L9a15

L9a14

L9a16

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a15 at Knotilus!
	L9a15 is $9_{15}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a15's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,8,15,7 X18,12,5,11 X8,18,9,17 X16,10,17,9 X10,16,11,15 X2536 X4,14,1,13
Gauss code	{1, -8, 2, -9}, {8, -1, 3, -5, 6, -7, 4, -2, 9, -3, 7, -6, 5, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}-t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{17/2}-2q^{15/2}+4q^{13/2}-6q^{11/2}+7q^{9/2}-7q^{7/2}+5q^{5/2}-5q^{3/2}+2{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+2za^{-7}+a^{-7}z^{-1}-z^{5}a^{-5}-3z^{3}a^{-5}-4za^{-5}-2a^{-5}z^{-1}-z^{5}a^{-3}-2z^{3}a^{-3}+z^{3}a^{-1}+2za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-10}-2z^{2}a^{-10}+2z^{5}a^{-9}-3z^{3}a^{-9}+3z^{6}a^{-8}-6z^{4}a^{-8}+6z^{2}a^{-8}-2a^{-8}+3z^{7}a^{-7}-7z^{5}a^{-7}+10z^{3}a^{-7}-4za^{-7}+a^{-7}z^{-1}+z^{8}a^{-6}+3z^{6}a^{-6}-11z^{4}a^{-6}+15z^{2}a^{-6}-5a^{-6}+5z^{7}a^{-5}-12z^{5}a^{-5}+13z^{3}a^{-5}-7za^{-5}+2a^{-5}z^{-1}+z^{8}a^{-4}+2z^{6}a^{-4}-8z^{4}a^{-4}+7z^{2}a^{-4}-3a^{-4}+2z^{7}a^{-3}-2z^{5}a^{-3}-3z^{3}a^{-3}+2z^{6}a^{-2}-4z^{4}a^{-2}+a^{-2}+z^{5}a^{-1}-3z^{3}a^{-1}+3za^{-1}-a^{-1}z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 χ 18                   1 -1 16                 1   1 14               3 1   -2 12             3 1     2 10           4 3       -1 8         3 3         0 6       2 4           2 4     3 3             0 2   1 4               3 0   1                 -1 -2 1                   1

Integral Khovanov Homology (db, data source)

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

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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