L11a365

L11a364

L11a366

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Link Presentations

[edit Notes on L11a365's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{\left(t(2)^2 t(1)^2-t(2) t(1)^2+t(2) t(1)-t(1)+1\right) \left(t(1)^2 t(2)^2-t(1) t(2)^2+t(1) t(2)-t(2)+1\right)}{t(1)^2 t(2)^2}$ (db)
Jones polynomial	$q^{9/2}-2 q^{7/2}+3 q^{5/2}-5 q^{3/2}+6 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{7}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^3 z^7+6 a^3 z^5+11 a^3 z^3+6 a^3 z+a^3 z^{-1} -a z^9-8 a z^7+z^7 a^{-1} -23 a z^5+6 z^5 a^{-1} -28 a z^3+11 z^3 a^{-1} -13 a z+6 z a^{-1} -a z^{-1}$ (db)
Kauffman polynomial	$a^7 z^5-3 a^7 z^3+a^7 z+2 a^6 z^6-6 a^6 z^4+3 a^6 z^2+2 a^5 z^7-5 a^5 z^5+2 a^5 z^3+2 a^4 z^8-6 a^4 z^6+z^6 a^{-4} +6 a^4 z^4-4 z^4 a^{-4} -a^4 z^2+3 z^2 a^{-4} +2 a^3 z^9-9 a^3 z^7+2 z^7 a^{-3} +18 a^3 z^5-8 z^5 a^{-3} -14 a^3 z^3+7 z^3 a^{-3} +6 a^3 z-z a^{-3} -a^3 z^{-1} +a^2 z^{10}-3 a^2 z^8+2 z^8 a^{-2} +a^2 z^6-7 z^6 a^{-2} +9 a^2 z^4+5 z^4 a^{-2} -6 a^2 z^2-z^2 a^{-2} +a^2+4 a z^9+2 z^9 a^{-1} -22 a z^7-9 z^7 a^{-1} +48 a z^5+16 z^5 a^{-1} -42 a z^3-16 z^3 a^{-1} +14 a z+6 z a^{-1} -a z^{-1} +z^{10}-3 z^8+z^6+6 z^4-6 z^2$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     1   1 6                   2 1   -1 4                 3 1     2 2               3 2       -1 0             5 3         2 -2           3 4           1 -4         4 4             0 -6       2 4               2 -8     1 3                 -2 -10   1 2                   1 -12   1                     -1 -14 1                       1

Integral Khovanov Homology (db, data source)

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

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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