L11a279

L11a278

L11a280

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a279 at Knotilus!

Link Presentations

[edit Notes on L11a279's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X8,9,1,10 X20,13,21,14 X14,6,15,5 X4,20,5,19 X18,8,19,7 X6,16,7,15 X22,17,9,18 X16,21,17,22
Gauss code	{1, -2, 3, -7, 6, -9, 8, -4}, {4, -1, 2, -3, 5, -6, 9, -11, 10, -8, 7, -5, 11, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(1)-1) (t(2)-1) \left(t(1)^2 t(2)^4-t(1)^2 t(2)^3+2 t(1) t(2)^3+t(1)^2 t(2)^2-t(1) t(2)^2+t(2)^2+2 t(1) t(2)-t(2)+1\right)}{t(1)^{3/2} t(2)^{5/2}}$ (db)
Jones polynomial	$\frac{12}{q^{9/2}}-\frac{14}{q^{7/2}}-q^{5/2}+\frac{13}{q^{5/2}}+3 q^{3/2}-\frac{12}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{3}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{9}{q^{11/2}}-6 \sqrt{q}+\frac{8}{\sqrt{q}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^5 z^7-5 a^5 z^5-8 a^5 z^3-5 a^5 z-a^5 z^{-1} +a^3 z^9+7 a^3 z^7+18 a^3 z^5+21 a^3 z^3+11 a^3 z+3 a^3 z^{-1} -a z^7-5 a z^5-8 a z^3-6 a z-2 a z^{-1}$ (db)
Kauffman polynomial	$-z^4 a^{10}+z^2 a^{10}-3 z^5 a^9+3 z^3 a^9-5 z^6 a^8+6 z^4 a^8-2 z^2 a^8-6 z^7 a^7+9 z^5 a^7-6 z^3 a^7+z a^7-6 z^8 a^6+12 z^6 a^6-12 z^4 a^6+6 z^2 a^6-a^6-5 z^9 a^5+13 z^7 a^5-17 z^5 a^5+14 z^3 a^5-6 z a^5+a^5 z^{-1} -2 z^{10} a^4-z^8 a^4+18 z^6 a^4-27 z^4 a^4+16 z^2 a^4-3 a^4-9 z^9 a^3+36 z^7 a^3-52 z^5 a^3+39 z^3 a^3-15 z a^3+3 a^3 z^{-1} -2 z^{10} a^2+2 z^8 a^2+13 z^6 a^2-20 z^4 a^2+10 z^2 a^2-3 a^2-4 z^9 a+16 z^7 a-19 z^5 a+12 z^3 a-7 z a+2 a z^{-1} -3 z^8+12 z^6-12 z^4+3 z^2-z^7 a^{-1} +4 z^5 a^{-1} -4 z^3 a^{-1} +z a^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                       1 1 4                     2   -2 2                   4 1   3 0                 4 2     -2 -2               8 4       4 -4             7 6         -1 -6           7 6           1 -8         5 7             2 -10       4 7               -3 -12     2 5                 3 -14   1 4                   -3 -16   2                     2 -18 1                       -1

Integral Khovanov Homology (db, data source)

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

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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