L10n89

L10n88

L10n90

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Link Presentations

[edit Notes on L10n89's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,14,8,15 X20,15,17,16 X18,11,19,12 X12,17,13,18 X16,19,5,20 X13,8,14,9 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {6, -5, 7, -4}, {9, -1, -3, 8, 10, -2, 5, -6, -8, 3, 4, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(3)-1) \left(t(1) t(3)^2+t(2) t(3)^2-t(3)^2-t(1) t(3)+2 t(1) t(2) t(3)-t(2) t(3)+2 t(3)+t(1)-t(1) t(2)+t(2)\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db)
Jones polynomial	$2 q^{-10} -4 q^{-9} +7 q^{-8} -8 q^{-7} +9 q^{-6} -7 q^{-5} +7 q^{-4} -3 q^{-3} + q^{-2}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{10} z^{-2} +2 a^{10}-5 z^2 a^8-2 a^8 z^{-2} -8 a^8+3 z^4 a^6+8 z^2 a^6+a^6 z^{-2} +6 a^6+z^4 a^4+z^2 a^4$ (db)
Kauffman polynomial	$3 a^{12} z^4-5 a^{12} z^2+2 a^{12}+a^{11} z^7+a^{11} z^5-2 a^{11} z^3+a^{10} z^8+2 a^{10} z^6-3 a^{10} z^4+a^{10} z^{-2} -2 a^{10}+5 a^9 z^7-3 a^9 z^5-8 a^9 z^3+8 a^9 z-2 a^9 z^{-1} +a^8 z^8+8 a^8 z^6-18 a^8 z^4+16 a^8 z^2+2 a^8 z^{-2} -9 a^8+4 a^7 z^7-a^7 z^5-8 a^7 z^3+8 a^7 z-2 a^7 z^{-1} +6 a^6 z^6-11 a^6 z^4+10 a^6 z^2+a^6 z^{-2} -6 a^6+3 a^5 z^5-2 a^5 z^3+a^4 z^4-a^4 z^2$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                 1 1 -5               3 1 -2 -7             4     4 -9           3 3     0 -11         6 4       2 -13       4 5         1 -15     3 4           -1 -17   1 4             3 -19 1 3               -2 -21 2                 2

Integral Khovanov Homology (db, data source)

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

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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