L10a95

L10a94

L10a96

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Link Presentations

[edit Notes on L10a95's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(t(1)-1) (t(2)-1) (t(1)+t(2)-1) (t(2) t(1)-t(1)-t(2))}{t(1)^{3/2} t(2)^{3/2}}$ (db)
Jones polynomial	$-\frac{5}{q^{9/2}}-q^{7/2}+\frac{9}{q^{7/2}}+3 q^{5/2}-\frac{12}{q^{5/2}}-6 q^{3/2}+\frac{11}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{3}{q^{11/2}}+9 \sqrt{q}-\frac{12}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^5 z^3+a^5 z-a^3 z^5-2 a^3 z^3-3 a^3 z-z a^{-3} -a z^5+2 z^3 a^{-1} +2 a z+a z^{-1} +z a^{-1} - a^{-1} z^{-1}$ (db)
Kauffman polynomial	$a^7 z^5-2 a^7 z^3+3 a^6 z^6-7 a^6 z^4+4 a^6 z^2+4 a^5 z^7-8 a^5 z^5+6 a^5 z^3-3 a^5 z+3 a^4 z^8-a^4 z^6-7 a^4 z^4+8 a^4 z^2+a^3 z^9+7 a^3 z^7-19 a^3 z^5+z^5 a^{-3} +21 a^3 z^3-2 z^3 a^{-3} -10 a^3 z+z a^{-3} +6 a^2 z^8-7 a^2 z^6+3 z^6 a^{-2} -a^2 z^4-6 z^4 a^{-2} +5 a^2 z^2+3 z^2 a^{-2} +a z^9+7 a z^7+4 z^7 a^{-1} -17 a z^5-6 z^5 a^{-1} +17 a z^3+2 z^3 a^{-1} -10 a z-2 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +3 z^8-7 z^4+4 z^2-1$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                     1 1 6                   2   -2 4                 4 1   3 2               5 2     -3 0             7 4       3 -2           6 7         1 -4         6 5           1 -6       3 6             3 -8     2 6               -4 -10   1 3                 2 -12   2                   -2 -14 1                     1

Integral Khovanov Homology (db, data source)

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

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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