L9a13

L9a12

L9a14

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a13 at Knotilus!
	L9a13 is $9_{16}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a13's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X18,13,5,14 X14,7,15,8 X8,17,9,18 X16,9,17,10 X10,15,11,16 X2536 X4,11,1,12
Gauss code	{1, -8, 2, -9}, {8, -1, 4, -5, 6, -7, 9, -2, 3, -4, 7, -6, 5, -3}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in 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 u} , 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 v} , $w$, ...)	$-{\frac {2t(2)^{3}+4t(1)t(2)^{2}-5t(2)^{2}-5t(1)t(2)+4t(2)+2t(1)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones 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 -\frac{1}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{8}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{4}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{21/2}}} (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+3a^{9}z+2a^{9}z^{-1}-2a^{7}z^{3}-a^{7}z-3a^{5}z^{3}-4a^{5}z-a^{5}z^{-1}-a^{3}z^{3}$ (db)
Kauffman polynomial	$-z^{6}a^{12}+4z^{4}a^{12}-5z^{2}a^{12}+2a^{12}-2z^{7}a^{11}+7z^{5}a^{11}-7z^{3}a^{11}+2za^{11}-a^{11}z^{-1}-z^{8}a^{10}-2z^{6}a^{10}+15z^{4}a^{10}-17z^{2}a^{10}+5a^{10}-6z^{7}a^{9}+16z^{5}a^{9}-12z^{3}a^{9}+6za^{9}-2a^{9}z^{-1}-z^{8}a^{8}-7z^{6}a^{8}+21z^{4}a^{8}-13z^{2}a^{8}+3a^{8}-4z^{7}a^{7}+3z^{5}a^{7}+3z^{3}a^{7}-6z^{6}a^{6}+7z^{4}a^{6}-z^{2}a^{6}-a^{6}-6z^{5}a^{5}+7z^{3}a^{5}-4za^{5}+a^{5}z^{-1}-3z^{4}a^{4}-z^{3}a^{3}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                   1 1 -4                 3 1 -2 -6               3     3 -8             3 3     0 -10           5 3       2 -12         3 4         1 -14       3 4           -1 -16     1 3             2 -18   1 3               -2 -20   1                 1 -22 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-9$ 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}} $r=-8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ 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=-7} ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ 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=-6} ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ 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=-4} ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=-3$ 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}\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} ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ 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}} 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=-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^{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}} $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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.

L9a12 L9a14