L11a278

L11a277

L11a279

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Link Presentations

[edit Notes on L11a278's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X8,9,1,10 X4,13,5,14 X20,17,21,18 X18,8,19,7 X6,20,7,19 X22,15,9,16 X16,21,17,22
Gauss code	{1, -2, 3, -6, 4, -9, 8, -5}, {5, -1, 2, -3, 6, -4, 10, -11, 7, -8, 9, -7, 11, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{2 t(2)^3 t(1)^3-2 t(2)^2 t(1)^3-2 t(2)^3 t(1)^2+3 t(2)^2 t(1)^2-2 t(2) t(1)^2-2 t(2)^2 t(1)+3 t(2) t(1)-2 t(1)-2 t(2)+2}{t(1)^{3/2} t(2)^{3/2}}$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{6}{q^{11/2}}-\frac{7}{q^{13/2}}+\frac{6}{q^{15/2}}-\frac{5}{q^{17/2}}+\frac{3}{q^{19/2}}-\frac{2}{q^{21/2}}+\frac{1}{q^{23/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^9 \left(-z^5\right)-4 a^9 z^3-3 a^9 z+a^7 z^7+5 a^7 z^5+7 a^7 z^3+3 a^7 z+a^5 z^7+5 a^5 z^5+7 a^5 z^3+4 a^5 z+a^5 z^{-1} -a^3 z^5-5 a^3 z^3-6 a^3 z-a^3 z^{-1}$ (db)
Kauffman polynomial	$a^{14} z^4-2 a^{14} z^2+2 a^{13} z^5-4 a^{13} z^3+a^{13} z+2 a^{12} z^6-3 a^{12} z^4+a^{12} z^2+2 a^{11} z^7-4 a^{11} z^5+4 a^{11} z^3+2 a^{10} z^8-6 a^{10} z^6+8 a^{10} z^4-a^{10} z^2+2 a^9 z^9-9 a^9 z^7+17 a^9 z^5-11 a^9 z^3+2 a^9 z+a^8 z^{10}-4 a^8 z^8+7 a^8 z^6-6 a^8 z^4+a^8 z^2+3 a^7 z^9-15 a^7 z^7+27 a^7 z^5-22 a^7 z^3+5 a^7 z+a^6 z^{10}-5 a^6 z^8+11 a^6 z^6-16 a^6 z^4+8 a^6 z^2+a^5 z^9-3 a^5 z^7-2 a^5 z^5+8 a^5 z^3-5 a^5 z+a^5 z^{-1} +a^4 z^8-4 a^4 z^6+2 a^4 z^4+3 a^4 z^2-a^4+a^3 z^7-6 a^3 z^5+11 a^3 z^3-7 a^3 z+a^3 z^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 0                       1 1 -2                         0 -4                   3 1   2 -6                 2 1     -1 -8               3 2       1 -10             3 2         -1 -12           4 3           1 -14         3 4             1 -16       2 3               -1 -18     1 3                 2 -20   1 2                   -1 -22   1                     1 -24 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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