L11n403

L11n402

L11n404

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Link Presentations

[edit Notes on L11n403's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,17,12,16 X7,20,8,21 X19,8,20,9 X21,15,22,14 X15,19,16,22 X13,5,14,18 X17,13,18,12 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, -6, 7}, {10, -1, -4, 5, 11, -2, -3, 9, -8, 6, -7, 3, -9, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(2)-1)(t(3)-1)\left(t(3)^{4}-t(1)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$-q^{5}+q^{-5}+q^{4}-q^{-4}+4q^{-3}-q^{2}-3q^{-2}+2q+4q^{-1}-2$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}-2a^{2}z^{4}+5z^{4}+a^{4}z^{2}-8a^{2}z^{2}-z^{2}a^{-4}+8z^{2}+3a^{4}-10a^{2}-a^{-4}+8+2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}+4z^{-2}$ (db)
Kauffman polynomial	$z^{5}a^{-5}-4z^{3}a^{-5}+2za^{-5}+a^{4}z^{8}-7a^{4}z^{6}+z^{6}a^{-4}+18a^{4}z^{4}-5z^{4}a^{-4}-21a^{4}z^{2}+4z^{2}a^{-4}-2a^{4}z^{-2}+11a^{4}-a^{-4}+a^{3}z^{9}-4a^{3}z^{7}-a^{3}z^{5}-z^{5}a^{-3}+17a^{3}z^{3}+z^{3}a^{-3}-18a^{3}z-2za^{-3}+5a^{3}z^{-1}+a^{-3}z^{-1}+5a^{2}z^{8}-30a^{2}z^{6}+58a^{2}z^{4}-3z^{4}a^{-2}-51a^{2}z^{2}+z^{2}a^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}+24a^{2}+2a^{-2}+az^{9}+4z^{7}a^{-1}-24az^{5}-25z^{5}a^{-1}+52az^{3}+40z^{3}a^{-1}-37az-23za^{-1}+9az^{-1}+5a^{-1}z^{-1}+4z^{8}-24z^{6}+42z^{4}-33z^{2}-4z^{-2}+17$ (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 4 5 χ 11                       1 -1 9                         0 7                 1 1 1   1 5               2 1       -1 3             2 2 1       1 1           3 4 1         0 -1         1 2 3           2 -3       2 3               1 -5     2 1                 1 -7   1 4                   3 -9                         0 -11 1                       1

Integral Khovanov Homology (db, data source)

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

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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