L11n144

L11n143

L11n145

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Link Presentations

[edit Notes on L11n144's Link Presentations]

Planar diagram presentation	X8192 X11,19,12,18 X3,10,4,11 X2,17,3,18 X12,5,13,6 X6718 X16,10,17,9 X20,16,21,15 X22,14,7,13 X14,22,15,21 X19,4,20,5
Gauss code	{1, -4, -3, 11, 5, -6}, {6, -1, 7, 3, -2, -5, 9, -10, 8, -7, 4, 2, -11, -8, 10, -9}

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} , ...)	$-{\frac {2t(1)t(2)^{4}-3t(1)t(2)^{3}+2t(2)^{3}-t(1)^{2}t(2)^{2}+3t(1)t(2)^{2}-t(2)^{2}+2t(1)^{2}t(2)-3t(1)t(2)+2t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {4}{q^{9/2}}}+{\frac {5}{q^{7/2}}}+q^{5/2}-{\frac {7}{q^{5/2}}}-2q^{3/2}+{\frac {6}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}+4{\sqrt {q}}-{\frac {6}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+2a^{5}z+a^{5}z^{-1}-a^{3}z^{5}-3a^{3}z^{3}-3a^{3}z-az^{5}-3az^{3}+z^{3}a^{-1}-4az-2az^{-1}+2za^{-1}+a^{-1}z^{-1}$ (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 -a^3 z^9-a z^9-2 a^4 z^8-3 a^2 z^8-z^8-3 a^5 z^7+2 a^3 z^7+5 a z^7-2 a^6 z^6+5 a^4 z^6+12 a^2 z^6+5 z^6-a^7 z^5+10 a^5 z^5-a^3 z^5-14 a z^5-2 z^5 a^{-1} +5 a^6 z^4-3 a^4 z^4-22 a^2 z^4-z^4 a^{-2} -15 z^4+3 a^7 z^3-11 a^5 z^3-4 a^3 z^3+14 a z^3+4 z^3 a^{-1} -a^6 z^2-2 a^4 z^2+13 a^2 z^2+3 z^2 a^{-2} +17 z^2-a^7 z+6 a^5 z+2 a^3 z-7 a z-2 z a^{-1} +a^4-3 a^2-2 a^{-2} -5-a^5 z^{-1} +2 a z^{-1} + a^{-1} z^{-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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                   1 -1 4                 1   1 2               3 1   -2 0             3 1     2 -2           4 4       0 -4         3 2         1 -6       2 4           2 -8     2 3             -1 -10     2               2 -12 1 2                 -1 -14 1                   1

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