L11n342

L11n341

L11n343

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Link Presentations

[edit Notes on L11n342's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,19,12,18 X7,14,8,15 X13,8,14,9 X15,21,16,20 X17,13,18,22 X21,17,22,16 X19,5,20,12 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 9}, {-5, 4, -6, 8, -7, 3, -9, 6, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(v-1) (w-1) \left(u v w-u w^2-u+v^2 w^2+v^2-v w\right)}{\sqrt{u} v^{3/2} w^{3/2}}$ (db)
Jones polynomial	$-q^5+ q^{-5} +q^4- q^{-4} +4 q^{-3} -q^2-3 q^{-2} +2 q+4 q^{-1} -2$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^6-2 a^2 z^4+5 z^4+a^4 z^2-8 a^2 z^2-z^2 a^{-4} +8 z^2+3 a^4-10 a^2- a^{-4} +8+2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} +4 z^{-2}$ (db)
Kauffman polynomial	$a^3 z^9+a z^9+a^4 z^8+5 a^2 z^8+4 z^8-4 a^3 z^7+4 z^7 a^{-1} -7 a^4 z^6-30 a^2 z^6+z^6 a^{-4} -24 z^6-a^3 z^5-24 a z^5-25 z^5 a^{-1} -z^5 a^{-3} +z^5 a^{-5} +18 a^4 z^4+58 a^2 z^4-3 z^4 a^{-2} -5 z^4 a^{-4} +42 z^4+17 a^3 z^3+52 a z^3+40 z^3 a^{-1} +z^3 a^{-3} -4 z^3 a^{-5} -21 a^4 z^2-51 a^2 z^2+z^2 a^{-2} +4 z^2 a^{-4} -33 z^2-18 a^3 z-37 a z-23 z a^{-1} -2 z a^{-3} +2 z a^{-5} +11 a^4+24 a^2+2 a^{-2} - a^{-4} +17+5 a^3 z^{-1} +9 a z^{-1} +5 a^{-1} z^{-1} + a^{-3} z^{-1} -2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} -4 z^{-2}$ (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                         0 7                 1 1 1   1 5               2 1       -1 3             2 2 1       1 1           3 4 1         0 -1         1 2 3           2 -3       2 3               1 -5     2 1                 1 -7   1 4                   3 -9                         0 -11 1                       1

Integral Khovanov Homology (db, data source)

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

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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