L11n321

L11n320

L11n322

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Link Presentations

[edit Notes on L11n321's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X3849 X2,16,3,15 X16,7,17,8 X9,18,10,19 X17,1,18,4 X19,13,20,22 X13,10,14,11 X21,5,22,12 X11,21,12,20
Gauss code	{1, -4, -3, 7}, {-2, -1, 5, 3, -6, 9, -11, 10}, {-9, 2, 4, -5, -7, 6, -8, 11, -10, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{-u v^2 w^3+2 u v^2 w^2-u v^2 w+u v w^3-4 u v w^2+2 u v w+u w^2-u w+v^3 w^2-v^3 w-2 v^2 w^2+4 v^2 w-v^2+v w^2-2 v w+v}{\sqrt{u} v^{3/2} w^{3/2}}$ (db)
Jones polynomial	$-2 q^3+5 q^2-6 q+9-8 q^{-1} +9 q^{-2} -6 q^{-3} +4 q^{-4} -2 q^{-5} + q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^2 z^6+a^4 z^4-5 a^2 z^4+3 z^4+3 a^4 z^2-11 a^2 z^2-2 z^2 a^{-2} +9 z^2+3 a^4-9 a^2-2 a^{-2} +8+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2}$ (db)
Kauffman polynomial	$a^6 z^6-4 a^6 z^4+4 a^6 z^2-a^6+2 a^5 z^7-7 a^5 z^5+6 a^5 z^3-a^5 z+2 a^4 z^8-5 a^4 z^6+2 a^4 z^4-2 a^4 z^2-a^4 z^{-2} +3 a^4+a^3 z^9+a^3 z^7-10 a^3 z^5+13 a^3 z^3+3 z^3 a^{-3} -8 a^3 z-2 z a^{-3} +2 a^3 z^{-1} +5 a^2 z^8-16 a^2 z^6+z^6 a^{-2} +24 a^2 z^4+4 z^4 a^{-2} -23 a^2 z^2-7 z^2 a^{-2} -2 a^2 z^{-2} +11 a^2+3 a^{-2} +a z^9+2 a z^7+3 z^7 a^{-1} -9 a z^5-6 z^5 a^{-1} +15 a z^3+11 z^3 a^{-1} -12 a z-7 z a^{-1} +2 a z^{-1} +3 z^8-9 z^6+22 z^4-24 z^2- z^{-2} +11$ (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 χ 7                   2 -2 5                 3   3 3               3 2   -1 1             6 3     3 -1           5 6       1 -3         4 3         1 -5       2 5           3 -7     2 4             -2 -9   1 3               2 -11   1                 -1 -13 1                   1

Integral Khovanov Homology (db, data source)

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

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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