L11a76

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {4t(2)^{3}+6t(1)t(2)^{2}-9t(2)^{2}-9t(1)t(2)+6t(2)+4t(1)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {8}{q^{9/2}}}-{\frac {11}{q^{11/2}}}+{\frac {11}{q^{13/2}}}-{\frac {12}{q^{15/2}}}+{\frac {9}{q^{17/2}}}-{\frac {7}{q^{19/2}}}+{\frac {5}{q^{21/2}}}-{\frac {2}{q^{23/2}}}+{\frac {1}{q^{25/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{13}z^{-1}+3za^{11}+a^{11}z^{-1}-2z^{3}a^{9}+za^{9}+2a^{9}z^{-1}-4z^{3}a^{7}-4za^{7}-2a^{7}z^{-1}-3z^{3}a^{5}-2za^{5}-z^{3}a^{3}$ (db)
Kauffman polynomial	$a^{14}z^{8}-6a^{14}z^{6}+13a^{14}z^{4}-12a^{14}z^{2}+4a^{14}+2a^{13}z^{9}-10a^{13}z^{7}+15a^{13}z^{5}-6a^{13}z^{3}-a^{13}z^{-1}+a^{12}z^{10}+a^{12}z^{8}-23a^{12}z^{6}+48a^{12}z^{4}-34a^{12}z^{2}+9a^{12}+6a^{11}z^{9}-23a^{11}z^{7}+20a^{11}z^{5}+3a^{11}z^{3}-3a^{11}z-a^{11}z^{-1}+a^{10}z^{10}+7a^{10}z^{8}-38a^{10}z^{6}+46a^{10}z^{4}-19a^{10}z^{2}+4a^{10}+4a^{9}z^{9}-4a^{9}z^{7}-19a^{9}z^{5}+26a^{9}z^{3}-12a^{9}z+2a^{9}z^{-1}+7a^{8}z^{8}-13a^{8}z^{6}-2a^{8}z^{4}+5a^{8}z^{2}-2a^{8}+9a^{7}z^{7}-18a^{7}z^{5}+11a^{7}z^{3}-7a^{7}z+2a^{7}z^{-1}+8a^{6}z^{6}-10a^{6}z^{4}+2a^{6}z^{2}+6a^{5}z^{5}-5a^{5}z^{3}+2a^{5}z+3a^{4}z^{4}+a^{3}z^{3}$ (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

3

1

-2

-6

3

-8

5

3

-2

-10

6

3

-12

6

0

-14

6

5

1

-16

3

6

3

-18

4

6

-2

-20

1

3

2

-22

1

4

-3

-24

1

-26

1

-1

Integral Khovanov Homology

(db, data source)

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