L10a96

L10a95

L10a97

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Link Presentations

[edit Notes on L10a96's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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