L11n188

L11n187

L11n189

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Link Presentations

[edit Notes on L11n188's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X16,6,17,5 X22,13,7,14 X18,15,19,16 X14,21,15,22 X9,20,10,21 X4,18,5,17 X19,10,20,11 X2738 X6,11,1,12
Gauss code	{1, -10, 2, -8, 3, -11}, {10, -1, -7, 9, 11, -2, 4, -6, 5, -3, 8, -5, -9, 7, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, 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} , 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(2)^4-t(2)^4+2 t(1)^2 t(2)^3-5 t(1) t(2)^3+2 t(2)^3-3 t(1)^2 t(2)^2+7 t(1) t(2)^2-3 t(2)^2+2 t(1)^2 t(2)-5 t(1) t(2)+2 t(2)-t(1)^2+t(1)}{t(1) t(2)^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 -\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{12}{q^{7/2}}+\frac{12}{q^{9/2}}-\frac{11}{q^{11/2}}+\frac{8}{q^{13/2}}-\frac{5}{q^{15/2}}+\frac{2}{q^{17/2}}} (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 -2 a^7 z^3-3 a^7 z-a^7 z^{-1} +2 a^5 z^5+6 a^5 z^3+7 a^5 z+3 a^5 z^{-1} +a^3 z^5-4 a^3 z-2 a^3 z^{-1} -a z^3-a z} (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 a^{10} z^4-4 a^{10} z^2+a^9 z^7+3 a^9 z^5-6 a^9 z^3+a^9 z+2 a^8 z^8-2 a^8 z^4+2 a^8 z^2-a^8+a^7 z^9+5 a^7 z^7-12 a^7 z^5+12 a^7 z^3-5 a^7 z+a^7 z^{-1} +6 a^6 z^8-7 a^6 z^6-3 a^6 z^4+10 a^6 z^2-3 a^6+a^5 z^9+9 a^5 z^7-26 a^5 z^5+28 a^5 z^3-14 a^5 z+3 a^5 z^{-1} +4 a^4 z^8-4 a^4 z^6-3 a^4 z^4+5 a^4 z^2-3 a^4+5 a^3 z^7-10 a^3 z^5+8 a^3 z^3-7 a^3 z+2 a^3 z^{-1} +3 a^2 z^6-5 a^2 z^4+a^2 z^2+a z^5-2 a z^3+a z} (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 $j$, 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                   1 1 0                 2   -2 -2               5 1   4 -4             5 3     -2 -6           7 4       3 -8         6 6         0 -10       5 6           -1 -12     3 6             3 -14   2 5               -3 -16   3                 3 -18 2                   -2

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb {Z} }^{2}$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=0$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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