L11a416

L11a415

L11a417

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a416 at Knotilus!

Link Presentations

[edit Notes on L11a416's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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