L11n72

L11n71

L11n73

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n72 at Knotilus!

Link Presentations

[edit Notes on L11n72's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,16,8,17 X22,18,5,17 X18,12,19,11 X12,22,13,21 X20,14,21,13 X14,20,15,19 X15,8,16,9 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, 5, -6, 7, -8, -9, 3, 4, -5, 8, -7, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2uv^{3}-5uv^{2}+6uv-2u-2v^{3}+6v^{2}-5v+2}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+5q^{5/2}-8q^{3/2}+10{\sqrt {q}}-{\frac {10}{\sqrt {q}}}+{\frac {9}{q^{3/2}}}-{\frac {8}{q^{5/2}}}+{\frac {4}{q^{7/2}}}-{\frac {2}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{-1}+a^{3}z^{3}+z^{3}a^{-3}-a^{3}z-a^{3}z^{-1}+za^{-3}-az^{5}-z^{5}a^{-1}-az^{3}-2z^{3}a^{-1}-2za^{-1}$ (db)
Kauffman polynomial	$3a^{5}z^{3}-3a^{5}z+a^{5}z^{-1}+a^{4}z^{6}+z^{6}a^{-4}+2a^{4}z^{4}-3z^{4}a^{-4}+a^{4}z^{2}+2z^{2}a^{-4}-a^{4}+2a^{3}z^{7}+3z^{7}a^{-3}+a^{3}z^{5}-10z^{5}a^{-3}+9z^{3}a^{-3}-a^{3}z-za^{-3}+a^{3}z^{-1}+2a^{2}z^{8}+3z^{8}a^{-2}+a^{2}z^{6}-7z^{6}a^{-2}-2a^{2}z^{4}+2z^{4}a^{-2}+z^{2}a^{-2}+az^{9}+z^{9}a^{-1}+3az^{7}+4z^{7}a^{-1}-4az^{5}-15z^{5}a^{-1}-3az^{3}+9z^{3}a^{-1}+2az-za^{-1}+5z^{8}-8z^{6}+z^{4}-2z^{2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 2   2 6               3 1   -2 4             5 2     3 2           5 3       -2 0         5 5         0 -2       5 6           1 -4     3 4             -1 -6   1 5               4 -8 1 3                 -2 -10 2                   2

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}} 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=-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=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 r=-4} ${\mathbb {Z} }^{2}$ 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}} $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=0$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

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