L11a419

L11a418

L11a420

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Link Presentations

[edit Notes on L11a419's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,12,15,11 X22,16,11,15 X20,18,21,17 X16,8,17,7 X10,22,5,21 X8,19,9,20 X18,9,19,10 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 6, -8, 9, -7}, {3, -2, 11, -3, 4, -6, 5, -9, 8, -5, 7, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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