L11a128

L11a127

L11a129

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Link Presentations

[edit Notes on L11a128's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X16,10,17,9 X18,12,19,11 X10,18,11,17 X12,16,13,15 X22,19,5,20 X20,7,21,8 X8,21,9,22 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 8, -9, 3, -5, 4, -6, 11, -2, 6, -3, 5, -4, 7, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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