L9a24

L9a23

L9a25

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a24 at Knotilus!
	L9a24 is $9_{21}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a24's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X14,6,15,5 X16,11,17,12 X18,13,7,14 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, -4, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, 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 w} , ...)	${\frac {u^{2}v^{3}-u^{2}v^{2}+u^{2}v-u^{2}+uv^{4}-3uv^{3}+3uv^{2}-3uv+u-v^{4}+v^{3}-v^{2}+v}{uv^{2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {2}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {6}{q^{9/2}}}-{\frac {6}{q^{11/2}}}+{\frac {4}{q^{13/2}}}-{\frac {2}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{7}-2za^{7}-a^{7}z^{-1}+z^{5}a^{5}+3z^{3}a^{5}+4za^{5}+3a^{5}z^{-1}+z^{5}a^{3}+2z^{3}a^{3}-za^{3}-2a^{3}z^{-1}-z^{3}a-2za$ (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^{10} z^4-2 a^{10} z^2+2 a^9 z^5-3 a^9 z^3+3 a^8 z^6-6 a^8 z^4+5 a^8 z^2-a^8+3 a^7 z^7-8 a^7 z^5+13 a^7 z^3-7 a^7 z+a^7 z^{-1} +a^6 z^8+2 a^6 z^6-8 a^6 z^4+11 a^6 z^2-3 a^6+5 a^5 z^7-15 a^5 z^5+22 a^5 z^3-14 a^5 z+3 a^5 z^{-1} +a^4 z^8+a^4 z^6-6 a^4 z^4+6 a^4 z^2-3 a^4+2 a^3 z^7-4 a^3 z^5+3 a^3 z^3-5 a^3 z+2 a^3 z^{-1} +2 a^2 z^6-5 a^2 z^4+2 a^2 z^2+a z^5-3 a z^3+2 a z} (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                   1 1 0                 1   -1 -2               3 1   2 -4             3 2     -1 -6           4 2       2 -8         3 4         1 -10       3 3           0 -12     1 3             2 -14   1 3               -2 -16   1                 1 -18 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ 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=-7} 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}} $r=-6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ 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}} $r=-5$ 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} ${\mathbb {Z} }$ $r=-4$ 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^{3}} ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ 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} ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=-1$ 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}^{2}\oplus{\mathbb Z}_2^{3}} 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}} $r=0$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ 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} ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L9a23 L9a25