L9n27

L9n26

L9n28

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n27 at Knotilus!
	L9n27 is $9_{21}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n27's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X9,18,10,15 X8493 X5,17,6,16 X17,5,18,14 X15,10,16,11 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 $u$, $v$, $w$, ...)	0 (db)
Jones polynomial	$q^{3}-q^{2}+q+1+q^{-1}+q^{-2}+q^{-4}-q^{-5}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-z^{2}a^{4}-2a^{4}+z^{4}a^{2}+5z^{2}a^{2}+a^{2}z^{-2}+6a^{2}-z^{4}-5z^{2}-2z^{-2}-6+z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$az^{7}+z^{7}a^{-1}+a^{4}z^{6}+2a^{2}z^{6}+z^{6}a^{-2}+2z^{6}+a^{5}z^{5}+a^{3}z^{5}-5az^{5}-5z^{5}a^{-1}-5a^{4}z^{4}-13a^{2}z^{4}-5z^{4}a^{-2}-13z^{4}-4a^{5}z^{3}-6a^{3}z^{3}+2az^{3}+4z^{3}a^{-1}+6a^{4}z^{2}+22a^{2}z^{2}+6z^{2}a^{-2}+22z^{2}+2a^{5}z+6a^{3}z+6az+2za^{-1}-4a^{4}-12a^{2}-4a^{-2}-11-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                   1 1 5                     0 3               1 1   0 1           3 1       2 -1         1 4 1       2 -3       1 1 2         2 -5       1             1 -7   1 1 1             1 -9                     0 -11 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }^{4}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=3$ ${\mathbb {Z} }$ $r=4$ ${\mathbb {Z} }_{2}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\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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