L11n273

L11n272

L11n274

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Link Presentations

[edit Notes on L11n273's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,7,17,8 X8,15,5,16 X11,19,12,18 X17,9,18,22 X21,13,22,12 X13,21,14,20 X19,15,20,14 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, -5, 7, -8, 9, 4, -3, -6, 5, -9, 8, -7, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2t(1)t(3)^{3}+2t(2)t(3)^{3}-2t(3)^{3}-3t(1)t(3)^{2}-3t(2)t(3)^{2}+t(3)^{2}+3t(1)t(3)-t(1)t(2)t(3)+3t(2)t(3)-2t(1)+2t(1)t(2)-2t(2)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-2q^{3}+3q^{2}-6q+9-8q^{-1}+9q^{-2}-6q^{-3}+6q^{-4}-2q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-3z^{2}a^{4}-a^{4}z^{-2}-4a^{4}+2z^{4}a^{2}+3z^{2}a^{2}-2a^{2}z^{-2}-a^{2}+2z^{4}+5z^{2}+3z^{-2}+7-2z^{2}a^{-2}-a^{-2}z^{-2}-3a^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-4a^{6}z^{4}+6a^{6}z^{2}+a^{6}z^{-2}-4a^{6}+2a^{5}z^{7}-5a^{5}z^{5}+4a^{5}z-2a^{5}z^{-1}+2a^{4}z^{8}-2a^{4}z^{6}-8a^{4}z^{4}+6a^{4}z^{2}+a^{4}z^{-2}-3a^{4}+a^{3}z^{9}+3a^{3}z^{7}-17a^{3}z^{5}+18a^{3}z^{3}+3z^{3}a^{-3}-10a^{3}z-7za^{-3}+2a^{3}z^{-1}+2a^{-3}z^{-1}+6a^{2}z^{8}-19a^{2}z^{6}+z^{6}a^{-2}+19a^{2}z^{4}-8a^{2}z^{2}-z^{2}a^{-2}-2a^{2}z^{-2}-a^{-2}z^{-2}+4a^{2}+a^{-2}+az^{9}+5az^{7}+4z^{7}a^{-1}-29az^{5}-17z^{5}a^{-1}+50az^{3}+35z^{3}a^{-1}-34az-27za^{-1}+10az^{-1}+8a^{-1}z^{-1}+4z^{8}-15z^{6}+23z^{4}-9z^{2}-3z^{-2}+5$ (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 7                   2 -2 5                 1   1 3               5 2   -3 1             4 1     3 -1           5 6       1 -3         4 4 1       1 -5       2 5           3 -7     4 4             0 -9   1 5               4 -11   1                 -1 -13 1                   1

Integral Khovanov Homology (db, data source)

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

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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