L11a318

L11a317

L11a319

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Link Presentations

[edit Notes on L11a318's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(1)-1) (t(2)-1) \left(t(1) t(2)^4+t(1)^2 t(2)^3+t(1) t(2)^3+t(2)^3+t(1) t(2)^2+t(1)^2 t(2)+t(1) t(2)+t(2)+t(1)\right)}{t(1)^{3/2} t(2)^{5/2}}$ (db)
Jones polynomial	$-\frac{6}{q^{9/2}}+\frac{5}{q^{7/2}}-\frac{5}{q^{5/2}}-q^{3/2}+\frac{4}{q^{3/2}}+\frac{1}{q^{19/2}}-\frac{2}{q^{17/2}}+\frac{3}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{5}{q^{11/2}}+\sqrt{q}-\frac{3}{\sqrt{q}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-z^5 a^7-4 z^3 a^7-3 z a^7+z^7 a^5+5 z^5 a^5+6 z^3 a^5-a^5 z^{-1} +z^7 a^3+6 z^5 a^3+12 z^3 a^3+10 z a^3+3 a^3 z^{-1} -z^5 a-5 z^3 a-7 z a-2 a z^{-1}$ (db)
Kauffman polynomial	$a^{12} z^2+2 a^{11} z^3+3 a^{10} z^4-2 a^{10} z^2+4 a^9 z^5-6 a^9 z^3+5 a^8 z^6-12 a^8 z^4+3 a^8 z^2+6 a^7 z^7-22 a^7 z^5+20 a^7 z^3-6 a^7 z+5 a^6 z^8-21 a^6 z^6+23 a^6 z^4-8 a^6 z^2+a^6+3 a^5 z^9-12 a^5 z^7+8 a^5 z^5+5 a^5 z^3-a^5 z^{-1} +a^4 z^{10}-a^4 z^8-16 a^4 z^6+37 a^4 z^4-21 a^4 z^2+3 a^4+4 a^3 z^9-26 a^3 z^7+57 a^3 z^5-51 a^3 z^3+19 a^3 z-3 a^3 z^{-1} +a^2 z^{10}-6 a^2 z^8+10 a^2 z^6-a^2 z^4-7 a^2 z^2+3 a^2+a z^9-8 a z^7+23 a z^5-28 a z^3+13 a z-2 a z^{-1}$ (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 4 χ 4                       1 1 2                         0 0                   3 1   2 -2                 1       -1 -4               4 3       1 -6             3 3         0 -8           3 2           1 -10         2 3             1 -12       2 3               -1 -14     1 2                 1 -16   1 2                   -1 -18   1                     1 -20 1                       -1

Integral Khovanov Homology (db, data source)

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

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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