L10n63

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L10n64

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Link Presentations

[edit Notes on L10n63's Link Presentations]

Planar diagram presentation	X12,1,13,2 X7,17,8,16 X3948 X17,2,18,3 X5,14,6,15 X11,6,12,7 X9,18,10,19 X15,11,16,20 X10,13,1,14 X19,5,20,4
Gauss code	{1, 4, -3, 10, -5, 6, -2, 3, -7, -9}, {-6, -1, 9, 5, -8, 2, -4, 7, -10, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(u v+1) \left(u^2 v^2-2 u^2 v+u^2-2 u v^2+3 u v-2 u+v^2-2 v+1\right)}{u^{3/2} v^{3/2}}$ (db)
Jones polynomial	$-q^{7/2}+4 q^{5/2}-7 q^{3/2}+9 \sqrt{q}-\frac{11}{\sqrt{q}}+\frac{10}{q^{3/2}}-\frac{9}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{3}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^5 z+a^5 z^{-1} -a^3 z^5-3 a^3 z^3-3 a^3 z-a^3 z^{-1} +a z^7+4 a z^5-z^5 a^{-1} +4 a z^3-2 z^3 a^{-1}$ (db)
Kauffman polynomial	$-3 a^2 z^8-3 z^8-6 a^3 z^7-12 a z^7-6 z^7 a^{-1} -3 a^4 z^6-a^2 z^6-4 z^6 a^{-2} -2 z^6+13 a^3 z^5+26 a z^5+12 z^5 a^{-1} -z^5 a^{-3} +6 a^2 z^4+7 z^4 a^{-2} +13 z^4-6 a^5 z^3-18 a^3 z^3-18 a z^3-5 z^3 a^{-1} +z^3 a^{-3} -3 a^2 z^2-2 z^2 a^{-2} -5 z^2+6 a^5 z+9 a^3 z+4 a z+z a^{-1} +a^4-a^5 z^{-1} -a^3 z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 χ 8                 1 1 6               3   -3 4             4 1   3 2           5 3     -2 0         6 4       2 -2       5 6         1 -4     4 5           -1 -6   2 5             3 -8 1 4               -3 -10 3                 3

Integral Khovanov Homology (db, data source)

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

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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