L11a315

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a315 at Knotilus!
	[edit L11a315 Quick Notes]

[edit L11a315 Further Notes and Views]

Link Presentations

[edit Notes on L11a315's Link Presentations]

Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_22,17,9,18 X_20,9,21,10 X_6,13,7,14 X_14,7,15,8 X_8,15,1,16 X_4,19,5,20 X_18,5,19,6 X_16,21,17,22
Gauss code	{1, -2, 3, -9, 10, -6, 7, -8}, {5, -1, 2, -3, 6, -7, 8, -11, 4, -10, 9, -5, 11, -4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2u^{3}v^{4}-2u^{3}v^{3}+u^{3}v^{2}+u^{2}v^{5}-4u^{2}v^{4}+7u^{2}v^{3}-5u^{2}v^{2}+2u^{2}v+2uv^{4}-5uv^{3}+7uv^{2}-4uv+u+v^{3}-2v^{2}+2v}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	${\frac {2}{q^{9/2}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{29/2}}}-{\frac {3}{q^{27/2}}}+{\frac {6}{q^{25/2}}}-{\frac {11}{q^{23/2}}}+{\frac {14}{q^{21/2}}}-{\frac {15}{q^{19/2}}}+{\frac {15}{q^{17/2}}}-{\frac {13}{q^{15/2}}}+{\frac {9}{q^{13/2}}}-{\frac {6}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-z^{3}a^{13}-3za^{13}-a^{13}z^{-1}+3z^{5}a^{11}+12z^{3}a^{11}+13za^{11}+3a^{11}z^{-1}-2z^{7}a^{9}-10z^{5}a^{9}-16z^{3}a^{9}-10za^{9}-2a^{9}z^{-1}-z^{7}a^{7}-5z^{5}a^{7}-8z^{3}a^{7}-4za^{7}$ (db)
Kauffman polynomial	$a^{18}z^{4}-a^{18}z^{2}+3a^{17}z^{5}-3a^{17}z^{3}+a^{17}z+5a^{16}z^{6}-4a^{16}z^{4}+a^{16}z^{2}+7a^{15}z^{7}-9a^{15}z^{5}+5a^{15}z^{3}+a^{15}z+7a^{14}z^{8}-11a^{14}z^{6}+8a^{14}z^{4}-3a^{14}z^{2}+a^{14}+4a^{13}z^{9}-13a^{13}z^{5}+9a^{13}z^{3}+a^{13}z-a^{13}z^{-1}+a^{12}z^{10}+10a^{12}z^{8}-35a^{12}z^{6}+39a^{12}z^{4}-22a^{12}z^{2}+3a^{12}+7a^{11}z^{9}-19a^{11}z^{7}+20a^{11}z^{5}-24a^{11}z^{3}+16a^{11}z-3a^{11}z^{-1}+a^{10}z^{10}+5a^{10}z^{8}-26a^{10}z^{6}+32a^{10}z^{4}-17a^{10}z^{2}+3a^{10}+3a^{9}z^{9}-11a^{9}z^{7}+16a^{9}z^{5}-17a^{9}z^{3}+11a^{9}z-2a^{9}z^{-1}+2a^{8}z^{8}-7a^{8}z^{6}+6a^{8}z^{4}+a^{7}z^{7}-5a^{7}z^{5}+8a^{7}z^{3}-4a^{7}z$ (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

-6

1

-8

2

1

-1

-10

4

-12

5

2

-3

-14

8

4

-16

7

5

-2

-18

8

0

-20

6

7

1

-22

5

8

-3

-24

2

7

5

-26

1

4

-3

-28

2

-30

1

-1

Integral Khovanov Homology

(db, data source)

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