L9n28

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L9n27

L10a1

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n28 at Knotilus!
	L9n28 is $9_{20}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n28's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X18,10,15,9 X8493 X5,17,6,16 X17,5,18,14 X10,16,11,15 X2,12,3,11
Gauss code	{1, -9, 5, -3}, {8, 6, -7, -4}, {-6, -1, 2, -5, 4, -8, 9, -2, 3, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(w-1)^{2}(vw+1)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$3q^{5}-4q^{4}+6q^{3}-5q^{2}+6q-4+3q^{-1}-q^{-2}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+4z^{4}a^{-2}-z^{4}a^{-4}-z^{4}+5z^{2}a^{-2}-3z^{2}a^{-4}-2z^{2}+3a^{-2}-4a^{-4}+a^{-6}+a^{-2}z^{-2}-2a^{-4}z^{-2}+a^{-6}z^{-2}$ (db)
Kauffman polynomial	$2z^{7}a^{-1}+2z^{7}a^{-3}+8z^{6}a^{-2}+5z^{6}a^{-4}+3z^{6}+az^{5}-z^{5}a^{-1}+z^{5}a^{-3}+3z^{5}a^{-5}-21z^{4}a^{-2}-13z^{4}a^{-4}-8z^{4}-2az^{3}-7z^{3}a^{-1}-8z^{3}a^{-3}-3z^{3}a^{-5}+17z^{2}a^{-2}+18z^{2}a^{-4}+6z^{2}a^{-6}+5z^{2}+az+3za^{-1}+7za^{-3}+5za^{-5}-7a^{-2}-10a^{-4}-5a^{-6}-1-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-2}z^{-2}+2a^{-4}z^{-2}+a^{-6}z^{-2}$ (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   \ -3 -2 -1 0 1 2 3 4 χ 11               3 3 9             3 2 -1 7           3 1   2 5         2 3     1 3       4 3       1 1     2 4         2 -1   1 2           -1 -3   2             2 -5 1               -1

Integral Khovanov Homology (db, data source)

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

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.

L9n27 L10a1