L11a439

L11a438

L11a440

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a439 at Knotilus!

Link Presentations

[edit Notes on L11a439's Link Presentations]

Planar diagram presentation	X6172 X14,6,15,5 X10,4,11,3 X2,16,3,15 X20,18,21,17 X16,9,17,10 X8,20,9,19 X18,8,19,7 X22,12,13,11 X12,14,5,13 X4,21,1,22
Gauss code	{1, -4, 3, -11}, {2, -1, 8, -7, 6, -3, 9, -10}, {10, -2, 4, -6, 5, -8, 7, -5, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(1)-1) (t(2)-1)^2 (t(3)-1)^2 (t(2) t(3)+1)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}}$ (db)
Jones polynomial	$q^{10}-4 q^9+9 q^8-14 q^7+18 q^6-20 q^5+21 q^4-16 q^3+13 q^2-7 q+4- q^{-1}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^4 a^{-8} +2 z^2 a^{-8} + a^{-8} -2 z^6 a^{-6} -7 z^4 a^{-6} -7 z^2 a^{-6} + a^{-6} z^{-2} -2 a^{-6} +z^8 a^{-4} +5 z^6 a^{-4} +9 z^4 a^{-4} +6 z^2 a^{-4} -2 a^{-4} z^{-2} - a^{-4} -z^6 a^{-2} -3 z^4 a^{-2} -z^2 a^{-2} + a^{-2} z^{-2} +2 a^{-2}$ (db)
Kauffman polynomial	$z^4 a^{-12} +4 z^5 a^{-11} -z^3 a^{-11} +9 z^6 a^{-10} -8 z^4 a^{-10} +3 z^2 a^{-10} - a^{-10} +13 z^7 a^{-9} -18 z^5 a^{-9} +9 z^3 a^{-9} -2 z a^{-9} +12 z^8 a^{-8} -15 z^6 a^{-8} +4 z^2 a^{-8} - a^{-8} +7 z^9 a^{-7} +z^7 a^{-7} -27 z^5 a^{-7} +21 z^3 a^{-7} -6 z a^{-7} +2 z^{10} a^{-6} +15 z^8 a^{-6} -49 z^6 a^{-6} +38 z^4 a^{-6} -9 z^2 a^{-6} + a^{-6} z^{-2} - a^{-6} +12 z^9 a^{-5} -28 z^7 a^{-5} +8 z^5 a^{-5} +10 z^3 a^{-5} -2 z a^{-5} -2 a^{-5} z^{-1} +2 z^{10} a^{-4} +7 z^8 a^{-4} -40 z^6 a^{-4} +46 z^4 a^{-4} -15 z^2 a^{-4} +2 a^{-4} z^{-2} -2 a^{-4} +5 z^9 a^{-3} -15 z^7 a^{-3} +10 z^5 a^{-3} +z^3 a^{-3} +2 z a^{-3} -2 a^{-3} z^{-1} +4 z^8 a^{-2} -15 z^6 a^{-2} +17 z^4 a^{-2} -5 z^2 a^{-2} + a^{-2} z^{-2} -2 a^{-2} +z^7 a^{-1} -3 z^5 a^{-1} +2 z^3 a^{-1}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 21                       1 1 19                     3   -3 17                   6 1   5 15                 8 3     -5 13               10 6       4 11             12 10         -2 9           9 8           1 7         7 12             5 5       6 9               -3 3     3 9                 6 1   1 4                   -3 -1   3                     3 -3 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=3$ $i=5$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=4$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=8$ ${\mathbb Z}_2$ ${\mathbb Z}$

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.

L11a438 L11a440