L11a235

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a235 at Knotilus!
	[edit L11a235 Quick Notes]

[edit L11a235 Further Notes and Views]

Link Presentations

[edit Notes on L11a235's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_20,13,21,14 X_16,9,17,10 X_10,19,11,20 X_22,15,7,16 X_14,21,15,22 X_18,5,19,6 X₂₇₃₈ X_4,11,5,12 X_6,17,1,18
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 4, -5, 10, -2, 3, -7, 6, -4, 11, -8, 5, -3, 7, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(2)^{6}-3t(1)t(2)^{5}+3t(2)^{5}-3t(1)^{2}t(2)^{4}+6t(1)t(2)^{4}-4t(2)^{4}+4t(1)^{2}t(2)^{3}-7t(1)t(2)^{3}+4t(2)^{3}-4t(1)^{2}t(2)^{2}+6t(1)t(2)^{2}-3t(2)^{2}+3t(1)^{2}t(2)-3t(1)t(2)-t(1)^{2}}{t(1)t(2)^{3}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {11}{q^{11/2}}}-{\frac {16}{q^{13/2}}}+{\frac {17}{q^{15/2}}}-{\frac {18}{q^{17/2}}}+{\frac {15}{q^{19/2}}}-{\frac {11}{q^{21/2}}}+{\frac {7}{q^{23/2}}}-{\frac {3}{q^{25/2}}}+{\frac {1}{q^{27/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}(-z)-2a^{13}z^{-1}+4a^{11}z^{3}+10a^{11}z+5a^{11}z^{-1}-3a^{9}z^{5}-9a^{9}z^{3}-8a^{9}z-3a^{9}z^{-1}-3a^{7}z^{5}-8a^{7}z^{3}-4a^{7}z-a^{5}z^{5}-2a^{5}z^{3}$ (db)
Kauffman polynomial	$a^{16}z^{6}-3a^{16}z^{4}+3a^{16}z^{2}-a^{16}+3a^{15}z^{7}-8a^{15}z^{5}+5a^{15}z^{3}+4a^{14}z^{8}-7a^{14}z^{6}-2a^{14}z^{4}+4a^{14}z^{2}+4a^{13}z^{9}-7a^{13}z^{7}+5a^{13}z^{5}-10a^{13}z^{3}+7a^{13}z-2a^{13}z^{-1}+2a^{12}z^{10}+2a^{12}z^{8}-12a^{12}z^{6}+17a^{12}z^{4}-17a^{12}z^{2}+5a^{12}+10a^{11}z^{9}-31a^{11}z^{7}+51a^{11}z^{5}-47a^{11}z^{3}+22a^{11}z-5a^{11}z^{-1}+2a^{10}z^{10}+5a^{10}z^{8}-23a^{10}z^{6}+36a^{10}z^{4}-21a^{10}z^{2}+5a^{10}+6a^{9}z^{9}-15a^{9}z^{7}+23a^{9}z^{5}-18a^{9}z^{3}+11a^{9}z-3a^{9}z^{-1}+7a^{8}z^{8}-16a^{8}z^{6}+15a^{8}z^{4}-3a^{8}z^{2}+6a^{7}z^{7}-14a^{7}z^{5}+12a^{7}z^{3}-4a^{7}z+3a^{6}z^{6}-5a^{6}z^{4}+a^{5}z^{5}-2a^{5}z^{3}$ (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

7

3

-4

-12

9

4

5

-14

8

7

-1

-16

10

9

1

-18

6

9

3

-20

5

9

-4

-22

2

6

4

-24

1

5

-4

-26

2

-28

1

-1

Integral Khovanov Homology

(db, data source)

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