L11n273

L11n272

L11n274

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Link Presentations

[edit Notes on L11n273's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,7,17,8 X8,15,5,16 X11,19,12,18 X17,9,18,22 X21,13,22,12 X13,21,14,20 X19,15,20,14 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, -5, 7, -8, 9, 4, -3, -6, 5, -9, 8, -7, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{2 t(1) t(3)^3+2 t(2) t(3)^3-2 t(3)^3-3 t(1) t(3)^2-3 t(2) t(3)^2+t(3)^2+3 t(1) t(3)-t(1) t(2) t(3)+3 t(2) t(3)-2 t(1)+2 t(1) t(2)-2 t(2)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db)
Jones polynomial	$-2 q^3+3 q^2-6 q+9-8 q^{-1} +9 q^{-2} -6 q^{-3} +6 q^{-4} -2 q^{-5} + q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^6 z^{-2} +a^6-3 z^2 a^4-a^4 z^{-2} -4 a^4+2 z^4 a^2+3 z^2 a^2-2 a^2 z^{-2} -a^2+2 z^4+5 z^2+3 z^{-2} +7-2 z^2 a^{-2} - a^{-2} z^{-2} -3 a^{-2}$ (db)
Kauffman polynomial	$a^6 z^6-4 a^6 z^4+6 a^6 z^2+a^6 z^{-2} -4 a^6+2 a^5 z^7-5 a^5 z^5+4 a^5 z-2 a^5 z^{-1} +2 a^4 z^8-2 a^4 z^6-8 a^4 z^4+6 a^4 z^2+a^4 z^{-2} -3 a^4+a^3 z^9+3 a^3 z^7-17 a^3 z^5+18 a^3 z^3+3 z^3 a^{-3} -10 a^3 z-7 z a^{-3} +2 a^3 z^{-1} +2 a^{-3} z^{-1} +6 a^2 z^8-19 a^2 z^6+z^6 a^{-2} +19 a^2 z^4-8 a^2 z^2-z^2 a^{-2} -2 a^2 z^{-2} - a^{-2} z^{-2} +4 a^2+ a^{-2} +a z^9+5 a z^7+4 z^7 a^{-1} -29 a z^5-17 z^5 a^{-1} +50 a z^3+35 z^3 a^{-1} -34 a z-27 z a^{-1} +10 a z^{-1} +8 a^{-1} z^{-1} +4 z^8-15 z^6+23 z^4-9 z^2-3 z^{-2} +5$ (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 3 χ 7                   2 -2 5                 1   1 3               5 2   -3 1             4 1     3 -1           5 6       1 -3         4 4 1       1 -5       2 5           3 -7     4 4             0 -9   1 5               4 -11   1                 -1 -13 1                   1

Integral Khovanov Homology (db, data source)

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

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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