L11a378

L11a377

L11a379

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Link Presentations

[edit Notes on L11a378's Link Presentations]

Planar diagram presentation	X12,1,13,2 X2,13,3,14 X14,3,15,4 X4,11,5,12 X16,8,17,7 X18,6,19,5 X22,16,11,15 X6,18,7,17 X8,22,9,21 X20,10,21,9 X10,20,1,19
Gauss code	{1, -2, 3, -4, 6, -8, 5, -9, 10, -11}, {4, -1, 2, -3, 7, -5, 8, -6, 11, -10, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{t(2)^2 t(1)^4-t(2) t(1)^4+2 t(2)^3 t(1)^3-4 t(2)^2 t(1)^3+3 t(2) t(1)^3-t(1)^3+t(2)^4 t(1)^2-4 t(2)^3 t(1)^2+5 t(2)^2 t(1)^2-4 t(2) t(1)^2+t(1)^2-t(2)^4 t(1)+3 t(2)^3 t(1)-4 t(2)^2 t(1)+2 t(2) t(1)-t(2)^3+t(2)^2}{t(1)^2 t(2)^2}$ (db)
Jones polynomial	$4 q^{9/2}-\frac{1}{q^{9/2}}-8 q^{7/2}+\frac{3}{q^{7/2}}+9 q^{5/2}-\frac{6}{q^{5/2}}-12 q^{3/2}+\frac{9}{q^{3/2}}+q^{13/2}-2 q^{11/2}+12 \sqrt{q}-\frac{11}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a z^5-2 z^5 a^{-1} -z^5 a^{-3} +a^3 z^3-a z^3-5 z^3 a^{-1} -2 z^3 a^{-3} +z^3 a^{-5} +a^3 z+a z-2 z a^{-1} -z a^{-3} +2 z a^{-5} + a^{-1} z^{-1} - a^{-3} z^{-1}$ (db)
Kauffman polynomial	$z^8 a^{-6} -6 z^6 a^{-6} +12 z^4 a^{-6} -8 z^2 a^{-6} +2 z^9 a^{-5} -11 z^7 a^{-5} +21 z^5 a^{-5} +a^5 z^3-17 z^3 a^{-5} +6 z a^{-5} +z^{10} a^{-4} -16 z^6 a^{-4} +3 a^4 z^4+31 z^4 a^{-4} -15 z^2 a^{-4} +6 z^9 a^{-3} -24 z^7 a^{-3} +6 a^3 z^5+28 z^5 a^{-3} -4 a^3 z^3-14 z^3 a^{-3} +a^3 z+8 z a^{-3} - a^{-3} z^{-1} +z^{10} a^{-2} +7 z^8 a^{-2} +9 a^2 z^6-36 z^6 a^{-2} -13 a^2 z^4+40 z^4 a^{-2} +5 a^2 z^2-13 z^2 a^{-2} + a^{-2} +4 z^9 a^{-1} +10 a z^7-3 z^7 a^{-1} -20 a z^5-19 z^5 a^{-1} +10 a z^3+18 z^3 a^{-1} -3 a z-2 z a^{-1} - a^{-1} z^{-1} +8 z^8-17 z^6+5 z^4-z^2$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 14                       1 -1 12                     1   1 10                   3 1   -2 8                 5 1     4 6               4 3       -1 4             8 5         3 2           5 5           0 0         6 7             -1 -2       4 6               2 -4     2 5                 -3 -6   1 4                   3 -8   2                     -2 -10 1                       1

Integral Khovanov Homology (db, data source)

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

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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