L11a228

L11a227

L11a229

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Link Presentations

[edit Notes on L11a228's Link Presentations]

Planar diagram presentation	X8192 X16,6,17,5 X20,10,21,9 X10,22,11,21 X18,16,19,15 X14,20,15,19 X2,11,3,12 X12,3,13,4 X4758 X22,14,7,13 X6,18,1,17
Gauss code	{1, -7, 8, -9, 2, -11}, {9, -1, 3, -4, 7, -8, 10, -6, 5, -2, 11, -5, 6, -3, 4, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{2 t(1)^2 t(2)^4-2 t(1) t(2)^4-5 t(1)^2 t(2)^3+9 t(1) t(2)^3-3 t(2)^3+6 t(1)^2 t(2)^2-13 t(1) t(2)^2+6 t(2)^2-3 t(1)^2 t(2)+9 t(1) t(2)-5 t(2)-2 t(1)+2}{t(1) t(2)^2}$ (db)
Jones polynomial	$q^{15/2}-4 q^{13/2}+9 q^{11/2}-14 q^{9/2}+19 q^{7/2}-22 q^{5/2}+21 q^{3/2}-19 \sqrt{q}+\frac{13}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^5 a^{-5} +2 z^3 a^{-5} +z a^{-5} -z^7 a^{-3} -3 z^5 a^{-3} -2 z^3 a^{-3} +2 z a^{-3} + a^{-3} z^{-1} -z^7 a^{-1} +a z^5-4 z^5 a^{-1} +3 a z^3-8 z^3 a^{-1} +4 a z-8 z a^{-1} +2 a z^{-1} -3 a^{-1} z^{-1}$ (db)
Kauffman polynomial	$z^6 a^{-8} -2 z^4 a^{-8} +z^2 a^{-8} +4 z^7 a^{-7} -9 z^5 a^{-7} +5 z^3 a^{-7} +7 z^8 a^{-6} -16 z^6 a^{-6} +10 z^4 a^{-6} -2 z^2 a^{-6} +6 z^9 a^{-5} -8 z^7 a^{-5} -z^5 a^{-5} -z^3 a^{-5} +z a^{-5} +2 z^{10} a^{-4} +10 z^8 a^{-4} -26 z^6 a^{-4} +11 z^4 a^{-4} +2 z^2 a^{-4} - a^{-4} +11 z^9 a^{-3} -15 z^7 a^{-3} +a^3 z^5-2 z^5 a^{-3} -2 a^3 z^3+9 z^3 a^{-3} +a^3 z-5 z a^{-3} + a^{-3} z^{-1} +2 z^{10} a^{-2} +10 z^8 a^{-2} +3 a^2 z^6-20 z^6 a^{-2} -4 a^2 z^4+5 z^4 a^{-2} +a^2 z^2+8 z^2 a^{-2} -3 a^{-2} +5 z^9 a^{-1} +6 a z^7+3 z^7 a^{-1} -10 a z^5-21 z^5 a^{-1} +10 a z^3+27 z^3 a^{-1} -7 a z-14 z a^{-1} +2 a z^{-1} +3 a^{-1} z^{-1} +7 z^8-8 z^6+2 z^4+4 z^2-3$ (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 16                       1 -1 14                     3   3 12                   6 1   -5 10                 8 3     5 8               11 6       -5 6             11 8         3 4           10 11           1 2         9 11             -2 0       5 11               6 -2     3 8                 -5 -4   1 6                   5 -6   2                     -2 -8 1                       1

Integral Khovanov Homology (db, data source)

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

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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