L11n329

L11n328

L11n330

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Link Presentations

[edit Notes on L11n329's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X3849 X15,2,16,3 X16,7,17,8 X13,18,14,19 X9,13,10,22 X11,21,12,20 X19,5,20,12 X21,11,22,10 X4,17,1,18
Gauss code	{1, 4, -3, -11}, {-2, -1, 5, 3, -7, 10, -8, 9}, {-6, 2, -4, -5, 11, 6, -9, 8, -10, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{-t(1) t(3)^2 t(2)^3+t(1) t(3) t(2)^3-t(1) t(3)^3 t(2)^2+t(1) t(3)^2 t(2)^2-t(2)^2+t(1) t(3)^3 t(2)-t(3) t(2)+t(2)-t(3)^2+t(3)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}}$ (db)
Jones polynomial	$- q^{-7} +2 q^{-6} -2 q^{-5} +3 q^{-4} -2 q^{-3} +q^2+4 q^{-2} -q-2 q^{-1} +2$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^8+a^6 z^4+4 a^6 z^2+2 a^6-a^4 z^6-4 a^4 z^4-2 a^4 z^2+a^4 z^{-2} +a^4-a^2 z^6-5 a^2 z^4-7 a^2 z^2-2 a^2 z^{-2} -5 a^2+z^4+4 z^2+ z^{-2} +3$ (db)
Kauffman polynomial	$z a^9+2 z^2 a^8-a^8+z^5 a^7-2 z^3 a^7+z a^7+2 z^6 a^6-7 z^4 a^6+5 z^2 a^6+2 z^7 a^5-8 z^5 a^5+8 z^3 a^5-3 z a^5+2 z^8 a^4-10 z^6 a^4+17 z^4 a^4-18 z^2 a^4-a^4 z^{-2} +9 a^4+z^9 a^3-4 z^7 a^3+z^5 a^3+8 z^3 a^3-8 z a^3+2 a^3 z^{-1} +3 z^8 a^2-19 z^6 a^2+40 z^4 a^2-35 z^2 a^2-2 a^2 z^{-2} +13 a^2+z^9 a-6 z^7 a+10 z^5 a-2 z^3 a-5 z a+2 a z^{-1} +z^8-7 z^6+16 z^4-14 z^2- z^{-2} +6$ (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 5                   1 1 3                     0 1               2 1   1 -1           1 1       0 -3           4 2       2 -5       1 2 3         2 -7       2 1           1 -9   1 2 2             1 -11   2 2               0 -13   1                 1 -15 1                   -1

Integral Khovanov Homology (db, data source)

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

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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