L10n30

L10n29

L10n31

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n30 at Knotilus!

Link Presentations

[edit Notes on L10n30's Link Presentations]

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

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} , ...)	$-{\frac {t(1)t(2)^{5}-3t(1)t(2)^{4}+t(2)^{4}+4t(1)t(2)^{3}-2t(2)^{3}-2t(1)t(2)^{2}+4t(2)^{2}+t(1)t(2)-3t(2)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	${\frac {6}{q^{9/2}}}-{\frac {8}{q^{7/2}}}+{\frac {7}{q^{5/2}}}+q^{3/2}-{\frac {7}{q^{3/2}}}+{\frac {2}{q^{13/2}}}-{\frac {5}{q^{11/2}}}-3{\sqrt {q}}+{\frac {5}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+z^{5}a^{5}+4z^{3}a^{5}+6za^{5}+4a^{5}z^{-1}-z^{7}a^{3}-5z^{5}a^{3}-9z^{3}a^{3}-9za^{3}-4a^{3}z^{-1}+z^{5}a+3z^{3}a+2za+az^{-1}$ (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 -3 z^2 a^8+a^8-z^5 a^7-5 z^3 a^7+4 z a^7-a^7 z^{-1} -5 z^6 a^6+11 z^4 a^6-13 z^2 a^6+4 a^6-6 z^7 a^5+18 z^5 a^5-24 z^3 a^5+16 z a^5-4 a^5 z^{-1} -2 z^8 a^4-3 z^6 a^4+21 z^4 a^4-21 z^2 a^4+7 a^4-9 z^7 a^3+29 z^5 a^3-28 z^3 a^3+15 z a^3-4 a^3 z^{-1} -2 z^8 a^2+z^6 a^2+13 z^4 a^2-14 z^2 a^2+4 a^2-3 z^7 a+10 z^5 a-9 z^3 a+3 z a-a z^{-1} -z^6+3 z^4-3 z^2+1} (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   \ -5 -4 -3 -2 -1 0 1 2 3 χ 4                 1 -1 2               2   2 0             3 1   -2 -2           4 2     2 -4         4 4       0 -6       4 3         1 -8     2 4           2 -10   3 4             -1 -12   3               3 -14 2                 -2

Integral Khovanov Homology (db, data source)

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

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.

L10n29 L10n31