L11a129

L11a128

L11a130

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Link Presentations

[edit Notes on L11a129's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X18,10,19,9 X16,12,17,11 X12,16,13,15 X10,18,11,17 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, -6, 4, -5, 11, -2, 5, -4, 6, -3, 7, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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