L11n412

L11n411

L11n413

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n412 at Knotilus!

Link Presentations

[edit Notes on L11n412's Link Presentations]

Planar diagram presentation	X8192 X5,15,6,14 X10,3,11,4 X13,5,14,4 X2738 X6,9,1,10 X11,18,12,19 X17,12,18,7 X15,20,16,21 X19,22,20,13 X21,16,22,17
Gauss code	{1, -5, 3, 4, -2, -6}, {5, -1, 6, -3, -7, 8}, {-4, 2, -9, 11, -8, 7, -10, 9, -11, 10}

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} , $v$, 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} , ...)	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{t(1)^2 t(3)^3-2 t(1)^2 t(3)^2+t(1) t(2)^2 t(3)^2-2 t(2)^2 t(3)^2+t(1) t(3)^2+t(1)^2 t(2) t(3)^2-2 t(1) t(2) t(3)^2+t(2) t(3)^2+2 t(1)^2 t(3)-t(1) t(2)^2 t(3)+2 t(2)^2 t(3)-t(1) t(3)-t(1)^2 t(2) t(3)+2 t(1) t(2) t(3)-t(2) t(3)-t(2)^2}{t(1) t(2) t(3)^{3/2}}} (db)
Jones polynomial	$-q^{-10}+2q^{-9}-4q^{-8}+6q^{-7}-6q^{-6}+8q^{-5}-6q^{-4}+6q^{-3}-3q^{-2}+2q^{-1}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{10}z^{-2}-a^{10}+3z^{2}a^{8}+4a^{8}z^{-2}+6a^{8}-2z^{4}a^{6}-6z^{2}a^{6}-5a^{6}z^{-2}-9a^{6}-z^{4}a^{4}+2a^{4}z^{-2}+2a^{4}+2z^{2}a^{2}+2a^{2}$ (db)
Kauffman polynomial	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a^{11}z^{7}-5a^{11}z^{5}+8a^{11}z^{3}-5a^{11}z+a^{11}z^{-1}+2a^{10}z^{8}-9a^{10}z^{6}+12a^{10}z^{4}-7a^{10}z^{2}-a^{10}z^{-2}+4a^{10}+a^{9}z^{9}+a^{9}z^{7}-20a^{9}z^{5}+35a^{9}z^{3}-21a^{9}z+5a^{9}z^{-1}+6a^{8}z^{8}-26a^{8}z^{6}+38a^{8}z^{4}-32a^{8}z^{2}-4a^{8}z^{-2}+17a^{8}+a^{7}z^{9}+4a^{7}z^{7}-28a^{7}z^{5}+44a^{7}z^{3}-33a^{7}z+9a^{7}z^{-1}+4a^{6}z^{8}-15a^{6}z^{6}+26a^{6}z^{4}-32a^{6}z^{2}-5a^{6}z^{-2}+20a^{6}+4a^{5}z^{7}-12a^{5}z^{5}+18a^{5}z^{3}-16a^{5}z+5a^{5}z^{-1}+2a^{4}z^{6}-4a^{4}z^{2}-2a^{4}z^{-2}+6a^{4}+a^{3}z^{5}+a^{3}z^{3}+a^{3}z+3a^{2}z^{2}-2a^{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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -1                   2 2 -3                 3 2 -1 -5               3     3 -7             4 4     0 -9           4 2       2 -11         2 4         2 -13       4 4           0 -15     1 3             2 -17   1 3               -2 -19   1                 1 -21 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ 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} 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=-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}^{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}} 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} 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}} 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}^{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 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}^{4}\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}^{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=-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}^{4}\oplus{\mathbb Z}_2^{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}^{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 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}^{2}\oplus{\mathbb Z}_2^{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}^{4}} $r=-2$ 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}^{4}\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}} 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}} 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} 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}} 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}}

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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