L11a73

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_20,13,21,14 X_16,7,17,8 X_8,19,9,20 X_18,9,19,10 X_10,17,11,18 X_22,15,5,16 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, 3, -9, 8, -4, 7, -6, 5, -3, 9, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(2)^{5}+4t(1)t(2)^{4}-5t(2)^{4}-7t(1)t(2)^{3}+7t(2)^{3}+7t(1)t(2)^{2}-7t(2)^{2}-5t(1)t(2)+4t(2)+2t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {10}{q^{11/2}}}-{\frac {15}{q^{13/2}}}+{\frac {16}{q^{15/2}}}-{\frac {16}{q^{17/2}}}+{\frac {13}{q^{19/2}}}-{\frac {10}{q^{21/2}}}+{\frac {6}{q^{23/2}}}-{\frac {2}{q^{25/2}}}+{\frac {1}{q^{27/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}(-z)-2a^{13}z^{-1}+3a^{11}z^{3}+7a^{11}z+4a^{11}z^{-1}-2a^{9}z^{5}-4a^{9}z^{3}-a^{9}z-a^{9}z^{-1}-3a^{7}z^{5}-9a^{7}z^{3}-7a^{7}z-a^{7}z^{-1}-a^{5}z^{5}-2a^{5}z^{3}$ (db)
Kauffman polynomial	$a^{16}z^{6}-4a^{16}z^{4}+5a^{16}z^{2}-2a^{16}+2a^{15}z^{7}-5a^{15}z^{5}+2a^{15}z^{3}+a^{15}z+3a^{14}z^{8}-6a^{14}z^{6}+a^{14}z^{4}+2a^{14}z^{2}-a^{14}+3a^{13}z^{9}-6a^{13}z^{7}+9a^{13}z^{5}-16a^{13}z^{3}+9a^{13}z-2a^{13}z^{-1}+a^{12}z^{10}+6a^{12}z^{8}-22a^{12}z^{6}+32a^{12}z^{4}-25a^{12}z^{2}+6a^{12}+7a^{11}z^{9}-18a^{11}z^{7}+29a^{11}z^{5}-31a^{11}z^{3}+17a^{11}z-4a^{11}z^{-1}+a^{10}z^{10}+9a^{10}z^{8}-30a^{10}z^{6}+41a^{10}z^{4}-22a^{10}z^{2}+5a^{10}+4a^{9}z^{9}-4a^{9}z^{7}-a^{9}z^{5}+4a^{9}z^{3}+2a^{9}z-a^{9}z^{-1}+6a^{8}z^{8}-12a^{8}z^{6}+9a^{8}z^{4}-a^{8}+6a^{7}z^{7}-15a^{7}z^{5}+15a^{7}z^{3}-7a^{7}z+a^{7}z^{-1}+3a^{6}z^{6}-5a^{6}z^{4}+a^{5}z^{5}-2a^{5}z^{3}$ (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

-4

1

-6

3

1

-2

-8

4

-10

6

3

-3

-12

9

4

5

-14

8

7

-1

-16

8

0

-18

5

8

3

-20

5

8

-3

-22

1

5

4

-24

1

5

-4

-26

1

-28

1

-1

Integral Khovanov Homology

(db, data source)

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