L9n15

L9n14

L9n16

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

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[edit L9n15 Quick Notes]

This is a Seifert-fibered link: it is the trefoil plus the core of one solid torus.

L9n15 is $9_{49}^{2}$ in the Rolfsen table of links.

[edit L9n15 Further Notes and Views]

The Triune sculpture in Philadelpdia by Robert Engman, showing a Seifert surface for L9n15. Photo by sameold2008

Link Presentations

[edit Notes on L9n15's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_16,11,17,12 X_3,10,4,11 X_2,15,3,16 X_12,5,13,6 X₆₇₁₈ X_9,14,10,15 X_13,18,14,7 X_17,4,18,5
Gauss code	{1, -4, -3, 9, 5, -6}, {6, -1, -7, 3, 2, -5, -8, 7, 4, -2, -9, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(2)^{4}+t(1)t(2)^{2}+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{11}(-z)-a^{11}z^{-1}+a^{9}z^{5}+6a^{9}z^{3}+9a^{9}z+3a^{9}z^{-1}-a^{7}z^{7}-7a^{7}z^{5}-15a^{7}z^{3}-11a^{7}z-2a^{7}z^{-1}$ (db)
Kauffman polynomial	$a^{12}+za^{11}-a^{11}z^{-1}-z^{6}a^{10}+6z^{4}a^{10}-9z^{2}a^{10}+3a^{10}-z^{7}a^{9}+7z^{5}a^{9}-15z^{3}a^{9}+12za^{9}-3a^{9}z^{-1}-z^{6}a^{8}+6z^{4}a^{8}-9z^{2}a^{8}+3a^{8}-z^{7}a^{7}+7z^{5}a^{7}-15z^{3}a^{7}+11za^{7}-2a^{7}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

χ

-6

1

-8

1

-10

1

-12

1

-14

1

0

-16

1

0

-18

1

0

Integral Khovanov Homology

(db, data source)

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