L11a292

L11a291

L11a293

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Link Presentations

[edit Notes on L11a292'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 X14,8,15,7 X16,6,17,5 X6,16,7,15 X4,20,5,19
Gauss code	{1, -2, 3, -11, 9, -10, 8, -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 $u$, $v$, $w$, ...)	$\frac{(t(1)-1) (t(2)-1) \left(t(1) t(2)^4+t(1)^2 t(2)^3-t(1) t(2)^3+t(2)^3-t(1)^2 t(2)^2+3 t(1) t(2)^2-t(2)^2+t(1)^2 t(2)-t(1) t(2)+t(2)+t(1)\right)}{t(1)^{3/2} t(2)^{5/2}}$ (db)
Jones polynomial	$3 q^{9/2}-\frac{3}{q^{9/2}}-6 q^{7/2}+\frac{7}{q^{7/2}}+10 q^{5/2}-\frac{12}{q^{5/2}}-14 q^{3/2}+\frac{14}{q^{3/2}}-q^{11/2}+\frac{1}{q^{11/2}}+16 \sqrt{q}-\frac{17}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^3 z^5-z^5 a^{-3} -3 a^3 z^3-3 z^3 a^{-3} -3 a^3 z-2 z a^{-3} +a z^7+z^7 a^{-1} +4 a z^5+4 z^5 a^{-1} +6 a z^3+5 z^3 a^{-1} +4 a z+z a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db)
Kauffman polynomial	$-2 z^{10} a^{-2} -2 z^{10}-5 a z^9-9 z^9 a^{-1} -4 z^9 a^{-3} -8 a^2 z^8+z^8 a^{-2} -3 z^8 a^{-4} -4 z^8-9 a^3 z^7+2 a z^7+27 z^7 a^{-1} +15 z^7 a^{-3} -z^7 a^{-5} -6 a^4 z^6+11 a^2 z^6+10 z^6 a^{-2} +12 z^6 a^{-4} +15 z^6-3 a^5 z^5+16 a^3 z^5+14 a z^5-28 z^5 a^{-1} -19 z^5 a^{-3} +4 z^5 a^{-5} -a^6 z^4+6 a^4 z^4-2 a^2 z^4-10 z^4 a^{-2} -14 z^4 a^{-4} -5 z^4+2 a^5 z^3-16 a^3 z^3-17 a z^3+17 z^3 a^{-1} +12 z^3 a^{-3} -4 z^3 a^{-5} +a^6 z^2-3 a^4 z^2-4 a^2 z^2+3 z^2 a^{-2} +4 z^2 a^{-4} -z^2+6 a^3 z+8 a z-2 z a^{-1} -4 z a^{-3} +1-a z^{-1} - a^{-1} z^{-1}$ (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 12                       1 1 10                     2   -2 8                   4 1   3 6                 6 2     -4 4               8 4       4 2             8 6         -2 0           9 8           1 -2         7 10             3 -4       5 7               -2 -6     2 7                 5 -8   1 5                   -4 -10   2                     2 -12 1                       -1

Integral Khovanov Homology (db, data source)

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

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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