L11n413

L11n412

L11n414

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Link Presentations

[edit Notes on L11n413's Link Presentations]

Planar diagram presentation	X8192 X5,15,6,14 X10,3,11,4 X13,5,14,4 X2738 X6,9,1,10 X11,18,12,19 X17,12,18,7 X20,16,21,15 X22,20,13,19 X16,22,17,21
Gauss code	{1, -5, 3, 4, -2, -6}, {5, -1, 6, -3, -7, 8}, {-4, 2, 9, -11, -8, 7, 10, -9, 11, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{t(1)^2 t(3)^3-t(1) t(2)^2 t(3)^2-t(1) t(3)^2-t(1)^2 t(2) t(3)^2+2 t(1) t(2) t(3)^2-t(2) t(3)^2+t(1) t(2)^2 t(3)+t(1) t(3)+t(1)^2 t(2) t(3)-2 t(1) t(2) t(3)+t(2) t(3)-t(2)^2}{t(1) t(2) t(3)^{3/2}}$ (db)
Jones polynomial	$-q^5+2 q^4-3 q^3+4 q^2-4 q+4-2 q^{-1} +2 q^{-2} + q^{-3} + q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^2 a^4+2 a^4 z^{-2} +3 a^4-z^4 a^2-6 z^2 a^2-5 a^2 z^{-2} -9 a^2+z^4+3 z^2+4 z^{-2} +6+z^4 a^{-2} +2 z^2 a^{-2} - a^{-2} z^{-2} + a^{-2} -z^2 a^{-4} - a^{-4}$ (db)
Kauffman polynomial	$a^4 z^8+a^2 z^8+z^8 a^{-2} +z^8+a^3 z^7+2 a z^7+3 z^7 a^{-1} +2 z^7 a^{-3} -8 a^4 z^6-10 a^2 z^6-2 z^6 a^{-2} +2 z^6 a^{-4} -6 z^6-9 a^3 z^5-16 a z^5-14 z^5 a^{-1} -6 z^5 a^{-3} +z^5 a^{-5} +21 a^4 z^4+31 a^2 z^4-z^4 a^{-2} -6 z^4 a^{-4} +15 z^4+22 a^3 z^3+41 a z^3+26 z^3 a^{-1} +4 z^3 a^{-3} -3 z^3 a^{-5} -23 a^4 z^2-39 a^2 z^2+3 z^2 a^{-2} +3 z^2 a^{-4} -16 z^2-19 a^3 z-35 a z-19 z a^{-1} -2 z a^{-3} +z a^{-5} +11 a^4+22 a^2- a^{-4} +13+5 a^3 z^{-1} +9 a z^{-1} +5 a^{-1} z^{-1} + a^{-3} z^{-1} -2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} -4 z^{-2}$ (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 4 5 χ 11                       1 -1 9                     1   1 7                   2 1   -1 5                 2 1     1 3               2 2       0 1           1 3 2         0 -1           2 4           2 -3       1 2 1             0 -5         3               3 -7     1                   1 -9 1                       1 -11 1                       1

Integral Khovanov Homology (db, data source)

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

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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