L11a120

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {7uv^{2}-10uv+4u+4v^{3}-10v^{2}+7v}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	${\frac {9}{q^{9/2}}}-{\frac {6}{q^{7/2}}}+{\frac {3}{q^{5/2}}}-{\frac {1}{q^{3/2}}}+{\frac {1}{q^{25/2}}}-{\frac {2}{q^{23/2}}}+{\frac {5}{q^{21/2}}}-{\frac {8}{q^{19/2}}}+{\frac {11}{q^{17/2}}}-{\frac {13}{q^{15/2}}}+{\frac {12}{q^{13/2}}}-{\frac {13}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{13}z^{-1}+3za^{11}+a^{11}z^{-1}-2z^{3}a^{9}+2za^{9}+2a^{9}z^{-1}-5z^{3}a^{7}-6za^{7}-2a^{7}z^{-1}-3z^{3}a^{5}-za^{5}-z^{3}a^{3}$ (db)
Kauffman polynomial	$-z^{8}a^{14}+6z^{6}a^{14}-13z^{4}a^{14}+12z^{2}a^{14}-4a^{14}-2z^{9}a^{13}+10z^{7}a^{13}-16z^{5}a^{13}+8z^{3}a^{13}+a^{13}z^{-1}-z^{10}a^{12}-2z^{8}a^{12}+27z^{6}a^{12}-54z^{4}a^{12}+38z^{2}a^{12}-9a^{12}-7z^{9}a^{11}+26z^{7}a^{11}-22z^{5}a^{11}-2z^{3}a^{11}+2za^{11}+a^{11}z^{-1}-z^{10}a^{10}-11z^{8}a^{10}+54z^{6}a^{10}-65z^{4}a^{10}+26z^{2}a^{10}-4a^{10}-5z^{9}a^{9}+4z^{7}a^{9}+28z^{5}a^{9}-35z^{3}a^{9}+11za^{9}-2a^{9}z^{-1}-10z^{8}a^{8}+24z^{6}a^{8}-10z^{4}a^{8}-z^{2}a^{8}+2a^{8}-12z^{7}a^{7}+28z^{5}a^{7}-20z^{3}a^{7}+8za^{7}-2a^{7}z^{-1}-9z^{6}a^{6}+11z^{4}a^{6}-z^{2}a^{6}-6z^{5}a^{5}+4z^{3}a^{5}-za^{5}-3z^{4}a^{4}-z^{3}a^{3}$ (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

6

3

-3

-10

7

3

4

-12

6

7

1

-14

7

6

1

-16

4

6

2

-18

4

7

-3

-20

1

4

3

-22

1

4

-3

-24

1

-26

1

-1

Integral Khovanov Homology

(db, data source)

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