L10n39

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L10n40

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Link Presentations

[edit Notes on L10n39's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,10,15,9 X16,12,17,11 X10,16,11,15 X17,5,18,20 X7,19,8,18 X19,9,20,8 X2536 X4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, -7, 8, 3, -5, 4, -2, 10, -3, 5, -4, -6, 7, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(t(1)-1) (t(2)-1) \left(t(2)^2-t(2)+1\right) \left(t(2)^2+t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$-4 q^{9/2}+3 q^{7/2}-4 q^{5/2}+q^{3/2}-q^{17/2}+3 q^{15/2}-3 q^{13/2}+4 q^{11/2}-\sqrt{q}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$-z a^{-9} - a^{-9} z^{-1} +z^5 a^{-7} +5 z^3 a^{-7} +8 z a^{-7} +5 a^{-7} z^{-1} -z^7 a^{-5} -6 z^5 a^{-5} -13 z^3 a^{-5} -15 z a^{-5} -8 a^{-5} z^{-1} +z^5 a^{-3} +5 z^3 a^{-3} +8 z a^{-3} +4 a^{-3} z^{-1}$ (db)
Kauffman polynomial	$-z^8 a^{-4} -z^8 a^{-6} -z^7 a^{-3} -5 z^7 a^{-5} -4 z^7 a^{-7} +3 z^6 a^{-4} -2 z^6 a^{-6} -5 z^6 a^{-8} +6 z^5 a^{-3} +24 z^5 a^{-5} +16 z^5 a^{-7} -2 z^5 a^{-9} +3 z^4 a^{-4} +23 z^4 a^{-6} +20 z^4 a^{-8} -13 z^3 a^{-3} -35 z^3 a^{-5} -18 z^3 a^{-7} +4 z^3 a^{-9} -13 z^2 a^{-4} -33 z^2 a^{-6} -23 z^2 a^{-8} -3 z^2 a^{-10} +12 z a^{-3} +23 z a^{-5} +11 z a^{-7} -z a^{-9} -z a^{-11} +8 a^{-4} +14 a^{-6} +9 a^{-8} +2 a^{-10} -4 a^{-3} z^{-1} -8 a^{-5} z^{-1} -5 a^{-7} z^{-1} - a^{-9} z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 χ 18                 1 1 16               2   -2 14             2 2   0 12           3 2     -1 10         1 2 1     0 8       2 3         1 6     2 1           1 4   1 4             3 2                   0 0 1                 1

Integral Khovanov Homology (db, data source)

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

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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