L11a538

L11a537

L11a539

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Link Presentations

[edit Notes on L11a538's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X18,12,19,11 X22,19,17,20 X16,21,9,22 X20,15,21,16 X12,18,13,17 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {9, -5, 6, -8, 7, -6}, {11, -2, 5, -9, 4, -3, 8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(3)-1) (t(4)-1) \left(t(1) t(4)^2+t(2) t(4)^2-t(4)^2-3 t(1) t(4)+2 t(1) t(2) t(4)-3 t(2) t(4)+2 t(4)+t(1)-t(1) t(2)+t(2)\right)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} t(4)^{3/2}}$ (db)
Jones polynomial	$\frac{19}{q^{9/2}}-\frac{20}{q^{7/2}}+\frac{13}{q^{5/2}}-\frac{8}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{7}{q^{17/2}}-\frac{14}{q^{15/2}}+\frac{17}{q^{13/2}}-\frac{22}{q^{11/2}}-\sqrt{q}+\frac{3}{\sqrt{q}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11} z^{-1} +4 z a^9+4 a^9 z^{-1} +a^9 z^{-3} -6 z^3 a^7-11 z a^7-9 a^7 z^{-1} -3 a^7 z^{-3} +3 z^5 a^5+8 z^3 a^5+12 z a^5+10 a^5 z^{-1} +3 a^5 z^{-3} +z^5 a^3-z^3 a^3-4 z a^3-4 a^3 z^{-1} -a^3 z^{-3} -z^3 a-z a$ (db)
Kauffman polynomial	$a^{12} z^6-3 a^{12} z^4+3 a^{12} z^2-a^{12}+3 a^{11} z^7-8 a^{11} z^5+9 a^{11} z^3-6 a^{11} z+2 a^{11} z^{-1} +4 a^{10} z^8-4 a^{10} z^6-8 a^{10} z^4+14 a^{10} z^2-6 a^{10}+3 a^9 z^9+7 a^9 z^7-34 a^9 z^5+43 a^9 z^3-a^9 z^{-3} -30 a^9 z+11 a^9 z^{-1} +a^8 z^{10}+12 a^8 z^8-21 a^8 z^6-8 a^8 z^4+27 a^8 z^2+3 a^8 z^{-2} -18 a^8+7 a^7 z^9+9 a^7 z^7-59 a^7 z^5+79 a^7 z^3-3 a^7 z^{-3} -52 a^7 z+18 a^7 z^{-1} +a^6 z^{10}+15 a^6 z^8-26 a^6 z^6-a^6 z^4+27 a^6 z^2+6 a^6 z^{-2} -21 a^6+4 a^5 z^9+11 a^5 z^7-44 a^5 z^5+59 a^5 z^3-3 a^5 z^{-3} -40 a^5 z+14 a^5 z^{-1} +7 a^4 z^8-7 a^4 z^6-2 a^4 z^4+12 a^4 z^2+3 a^4 z^{-2} -9 a^4+6 a^3 z^7-10 a^3 z^5+12 a^3 z^3-a^3 z^{-3} -11 a^3 z+5 a^3 z^{-1} +3 a^2 z^6-4 a^2 z^4+a^2 z^2+a z^5-2 a z^3+a z$ (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 2                       1 1 0                     2   -2 -2                   6 1   5 -4                 9 4     -5 -6               11 4       7 -8             8 9         1 -10           14 11           3 -12         9 14             5 -14       5 8               -3 -16     2 9                 7 -18   1 5                   -4 -20   2                     2 -22 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-5$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-4$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{14}$ $r=-3$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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