L10a84

L10a83

L10a85

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Link Presentations

[edit Notes on L10a84's Link Presentations]

Planar diagram presentation	X8192 X14,9,15,10 X4758 X16,6,17,5 X18,16,19,15 X6,18,1,17 X20,11,7,12 X10,19,11,20 X2,14,3,13 X12,4,13,3
Gauss code	{1, -9, 10, -3, 4, -6}, {3, -1, 2, -8, 7, -10, 9, -2, 5, -4, 6, -5, 8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{t(1)^2 t(2)^4-2 t(1) t(2)^4+t(2)^4-2 t(1)^2 t(2)^3+6 t(1) t(2)^3-3 t(2)^3+3 t(1)^2 t(2)^2-7 t(1) t(2)^2+3 t(2)^2-3 t(1)^2 t(2)+6 t(1) t(2)-2 t(2)+t(1)^2-2 t(1)+1}{t(1) t(2)^2}$ (db)
Jones polynomial	$q^{9/2}-\frac{3}{q^{9/2}}-4 q^{7/2}+\frac{7}{q^{7/2}}+7 q^{5/2}-\frac{11}{q^{5/2}}-11 q^{3/2}+\frac{13}{q^{3/2}}+\frac{1}{q^{11/2}}+13 \sqrt{q}-\frac{15}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^3 z^5-3 a^3 z^3+z^3 a^{-3} -4 a^3 z+z a^{-3} -a^3 z^{-1} +a z^7+5 a z^5-2 z^5 a^{-1} +11 a z^3-6 z^3 a^{-1} +10 a z-6 z a^{-1} +3 a z^{-1} -2 a^{-1} z^{-1}$ (db)
Kauffman polynomial	$a^6 z^4-a^6 z^2+3 a^5 z^5-2 a^5 z^3+6 a^4 z^6+z^6 a^{-4} -7 a^4 z^4-2 z^4 a^{-4} +5 a^4 z^2+z^2 a^{-4} -a^4+8 a^3 z^7+4 z^7 a^{-3} -13 a^3 z^5-11 z^5 a^{-3} +13 a^3 z^3+8 z^3 a^{-3} -6 a^3 z-z a^{-3} +a^3 z^{-1} +6 a^2 z^8+5 z^8 a^{-2} -4 a^2 z^6-11 z^6 a^{-2} -7 a^2 z^4+3 z^4 a^{-2} +10 a^2 z^2+2 z^2 a^{-2} -3 a^2+2 a z^9+2 z^9 a^{-1} +11 a z^7+7 z^7 a^{-1} -35 a z^5-30 z^5 a^{-1} +33 a z^3+26 z^3 a^{-1} -15 a z-10 z a^{-1} +3 a z^{-1} +2 a^{-1} z^{-1} +11 z^8-22 z^6+6 z^4+5 z^2-3$ (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 5 χ 10                     1 -1 8                   3   3 6                 4 1   -3 4               7 3     4 2             7 5       -2 0           8 6         2 -2         6 8           2 -4       5 7             -2 -6     2 6               4 -8   1 5                 -4 -10   2                   2 -12 1                     -1

Integral Khovanov Homology (db, data source)

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

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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