L11a255

L11a254

L11a256

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Link Presentations

[edit Notes on L11a255's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(1)-1) (t(2)-1) (t(1) t(2)+1)^2 \left(t(2)^2-t(2)+1\right)}{t(1)^{3/2} t(2)^{5/2}}$ (db)
Jones polynomial	$-q^{13/2}+3 q^{11/2}-6 q^{9/2}+10 q^{7/2}-13 q^{5/2}+15 q^{3/2}-16 \sqrt{q}+\frac{12}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^7 a^{-3} -5 z^5 a^{-3} -8 z^3 a^{-3} -4 z a^{-3} +z^9 a^{-1} -a z^7+7 z^7 a^{-1} -5 a z^5+18 z^5 a^{-1} -8 a z^3+20 z^3 a^{-1} -4 a z+8 z a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db)
Kauffman polynomial	$z^5 a^{-7} -2 z^3 a^{-7} +3 z^6 a^{-6} -6 z^4 a^{-6} +z^2 a^{-6} +5 z^7 a^{-5} -11 z^5 a^{-5} +6 z^3 a^{-5} -z a^{-5} +6 z^8 a^{-4} +a^4 z^6-16 z^6 a^{-4} -3 a^4 z^4+18 z^4 a^{-4} +2 a^4 z^2-7 z^2 a^{-4} +5 z^9 a^{-3} +3 a^3 z^7-14 z^7 a^{-3} -9 a^3 z^5+20 z^5 a^{-3} +6 a^3 z^3-11 z^3 a^{-3} -a^3 z+4 z a^{-3} +2 z^{10} a^{-2} +4 a^2 z^8+z^8 a^{-2} -10 a^2 z^6-15 z^6 a^{-2} +4 a^2 z^4+29 z^4 a^{-2} -13 z^2 a^{-2} +4 a z^9+9 z^9 a^{-1} -11 a z^7-33 z^7 a^{-1} +13 a z^5+54 z^5 a^{-1} -13 a z^3-38 z^3 a^{-1} +4 a z+10 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +2 z^{10}-z^8-7 z^6+12 z^4-7 z^2-1$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     2   -2 10                   4 1   3 8                 6 2     -4 6               7 4       3 4             8 6         -2 2           8 7           1 0         6 10             4 -2       4 6               -2 -4     2 6                 4 -6   1 4                   -3 -8   2                     2 -10 1                       -1

Integral Khovanov Homology (db, data source)

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

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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