L10n92

L10n91

L10n93

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Link Presentations

[edit Notes on L10n92's Link Presentations]

Planar diagram presentation	X8192 X18,10,19,9 X17,6,18,1 X7,17,8,16 X3,10,4,11 X5,14,6,15 X13,4,14,5 X11,13,12,20 X15,7,16,12 X2,19,3,20
Gauss code	{1, -10, -5, 7, -6, 3}, {-4, -1, 2, 5, -8, 9}, {-7, 6, -9, 4, -3, -2, 10, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{t(1) t(2)^2 t(3)^3-t(1) t(2)^2 t(3)^2+t(2)^2 t(3)^2+t(1) t(2) t(3)^2-t(1)^2 t(3)+t(1) t(3)-t(1) t(2) t(3)-t(1)}{t(1) t(2) t(3)^{3/2}}$ (db)
Jones polynomial	$q^{-6} - q^{-5} +2 q^{-4} -2 q^{-3} +q^2+3 q^{-2} -q-2 q^{-1} +3$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^4 z^4+4 a^4 z^2+a^4 z^{-2} +4 a^4-a^2 z^6-6 a^2 z^4-12 a^2 z^2-2 a^2 z^{-2} -9 a^2+z^4+4 z^2+ z^{-2} +5$ (db)
Kauffman polynomial	$a^6 z^6-5 a^6 z^4+6 a^6 z^2-a^6+a^5 z^7-4 a^5 z^5+3 a^5 z^3+a^4 z^8-5 a^4 z^6+9 a^4 z^4-10 a^4 z^2-a^4 z^{-2} +5 a^4+2 a^3 z^7-9 a^3 z^5+13 a^3 z^3-9 a^3 z+2 a^3 z^{-1} +a^2 z^8-6 a^2 z^6+16 a^2 z^4-21 a^2 z^2+z^2 a^{-2} -2 a^2 z^{-2} +11 a^2- a^{-2} +a z^7-5 a z^5+11 a z^3+z^3 a^{-1} -9 a z+2 a z^{-1} +2 z^4-4 z^2- z^{-2} +5$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 5                 1 1 3               1 1 0 1             2     2 -1           2 3     1 -3         1         1 -5       1 2         1 -7     1 1           0 -9     1             1 -11 1 1               0 -13 1                 1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}$ ${\mathbb Z}$

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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