L11a34

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {4uv^{4}-8uv^{3}+8uv^{2}-5uv+2u+2v^{5}-5v^{4}+8v^{3}-8v^{2}+4v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {2}{q^{25/2}}}+{\frac {6}{q^{23/2}}}-{\frac {10}{q^{21/2}}}+{\frac {14}{q^{19/2}}}-{\frac {18}{q^{17/2}}}+{\frac {17}{q^{15/2}}}-{\frac {16}{q^{13/2}}}+{\frac {12}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-za^{13}-2a^{13}z^{-1}+3z^{3}a^{11}+7za^{11}+4a^{11}z^{-1}-2z^{5}a^{9}-4z^{3}a^{9}-2za^{9}-a^{9}z^{-1}-3z^{5}a^{7}-8z^{3}a^{7}-5za^{7}-a^{7}z^{-1}-z^{5}a^{5}-2z^{3}a^{5}-za^{5}$ (db)
Kauffman polynomial	$a^{16}z^{6}-4a^{16}z^{4}+5a^{16}z^{2}-2a^{16}+2a^{15}z^{7}-5a^{15}z^{5}+2a^{15}z^{3}+a^{15}z+3a^{14}z^{8}-6a^{14}z^{6}+2a^{14}z^{4}+a^{14}z^{2}-a^{14}+3a^{13}z^{9}-5a^{13}z^{7}+6a^{13}z^{5}-11a^{13}z^{3}+9a^{13}z-2a^{13}z^{-1}+a^{12}z^{10}+7a^{12}z^{8}-23a^{12}z^{6}+32a^{12}z^{4}-24a^{12}z^{2}+6a^{12}+7a^{11}z^{9}-13a^{11}z^{7}+17a^{11}z^{5}-22a^{11}z^{3}+16a^{11}z-4a^{11}z^{-1}+a^{10}z^{10}+10a^{10}z^{8}-25a^{10}z^{6}+28a^{10}z^{4}-18a^{10}z^{2}+5a^{10}+4a^{9}z^{9}-6a^{9}z^{5}+3a^{9}z^{3}+a^{9}z-a^{9}z^{-1}+6a^{8}z^{8}-6a^{8}z^{6}-2a^{8}z^{4}+3a^{8}z^{2}-a^{8}+6a^{7}z^{7}-11a^{7}z^{5}+10a^{7}z^{3}-6a^{7}z+a^{7}z^{-1}+3a^{6}z^{6}-4a^{6}z^{4}+a^{6}z^{2}+a^{5}z^{5}-2a^{5}z^{3}+a^{5}z$ (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

5

-10

7

3

-4

-12

9

5

4

-14

9

8

-1

-16

9

8

1

-18

5

9

4

-20

5

9

-4

-22

1

5

4

-24

1

5

-4

-26

1

-28

1

-1

Integral Khovanov Homology

(db, data source)

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