L9a25

L9a24

L9a26

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a25 at Knotilus!
	L9a25 is $9_{8}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a25's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X14,6,15,5 X18,11,7,12 X16,13,17,14 X12,17,13,18 X4,16,5,15 X2738 X6,9,1,10
Gauss code	{1, -8, 2, -7, 3, -9}, {8, -1, 9, -2, 4, -6, 5, -3, 7, -5, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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