L11n40

L11n39

L11n41

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Link Presentations

[edit Notes on L11n40's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X9,18,10,19 X3849 X5,14,6,15 X15,20,16,21 X17,22,18,5 X21,16,22,17 X19,10,20,11 X11,2,12,3
Gauss code	{1, 11, -5, -3}, {-6, -1, 2, 5, -4, 10, -11, -2, 3, 6, -7, 9, -8, 4, -10, 7, -9, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2uv^{5}-3uv^{4}+3uv^{3}-2uv^{2}+uv+v^{4}-2v^{3}+3v^{2}-3v+2}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}-{\frac {1}{q^{7/2}}}+{\frac {2}{q^{25/2}}}-{\frac {4}{q^{23/2}}}+{\frac {6}{q^{21/2}}}-{\frac {7}{q^{19/2}}}+{\frac {7}{q^{17/2}}}-{\frac {7}{q^{15/2}}}+{\frac {5}{q^{13/2}}}-{\frac {4}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-za^{13}-a^{13}z^{-1}+z^{5}a^{11}+4z^{3}a^{11}+4za^{11}+a^{11}z^{-1}-z^{7}a^{9}-4z^{5}a^{9}-2z^{3}a^{9}+4za^{9}+2a^{9}z^{-1}-z^{7}a^{7}-6z^{5}a^{7}-12z^{3}a^{7}-9za^{7}-2a^{7}z^{-1}$ (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 -3 z^2 a^{16}+a^{16}-z^5 a^{15}-3 z^3 a^{15}+z a^{15}-3 z^6 a^{14}+3 z^4 a^{14}-2 z^2 a^{14}-4 z^7 a^{13}+8 z^5 a^{13}-4 z^3 a^{13}-z a^{13}+a^{13} z^{-1} -3 z^8 a^{12}+7 z^6 a^{12}-7 z^4 a^{12}+9 z^2 a^{12}-3 a^{12}-z^9 a^{11}-z^7 a^{11}+7 z^5 a^{11}+z^3 a^{11}-4 z a^{11}+a^{11} z^{-1} -4 z^8 a^{10}+13 z^6 a^{10}-9 z^4 a^{10}+z^2 a^{10}-z^9 a^9+2 z^7 a^9+4 z^5 a^9-10 z^3 a^9+7 z a^9-2 a^9 z^{-1} -z^8 a^8+3 z^6 a^8+z^4 a^8-7 z^2 a^8+3 a^8-z^7 a^7+6 z^5 a^7-12 z^3 a^7+9 z a^7-2 a^7 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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                   1 1 -8                 1 1 0 -10               3     3 -12             2 1     -1 -14           5 3       2 -16         3 3         0 -18       4 4           0 -20     2 3             1 -22   2 4               -2 -24   2                 2 -26 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}} $i=-8$ $i=-6$ $r=-9$ ${\mathbb {Z} }^{2}$ $r=-8$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-7$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ 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}} $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ 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}^{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 r=-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}\oplus{\mathbb Z}_2^{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}^{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 r=-2} ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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