L11a213

L11a212

L11a214

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Link Presentations

[edit Notes on L11a213's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X18,10,19,9 X22,20,7,19 X20,14,21,13 X14,22,15,21 X10,16,11,15 X16,6,17,5 X2738 X4,11,5,12 X6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 3, -7, 10, -2, 5, -6, 7, -8, 11, -3, 4, -5, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{t(1)^2 t(2)^4-3 t(1) t(2)^4+2 t(2)^4-4 t(1)^2 t(2)^3+7 t(1) t(2)^3-5 t(2)^3+6 t(1)^2 t(2)^2-9 t(1) t(2)^2+6 t(2)^2-5 t(1)^2 t(2)+7 t(1) t(2)-4 t(2)+2 t(1)^2-3 t(1)+1}{t(1) t(2)^2}$ (db)
Jones polynomial	$q^{17/2}-4 q^{15/2}+9 q^{13/2}-14 q^{11/2}+19 q^{9/2}-21 q^{7/2}+20 q^{5/2}-18 q^{3/2}+12 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^3 a^{-7} +z a^{-7} -2 z^5 a^{-5} -4 z^3 a^{-5} -z a^{-5} + a^{-5} z^{-1} +z^7 a^{-3} +3 z^5 a^{-3} +3 z^3 a^{-3} -2 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-5 z^3 a^{-1} +2 a z-3 z a^{-1} +a z^{-1}$ (db)
Kauffman polynomial	$-2 z^{10} a^{-2} -2 z^{10} a^{-4} -4 z^9 a^{-1} -12 z^9 a^{-3} -8 z^9 a^{-5} -6 z^8 a^{-2} -16 z^8 a^{-4} -13 z^8 a^{-6} -3 z^8-a z^7+9 z^7 a^{-1} +27 z^7 a^{-3} +4 z^7 a^{-5} -13 z^7 a^{-7} +33 z^6 a^{-2} +51 z^6 a^{-4} +19 z^6 a^{-6} -9 z^6 a^{-8} +10 z^6+4 a z^5+z^5 a^{-1} -9 z^5 a^{-3} +15 z^5 a^{-5} +17 z^5 a^{-7} -4 z^5 a^{-9} -35 z^4 a^{-2} -42 z^4 a^{-4} -8 z^4 a^{-6} +8 z^4 a^{-8} -z^4 a^{-10} -10 z^4-6 a z^3-10 z^3 a^{-1} -3 z^3 a^{-3} -8 z^3 a^{-5} -8 z^3 a^{-7} +z^3 a^{-9} +13 z^2 a^{-2} +18 z^2 a^{-4} +4 z^2 a^{-6} -3 z^2 a^{-8} +2 z^2+4 a z+3 z a^{-1} -2 z a^{-3} +z a^{-7} -3 a^{-2} -5 a^{-4} -2 a^{-6} +1-a z^{-1} +2 a^{-3} z^{-1} + a^{-5} z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                     3   3 14                   6 1   -5 12                 8 3     5 10               11 6       -5 8             10 8         2 6           10 11           1 4         8 10             -2 2       5 11               6 0     3 7                 -4 -2   1 6                   5 -4   2                     -2 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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