L9n24

L9n23

L9n25

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n24 at Knotilus!
	L9n24 is 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 9^3_{14}} in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n24's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X11,16,12,17 X9,11,10,18 X17,5,18,10 X15,1,16,4
Gauss code	{1, -4, -3, 9}, {-2, -1, 5, 3, -7, 8}, {-6, 2, 4, -5, -9, 6, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uvw^{2}-2uvw-uw^{2}+uw+v^{2}(-w)+v^{2}+2vw-v}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$2q^{2}-2q+4-3q^{-1}+4q^{-2}-2q^{-3}+2q^{-4}-q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{4}-a^{4}+z^{4}a^{2}+3z^{2}a^{2}+a^{2}z^{-2}+4a^{2}-3z^{2}-2z^{-2}-5+a^{-2}z^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$a^{5}z^{5}-3a^{5}z^{3}+a^{5}z+2a^{4}z^{6}-7a^{4}z^{4}+6a^{4}z^{2}-2a^{4}+a^{3}z^{7}-a^{3}z^{5}-4a^{3}z^{3}+3a^{3}z+4a^{2}z^{6}-14a^{2}z^{4}+17a^{2}z^{2}+3z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-8a^{2}-4a^{-2}+az^{7}-az^{5}+z^{5}a^{-1}-2az^{3}-z^{3}a^{-1}+5az+3za^{-1}-2az^{-1}-2a^{-1}z^{-1}+2z^{6}-7z^{4}+14z^{2}+2z^{-2}-9$ (db)

Khovanov Homology

The coefficients of the monomials 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 t^rq^j} are shown, along with their alternating sums 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 \chi} (fixed $j$, alternation over $r$).

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 χ 5               2 2 3             1 1 0 1           3 1   2 -1         2 3     1 -3       2 1       1 -5     1 3         2 -7   1 1           0 -9   1             1 -11 1               -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ 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}} 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 r=-3} ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ 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}^{2}} $r=-1$ 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}\oplus{\mathbb Z}_2^{2}} ${\mathbb {Z} }^{2}$ 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 r=0} ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ 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}^{3}} $r=1$ 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}\oplus{\mathbb Z}_2} ${\mathbb {Z} }$ 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 r=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}\oplus{\mathbb Z}_2} ${\mathbb {Z} }^{2}$

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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