L10a65

L10a64

L10a66

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Link Presentations

[edit Notes on L10a65's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X20,10,7,9 X2738 X4,20,5,19 X14,5,15,6 X18,11,19,12 X16,13,17,14 X12,17,13,18 X6,15,1,16
Gauss code	{1, -4, 2, -5, 6, -10}, {4, -1, 3, -2, 7, -9, 8, -6, 10, -8, 9, -7, 5, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{2 t(2)^2 t(1)^2-4 t(2) t(1)^2+2 t(1)^2-4 t(2)^2 t(1)+9 t(2) t(1)-4 t(1)+2 t(2)^2-4 t(2)+2}{t(1) t(2)}$ (db)
Jones polynomial	$-\frac{8}{q^{9/2}}+\frac{10}{q^{7/2}}+q^{5/2}-\frac{11}{q^{5/2}}-3 q^{3/2}+\frac{10}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{5}{q^{11/2}}+5 \sqrt{q}-\frac{9}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-z a^7+2 z^3 a^5+2 z a^5-z^5 a^3-z^3 a^3+a^3 z^{-1} -z^5 a-2 z^3 a-3 z a-a z^{-1} +z^3 a^{-1} +z a^{-1}$ (db)
Kauffman polynomial	$-a^5 z^9-a^3 z^9-3 a^6 z^8-6 a^4 z^8-3 a^2 z^8-3 a^7 z^7-5 a^5 z^7-6 a^3 z^7-4 a z^7-a^8 z^6+7 a^6 z^6+12 a^4 z^6-4 z^6+10 a^7 z^5+20 a^5 z^5+14 a^3 z^5+a z^5-3 z^5 a^{-1} +3 a^8 z^4-2 a^6 z^4-6 a^4 z^4+3 a^2 z^4-z^4 a^{-2} +3 z^4-9 a^7 z^3-17 a^5 z^3-7 a^3 z^3+5 a z^3+4 z^3 a^{-1} -2 a^8 z^2-a^6 z^2+2 a^4 z^2+z^2 a^{-2} +2 a^7 z+4 a^5 z-a^3 z-5 a z-2 z a^{-1} -a^2+a^3 z^{-1} +a z^{-1}$ (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             5 4       -1 -4           6 5         1 -6         5 6           1 -8       3 5             -2 -10     2 5               3 -12   1 3                 -2 -14   2                   2 -16 1                     -1

Integral Khovanov Homology (db, data source)

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

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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