L11n301

L11n300

L11n302

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n301 at Knotilus!

Link Presentations

[edit Notes on L11n301's Link Presentations]

Planar diagram presentation	X6172 X3,13,4,12 X13,21,14,20 X19,11,20,22 X15,5,16,10 X17,9,18,8 X7,17,8,16 X9,19,10,18 X21,15,22,14 X2536 X11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -7, 6, -8, 5}, {-11, 2, -3, 9, -5, 7, -6, 8, -4, 3, -9, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{t(2) t(3)^4-2 t(1) t(2)^2 t(3)^3+t(2)^2 t(3)^3+2 t(1) t(2) t(3)^3-3 t(2) t(3)^3+t(3)^3+2 t(1) t(2)^2 t(3)^2-t(2)^2 t(3)^2+t(1) t(3)^2-4 t(1) t(2) t(3)^2+4 t(2) t(3)^2-2 t(3)^2-t(1) t(2)^2 t(3)-t(1) t(3)+3 t(1) t(2) t(3)-2 t(2) t(3)+2 t(3)-t(1) t(2)}{\sqrt{t(1)} t(2) t(3)^2}$ (db)
Jones polynomial	$-q^{11}+3 q^{10}-6 q^9+9 q^8-11 q^7+12 q^6-10 q^5+9 q^4-4 q^3+3 q^2$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^2 a^{-10} - a^{-10} z^{-2} -2 a^{-10} +3 z^4 a^{-8} +10 z^2 a^{-8} +4 a^{-8} z^{-2} +11 a^{-8} -2 z^6 a^{-6} -10 z^4 a^{-6} -20 z^2 a^{-6} -5 a^{-6} z^{-2} -18 a^{-6} +3 z^4 a^{-4} +10 z^2 a^{-4} +2 a^{-4} z^{-2} +9 a^{-4}$ (db)
Kauffman polynomial	$z^5 a^{-13} -2 z^3 a^{-13} +z a^{-13} +3 z^6 a^{-12} -6 z^4 a^{-12} +3 z^2 a^{-12} - a^{-12} +4 z^7 a^{-11} -6 z^5 a^{-11} +z^3 a^{-11} -2 z a^{-11} + a^{-11} z^{-1} +3 z^8 a^{-10} -9 z^4 a^{-10} +6 z^2 a^{-10} - a^{-10} z^{-2} +z^9 a^{-9} +8 z^7 a^{-9} -26 z^5 a^{-9} +33 z^3 a^{-9} -19 z a^{-9} +5 a^{-9} z^{-1} +7 z^8 a^{-8} -17 z^6 a^{-8} +23 z^4 a^{-8} -19 z^2 a^{-8} -4 a^{-8} z^{-2} +13 a^{-8} +z^9 a^{-7} +7 z^7 a^{-7} -28 z^5 a^{-7} +48 z^3 a^{-7} -35 z a^{-7} +9 a^{-7} z^{-1} +4 z^8 a^{-6} -14 z^6 a^{-6} +32 z^4 a^{-6} -38 z^2 a^{-6} -5 a^{-6} z^{-2} +22 a^{-6} +3 z^7 a^{-5} -9 z^5 a^{-5} +18 z^3 a^{-5} -19 z a^{-5} +5 a^{-5} z^{-1} +6 z^4 a^{-4} -16 z^2 a^{-4} -2 a^{-4} z^{-2} +11 a^{-4}$ (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   \ 0 1 2 3 4 5 6 7 8 9 χ 23                   1 -1 21                 2   2 19               4 1   -3 17             5 2     3 15           7 5       -2 13         5 4         1 11       5 7           2 9     4 5             -1 7   1 6               5 5 2 3                 -1 3 3                   3

Integral Khovanov Homology (db, data source)

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

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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