a^{9}-14z^{5}a^{9}+13z^{3}a^{9}-5za^{9}+a^{9}z^{-1}-3z^{8}a^{8}+6z^{6}a^{8}-3z^{7}a^{7}+3z^{5}a^{7}+4z^{3}a^{7}-4za^{7}+a^{7}z^{-1}-3z^{6}a^{6}+2z^{4}a^{6}+z^{2}a^{6}-a^{6}-3z^{5}a^{5}+3z^{3}a^{5}-3za^{5}+a^{5}z^{-1}-2z^{4}a^{4}+z^{2}a^{4}-z^{3}a^{3}+za^{3}$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

1

-4

2

1

-1

-6

2

-8

3

2

-1

-10

4

2

-12

4

0

-14

4

3

1

-16

2

4

2

-18

3

4

-1

-20

1

2

1

-22

1

3

-2

-24

1

-26

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-11$	${\mathbb {Z} }$
$r=-10$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-9$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-7$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{4}$
$r=-3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-1$	${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$