L11a407

L11a406

L11a408

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Link Presentations

[edit Notes on L11a407's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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