L11n293

L11n292

L11n294

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Link Presentations

[edit Notes on L11n293's Link Presentations]

Planar diagram presentation	X6172 X11,18,12,19 X8493 X2,16,3,15 X16,7,17,8 X9,11,10,22 X4,17,1,18 X19,5,20,10 X5,12,6,13 X21,15,22,14 X13,21,14,20
Gauss code	{1, -4, 3, -7}, {-9, -1, 5, -3, -6, 8}, {-2, 9, -11, 10, 4, -5, 7, 2, -8, 11, -10, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(1)-1) (t(2)-1) (t(3)-1)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)}}$ (db)
Jones polynomial	$q^6-3 q^5+4 q^4-4 q^3+4 q^2-2 q+3+ q^{-1} - q^{-2} +2 q^{-3} - q^{-4}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^4 a^{-4} +2 z^2 a^{-4} + a^{-4} -z^6 a^{-2} -a^2 z^4-5 z^4 a^{-2} -3 a^2 z^2-8 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -a^2-3 a^{-2} +z^6+6 z^4+9 z^2-2 z^{-2} +3$ (db)
Kauffman polynomial	$z^6 a^{-6} -3 z^4 a^{-6} +z^2 a^{-6} +3 z^7 a^{-5} -11 z^5 a^{-5} +7 z^3 a^{-5} -2 z a^{-5} +3 z^8 a^{-4} -12 z^6 a^{-4} +10 z^4 a^{-4} -6 z^2 a^{-4} +2 a^{-4} +z^9 a^{-3} +a^3 z^7-z^7 a^{-3} -5 a^3 z^5-13 z^5 a^{-3} +5 a^3 z^3+20 z^3 a^{-3} -a^3 z-7 z a^{-3} +2 a^2 z^8+5 z^8 a^{-2} -12 a^2 z^6-30 z^6 a^{-2} +19 a^2 z^4+50 z^4 a^{-2} -13 a^2 z^2-30 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +2 a^2+6 a^{-2} +a z^9+2 z^9 a^{-1} -5 a z^7-10 z^7 a^{-1} +3 z^5 a^{-1} +11 a z^3+19 z^3 a^{-1} -5 a z-9 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +4 z^8-29 z^6+56 z^4-36 z^2+2 z^{-2} +7$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     2   -2 9                   2 1   1 7               1 3 2     0 5             1 3 2       0 3           2 3 3         2 1         2 6 3           1 -1       1 2 5             4 -3     1 2 1               0 -5   1 1 1                 1 -7   1                     1 -9 1                       -1

Integral Khovanov Homology (db, data source)

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

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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