L11n115

L11n114

L11n116

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Link Presentations

[edit Notes on L11n115's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X20,8,21,7 X22,17,5,18 X18,21,19,22 X13,17,14,16 X9,15,10,14 X15,11,16,10 X8,20,9,19 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, -7, 8, 11, -2, -6, 7, -8, 6, 4, -5, 9, -3, 5, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{t(2)^5+t(1) t(2)^4-4 t(2)^4-2 t(1) t(2)^3+3 t(2)^3+3 t(1) t(2)^2-2 t(2)^2-4 t(1) t(2)+t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$q^{7/2}-3 q^{5/2}+5 q^{3/2}-7 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z a^5+2 a^5 z^{-1} -3 z^3 a^3-8 z a^3-5 a^3 z^{-1} +2 z^5 a+8 z^3 a+11 z a+6 a z^{-1} -3 z^3 a^{-1} -7 z a^{-1} -4 a^{-1} z^{-1} +z a^{-3} + a^{-3} z^{-1}$ (db)
Kauffman polynomial	$a^5 z^7-5 a^5 z^5+9 a^5 z^3-7 a^5 z+2 a^5 z^{-1} +2 a^4 z^8-8 a^4 z^6+8 a^4 z^4-a^4 z^2+z^2 a^{-4} -a^4- a^{-4} +a^3 z^9+2 a^3 z^7-23 a^3 z^5+36 a^3 z^3+3 z^3 a^{-3} -22 a^3 z-3 z a^{-3} +5 a^3 z^{-1} + a^{-3} z^{-1} +6 a^2 z^8-22 a^2 z^6+2 z^6 a^{-2} +18 a^2 z^4-5 z^4 a^{-2} -2 a^2 z^2+8 z^2 a^{-2} -a^2-3 a^{-2} +a z^9+6 a z^7+5 z^7 a^{-1} -38 a z^5-20 z^5 a^{-1} +54 a z^3+30 z^3 a^{-1} -30 a z-18 z a^{-1} +6 a z^{-1} +4 a^{-1} z^{-1} +4 z^8-12 z^6+5 z^4+6 z^2-3$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 8                   1 -1 6                 2   2 4               3 1   -2 2             4 2     2 0           4 4       0 -2         3 4 1       0 -4       3 4           1 -6     2 3             -1 -8   1 4               3 -10   1                 -1 -12 1                   1

Integral Khovanov Homology (db, data source)

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

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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