L11a321

L11a320

L11a322

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Link Presentations

[edit Notes on L11a321's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(1)-1) (t(2)-1) (t(1) t(2)-t(2)+2) (2 t(2) t(1)-t(1)+1)}{t(1)^{3/2} t(2)^{3/2}}$ (db)
Jones polynomial	$-\frac{14}{q^{9/2}}-q^{7/2}+\frac{18}{q^{7/2}}+3 q^{5/2}-\frac{21}{q^{5/2}}-7 q^{3/2}+\frac{20}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{9}{q^{11/2}}+12 \sqrt{q}-\frac{18}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^5 z^5-2 a^5 z^3-a^5 z+a^3 z^7+3 a^3 z^5+2 a^3 z^3-a^3 z+a z^7+4 a z^5-z^5 a^{-1} +7 a z^3-3 z^3 a^{-1} +5 a z-3 z a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db)
Kauffman polynomial	$a^8 z^6-2 a^8 z^4+a^8 z^2+4 a^7 z^7-9 a^7 z^5+5 a^7 z^3-a^7 z+7 a^6 z^8-16 a^6 z^6+9 a^6 z^4-3 a^6 z^2+6 a^5 z^9-8 a^5 z^7-5 a^5 z^5+5 a^5 z^3-a^5 z+2 a^4 z^{10}+10 a^4 z^8-32 a^4 z^6+24 a^4 z^4-5 a^4 z^2+11 a^3 z^9-19 a^3 z^7+4 a^3 z^5+z^5 a^{-3} +9 a^3 z^3-2 z^3 a^{-3} -4 a^3 z+z a^{-3} +2 a^2 z^{10}+9 a^2 z^8-25 a^2 z^6+3 z^6 a^{-2} +21 a^2 z^4-5 z^4 a^{-2} -3 a^2 z^2+2 z^2 a^{-2} +5 a z^9-2 a z^7+5 z^7 a^{-1} -8 a z^5-7 z^5 a^{-1} +15 a z^3+4 z^3 a^{-1} -8 a z-3 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +6 z^8-7 z^6+3 z^4-1$ (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                     2   -2 4                   5 1   4 2                 7 2     -5 0               11 5       6 -2             11 9         -2 -4           10 9           1 -6         8 11             3 -8       6 10               -4 -10     3 8                 5 -12   1 6                   -5 -14   3                     3 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{11}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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