L11n140

L11n139

L11n141

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Link Presentations

[edit Notes on L11n140's Link Presentations]

Planar diagram presentation	X8192 X18,11,19,12 X3,10,4,11 X17,3,18,2 X12,5,13,6 X6718 X9,16,10,17 X15,20,16,21 X13,22,14,7 X21,14,22,15 X4,20,5,19
Gauss code	{1, 4, -3, -11, 5, -6}, {6, -1, -7, 3, 2, -5, -9, 10, -8, 7, -4, -2, 11, 8, -10, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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