L11n68

L11n67

L11n69

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Link Presentations

[edit Notes on L11n68's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,16,8,17 X17,22,18,5 X11,18,12,19 X13,20,14,21 X19,12,20,13 X21,14,22,15 X15,8,16,9 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, -5, 7, -6, 8, -9, 3, -4, 5, -7, 6, -8, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{u v^5-u v^3+u v^2-u v+u+v^5-v^4+v^3-v^2+1}{\sqrt{u} v^{5/2}}$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{17/2}}+\frac{2}{q^{19/2}}-\frac{2}{q^{21/2}}+\frac{2}{q^{23/2}}-\frac{1}{q^{25/2}}+\frac{1}{q^{27/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-z a^{13}-2 a^{13} z^{-1} +2 z^3 a^{11}+6 z a^{11}+4 a^{11} z^{-1} +z^3 a^9+z a^9-a^9 z^{-1} -z^7 a^7-7 z^5 a^7-14 z^3 a^7-8 z a^7-a^7 z^{-1}$ (db)
Kauffman polynomial	$-z^6 a^{16}+5 z^4 a^{16}-6 z^2 a^{16}+2 a^{16}-z^7 a^{15}+4 z^5 a^{15}-2 z^3 a^{15}-z a^{15}-z^8 a^{14}+4 z^6 a^{14}-2 z^4 a^{14}-2 z^2 a^{14}+a^{14}-z^9 a^{13}+6 z^7 a^{13}-13 z^5 a^{13}+16 z^3 a^{13}-10 z a^{13}+2 a^{13} z^{-1} -2 z^8 a^{12}+13 z^6 a^{12}-28 z^4 a^{12}+24 z^2 a^{12}-6 a^{12}-z^9 a^{11}+8 z^7 a^{11}-23 z^5 a^{11}+30 z^3 a^{11}-18 z a^{11}+4 a^{11} z^{-1} -z^8 a^{10}+8 z^6 a^{10}-20 z^4 a^{10}+18 z^2 a^{10}-5 a^{10}+z^5 a^9-2 z^3 a^9-z a^9+a^9 z^{-1} +z^4 a^8-2 z^2 a^8+a^8-z^7 a^7+7 z^5 a^7-14 z^3 a^7+8 z a^7-a^7 z^{-1}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                       1 1 -8                       1 1 -10                 1 1     0 -12               1         1 -14             2 3 1       0 -16           2             2 -18         1 3 1           1 -20       2 2               0 -22       1 1               0 -24   1 2                   -1 -26                         0 -28 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-8$ $i=-6$ $i=-4$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{2}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-7$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}_2$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

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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