L11n346

L11n345

L11n347

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Link Presentations

[edit Notes on L11n346's Link Presentations]

Planar diagram presentation	X6172 X3,15,4,14 X11,20,12,21 X7,18,8,19 X17,22,18,13 X9,17,10,16 X15,11,16,10 X19,12,20,5 X21,8,22,9 X2536 X13,1,14,4
Gauss code	{1, -10, -2, 11}, {10, -1, -4, 9, -6, 7, -3, 8}, {-11, 2, -7, 6, -5, 4, -8, 3, -9, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{t(1) t(2)^3-t(1) t(3) t(2)^3+t(1) t(3)^3 t(2)^2-t(3)^3 t(2)^2-3 t(1) t(3)^2 t(2)^2+3 t(3)^2 t(2)^2-2 t(1) t(2)^2+4 t(1) t(3) t(2)^2-2 t(3) t(2)^2+t(2)^2-t(1) t(3)^3 t(2)+2 t(3)^3 t(2)+2 t(1) t(3)^2 t(2)-4 t(3)^2 t(2)+t(1) t(2)-3 t(1) t(3) t(2)+3 t(3) t(2)-t(2)-t(3)^3+t(3)^2}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}}$ (db)
Jones polynomial	$q^3-3 q^2+7 q-10+13 q^{-1} -12 q^{-2} +13 q^{-3} -9 q^{-4} +6 q^{-5} -2 q^{-6}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^6-z^4 a^4+z^2 a^4+a^4 z^{-2} +3 a^4+z^6 a^2+2 z^4 a^2-2 a^2 z^{-2} -3 a^2-2 z^4-3 z^2+ z^{-2} +z^2 a^{-2} + a^{-2}$ (db)
Kauffman polynomial	$2 a^3 z^9+2 a z^9+6 a^4 z^8+10 a^2 z^8+4 z^8+5 a^5 z^7+6 a^3 z^7+4 a z^7+3 z^7 a^{-1} +a^6 z^6-15 a^4 z^6-25 a^2 z^6+z^6 a^{-2} -8 z^6-9 a^5 z^5-22 a^3 z^5-21 a z^5-8 z^5 a^{-1} +6 a^6 z^4+27 a^4 z^4+25 a^2 z^4-3 z^4 a^{-2} +z^4+3 a^7 z^3+15 a^5 z^3+23 a^3 z^3+17 a z^3+6 z^3 a^{-1} -6 a^6 z^2-26 a^4 z^2-23 a^2 z^2+3 z^2 a^{-2} -2 a^7 z-7 a^5 z-12 a^3 z-8 a z-z a^{-1} +3 a^6+11 a^4+11 a^2- a^{-2} +3+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - z^{-2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                   1 1 5                 2   -2 3               5 1   4 1             5 2     -3 -1           8 5       3 -3         7 8         1 -5       6 5           1 -7     3 7             4 -9   3 6               -3 -11   4                 4 -13 2                   -2

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-5$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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