L10n71

L10n70

L10n72

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Link Presentations

[edit Notes on L10n71's Link Presentations]

Planar diagram presentation	X6172 X3,12,4,13 X7,17,8,16 X9,11,10,20 X11,18,12,19 X15,9,16,8 X19,5,20,10 X17,14,18,15 X2536 X13,4,14,1
Gauss code	{1, -9, -2, 10}, {9, -1, -3, 6, -4, 7}, {-5, 2, -10, 8, -6, 3, -8, 5, -7, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{t(1) t(3)^2 t(2)^2+t(1) t(2)^2-2 t(1) t(3) t(2)^2-t(2)^2-2 t(1) t(3)^2 t(2)+2 t(1) t(3) t(2)-2 t(3) t(2)+2 t(2)+t(1) t(3)^2-t(3)^2+2 t(3)-1}{\sqrt{t(1)} t(2) t(3)}$ (db)
Jones polynomial	$q^3-2 q^2+5 q-4+7 q^{-1} -6 q^{-2} +5 q^{-3} -4 q^{-4} +2 q^{-5}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^6-a^4 z^4-3 a^4 z^2-2 a^4+a^2 z^6+4 a^2 z^4+5 a^2 z^2+a^2 z^{-2} +z^2 a^{-2} + a^{-2} z^{-2} +3 a^2+2 a^{-2} -2 z^4-6 z^2-2 z^{-2} -4$ (db)
Kauffman polynomial	$a^2 z^8+z^8+3 a^3 z^7+5 a z^7+2 z^7 a^{-1} +3 a^4 z^6+4 a^2 z^6+z^6 a^{-2} +2 z^6+a^5 z^5-6 a^3 z^5-13 a z^5-6 z^5 a^{-1} -5 a^4 z^4-17 a^2 z^4-4 z^4 a^{-2} -16 z^4+3 a^5 z^3+7 a^3 z^3+6 a z^3+2 z^3 a^{-1} +3 a^6 z^2+6 a^4 z^2+14 a^2 z^2+6 z^2 a^{-2} +17 z^2-2 a^5 z-6 a^3 z+4 z a^{-1} -a^6-a^4-3 a^2-4 a^{-2} -6-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 χ 7                 1 1 5               1   -1 3             4 1   3 1           2 3     1 -1         5 2       3 -3       2 3         1 -5     3 4           -1 -7   1 2             1 -9 1 3               -2 -11 2                 2

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-4$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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