L10a19

L10a18

L10a20

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Link Presentations

[edit Notes on L10a19's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,10,13,9 X18,15,19,16 X16,7,17,8 X8,17,9,18 X20,13,5,14 X14,19,15,20 X2536 X4,12,1,11
Gauss code	{1, -9, 2, -10}, {9, -1, 5, -6, 3, -2, 10, -3, 7, -8, 4, -5, 6, -4, 8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{2 (t(1)-1) (t(2)-1)^3}{\sqrt{t(1)} t(2)^{3/2}}$ (db)
Jones polynomial	$q^{5/2}-3 q^{3/2}+5 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{10}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{2}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^7 (-z)-a^7 z^{-1} +2 a^5 z^3+3 a^5 z+2 a^5 z^{-1} -a^3 z^5-a^3 z^3-a z^5-2 a z^3+z^3 a^{-1} -3 a z-a z^{-1} +z a^{-1}$ (db)
Kauffman polynomial	$a^8 z^6-4 a^8 z^4+5 a^8 z^2-2 a^8+2 a^7 z^7-6 a^7 z^5+5 a^7 z^3-2 a^7 z+a^7 z^{-1} +2 a^6 z^8-2 a^6 z^6-8 a^6 z^4+12 a^6 z^2-5 a^6+a^5 z^9+3 a^5 z^7-12 a^5 z^5+9 a^5 z^3-5 a^5 z+2 a^5 z^{-1} +5 a^4 z^8-7 a^4 z^6-4 a^4 z^4+8 a^4 z^2-3 a^4+a^3 z^9+5 a^3 z^7-10 a^3 z^5+3 a^3 z^3+3 a^2 z^8-4 a^2 z^4+z^4 a^{-2} +2 a^2 z^2-z^2 a^{-2} +a^2+4 a z^7-a z^5+3 z^5 a^{-1} -5 a z^3-4 z^3 a^{-1} +5 a z+2 z a^{-1} -a z^{-1} +4 z^6-3 z^4$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                     1 -1 4                   2   2 2                 3 1   -2 0               6 2     4 -2             6 5       -1 -4           5 4         1 -6         4 6           2 -8       4 5             -1 -10     1 4               3 -12   1 4                 -3 -14   1                   1 -16 1                     -1

Integral Khovanov Homology (db, data source)

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

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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