L11a75

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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_18,9,19,10 X_8,17,9,18 X_10,19,11,20 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, -6, 5, -7, 11, -2, 8, -9, 3, -4, 6, -5, 7, -3, 9, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(2)^{5}+3t(1)t(2)^{4}-4t(2)^{4}-4t(1)t(2)^{3}+4t(2)^{3}+4t(1)t(2)^{2}-4t(2)^{2}-4t(1)t(2)+3t(2)+2t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {4}{q^{9/2}}}+{\frac {2}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {2}{q^{25/2}}}+{\frac {5}{q^{23/2}}}-{\frac {7}{q^{21/2}}}+{\frac {9}{q^{19/2}}}-{\frac {11}{q^{17/2}}}+{\frac {10}{q^{15/2}}}-{\frac {10}{q^{13/2}}}+{\frac {6}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-za^{13}-2a^{13}z^{-1}+3z^{3}a^{11}+8za^{11}+4a^{11}z^{-1}-2z^{5}a^{9}-6z^{3}a^{9}-4za^{9}-a^{9}z^{-1}-2z^{5}a^{7}-6z^{3}a^{7}-4za^{7}-a^{7}z^{-1}-z^{5}a^{5}-3z^{3}a^{5}-za^{5}$ (db)
Kauffman polynomial	$-z^{6}a^{16}+4z^{4}a^{16}-5z^{2}a^{16}+2a^{16}-2z^{7}a^{15}+6z^{5}a^{15}-3z^{3}a^{15}-za^{15}-2z^{8}a^{14}+3z^{6}a^{14}+5z^{4}a^{14}-5z^{2}a^{14}+a^{14}-2z^{9}a^{13}+5z^{7}a^{13}-7z^{5}a^{13}+13z^{3}a^{13}-8za^{13}+2a^{13}z^{-1}-z^{10}a^{12}+7z^{6}a^{12}-17z^{4}a^{12}+19z^{2}a^{12}-6a^{12}-5z^{9}a^{11}+21z^{7}a^{11}-40z^{5}a^{11}+34z^{3}a^{11}-15za^{11}+4a^{11}z^{-1}-z^{10}a^{10}-z^{8}a^{10}+13z^{6}a^{10}-29z^{4}a^{10}+18z^{2}a^{10}-5a^{10}-3z^{9}a^{9}+11z^{7}a^{9}-17z^{5}a^{9}+6z^{3}a^{9}-2za^{9}+a^{9}z^{-1}-3z^{8}a^{8}+8z^{6}a^{8}-6z^{4}a^{8}-2z^{2}a^{8}+a^{8}-3z^{7}a^{7}+9z^{5}a^{7}-9z^{3}a^{7}+5za^{7}-a^{7}z^{-1}-2z^{6}a^{6}+5z^{4}a^{6}-z^{2}a^{6}-z^{5}a^{5}+3z^{3}a^{5}-za^{5}$ (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

2

1

-1

-8

2

-10

4

2

-2

-12

6

2

4

-14

5

0

-16

6

5

1

-18

3

5

2

-20

4

6

-2

-22

1

3

2

-24

1

4

-3

-26

1

-28

1

-1

Integral Khovanov Homology

(db, data source)

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