L11n438

L11n437

L11n439

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n438 at Knotilus!

Link Presentations

[edit Notes on L11n438's Link Presentations]

Planar diagram presentation	X6172 X2536 X11,19,12,18 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X19,13,20,22 X15,21,16,20 X21,17,22,16 X17,9,18,12
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 11}, {-7, 6, -9, 10, -11, 3, -8, 9, -10, 8}

A Braid Representative
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) t(4)^3-t(1) t(2) t(4)^3+t(3) t(4)^3-t(4)^3-2 t(1) t(4)^2+3 t(1) t(2) t(4)^2-t(2) t(4)^2+t(1) t(3) t(4)^2-2 t(1) t(2) t(3) t(4)^2+2 t(2) t(3) t(4)^2-3 t(3) t(4)^2+2 t(4)^2+2 t(1) t(4)-3 t(1) t(2) t(4)+t(2) t(4)-t(1) t(3) t(4)+2 t(1) t(2) t(3) t(4)-2 t(2) t(3) t(4)+3 t(3) t(4)-2 t(4)+t(1) t(2)-t(1) t(2) t(3)+t(2) t(3)-t(3)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} t(4)^{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 11 q^{9/2}-13 q^{7/2}+6 q^{5/2}-4 q^{3/2}+q^{21/2}-2 q^{19/2}+6 q^{17/2}-10 q^{15/2}+12 q^{13/2}-15 q^{11/2}} (db)
Signature	3 (db)
HOMFLY-PT 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 -3 z^5 a^{-5} +4 z^3 a^{-3} -12 z^3 a^{-5} +6 z^3 a^{-7} +9 z a^{-3} -22 z a^{-5} +17 z a^{-7} -4 z a^{-9} +7 a^{-3} z^{-1} -20 a^{-5} z^{-1} +20 a^{-7} z^{-1} -8 a^{-9} z^{-1} + a^{-11} z^{-1} +2 a^{-3} z^{-3} -7 a^{-5} z^{-3} +9 a^{-7} z^{-3} -5 a^{-9} z^{-3} + a^{-11} z^{-3} } (db)
Kauffman polynomial	$z^{6}a^{-12}-4z^{4}a^{-12}+6z^{2}a^{-12}+a^{-12}z^{-2}-4a^{-12}+2z^{7}a^{-11}-5z^{5}a^{-11}+4z^{3}a^{-11}-a^{-11}z^{-3}-2za^{-11}+2a^{-11}z^{-1}+2z^{8}a^{-10}+2z^{6}a^{-10}-21z^{4}a^{-10}+34z^{2}a^{-10}+7a^{-10}z^{-2}-24a^{-10}+z^{9}a^{-9}+8z^{7}a^{-9}-25z^{5}a^{-9}+22z^{3}a^{-9}-5a^{-9}z^{-3}-16za^{-9}+12a^{-9}z^{-1}+7z^{8}a^{-8}-46z^{4}a^{-8}+75z^{2}a^{-8}+18a^{-8}z^{-2}-58a^{-8}+z^{9}a^{-7}+16z^{7}a^{-7}-46z^{5}a^{-7}+50z^{3}a^{-7}-9a^{-7}z^{-3}-37za^{-7}+24a^{-7}z^{-1}+5z^{8}a^{-6}+5z^{6}a^{-6}-41z^{4}a^{-6}+73z^{2}a^{-6}+19a^{-6}z^{-2}-60a^{-6}+10z^{7}a^{-5}-26z^{5}a^{-5}+42z^{3}a^{-5}-7a^{-5}z^{-3}-39za^{-5}+23a^{-5}z^{-1}+6z^{6}a^{-4}-12z^{4}a^{-4}+26z^{2}a^{-4}+7a^{-4}z^{-2}-23a^{-4}+10z^{3}a^{-3}-2a^{-3}z^{-3}-16za^{-3}+9a^{-3}z^{-1}$ (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 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 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   \ 0 1 2 3 4 5 6 7 8 9 χ 22                   1 -1 20                 1   1 18               5 1   -4 16             5 1     4 14           7 5       -2 12         8 5         3 10       7 11           4 8     6 4             2 6     7               7 4 4 6                 -2 2 4                   4

Integral Khovanov Homology (db, data source)

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 \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} $i=2$ 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 i=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=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}^{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=1} ${\mathbb {Z} }^{6}$ 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 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}^{7}\oplus{\mathbb Z}_2^{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}^{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 r=3} ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ 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}^{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 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}^{11}\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}^{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 r=5} ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ 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}^{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 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}^{5}\oplus{\mathbb Z}_2^{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}^{5}} $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ 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=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}_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}}

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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