L9n15

L9n14

L9n16

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n15 at Knotilus!
	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.

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 X16,11,17,12 X3,10,4,11 X2,15,3,16 X12,5,13,6 X6718 X9,14,10,15 X13,18,14,7 X17,4,18,5
Gauss code	{1, -4, -3, 9, 5, -6}, {6, -1, -7, 3, 2, -5, -8, 7, 4, -2, -9, 8}

A Braid Representative
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 1 -8             1 1 -10         1     1 -12     1         1 -14     1 1       0 -16 1 1           0 -18 1 1           0

Integral Khovanov Homology (db, data source)

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

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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