L9n24

L9n23

L9n25

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n24 at Knotilus!
	L9n24 is $9_{14}^{3}$ 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	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 a^5 z^5-3 a^5 z^3+a^5 z+2 a^4 z^6-7 a^4 z^4+6 a^4 z^2-2 a^4+a^3 z^7-a^3 z^5-4 a^3 z^3+3 a^3 z+4 a^2 z^6-14 a^2 z^4+17 a^2 z^2+3 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -8 a^2-4 a^{-2} +a z^7-a z^5+z^5 a^{-1} -2 a z^3-z^3 a^{-1} +5 a z+3 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +2 z^6-7 z^4+14 z^2+2 z^{-2} -9} (db)

Khovanov Homology

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

\   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}} $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ 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}^{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}} $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$

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.

L9n23 L9n25