L11n432

L11n431

L11n433

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n432 at Knotilus!

Link Presentations

[edit Notes on L11n432's Link Presentations]

Planar diagram presentation	X8192 X14,4,15,3 X12,14,7,13 X2738 X22,10,13,9 X6,22,1,21 X15,20,16,21 X5,17,6,16 X11,19,12,18 X17,11,18,10 X19,5,20,4
Gauss code	{1, -4, 2, 11, -8, -6}, {4, -1, 5, 10, -9, -3}, {3, -2, -7, 8, -10, 9, -11, 7, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(t(1) t(2)+t(3)) \left(-t(1) t(3)^2+t(1) t(2) t(3)^2-t(2) t(3)^2+t(1)+t(2)-1\right)}{t(1) t(2) t(3)^{3/2}}$ (db)
Jones polynomial	$q^9-q^8+q^7+q^5+q^4+q^2-q+1$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^6 a^{-4} +z^4 a^{-2} -5 z^4 a^{-4} +4 z^2 a^{-2} -5 z^2 a^{-4} +z^2 a^{-8} +2 a^{-2} -3 a^{-6} + a^{-8} + a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2}$ (db)
Kauffman polynomial	$z^6 a^{-10} -5 z^4 a^{-10} +5 z^2 a^{-10} +z^7 a^{-9} -5 z^5 a^{-9} +5 z^3 a^{-9} +z^8 a^{-8} -7 z^6 a^{-8} +16 z^4 a^{-8} -17 z^2 a^{-8} - a^{-8} z^{-2} +7 a^{-8} +z^7 a^{-7} -7 z^5 a^{-7} +13 z^3 a^{-7} -9 z a^{-7} +2 a^{-7} z^{-1} +z^8 a^{-6} -8 z^6 a^{-6} +21 z^4 a^{-6} -25 z^2 a^{-6} -2 a^{-6} z^{-2} +11 a^{-6} +z^7 a^{-5} -7 z^5 a^{-5} +13 z^3 a^{-5} -9 z a^{-5} +2 a^{-5} z^{-1} +z^6 a^{-4} -5 z^4 a^{-4} +3 z^2 a^{-4} - a^{-4} z^{-2} +3 a^{-4} +z^7 a^{-3} -5 z^5 a^{-3} +5 z^3 a^{-3} +z^6 a^{-2} -5 z^4 a^{-2} +6 z^2 a^{-2} -2 a^{-2}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 19                     1 1 17                   1 1 0 15               1 1     0 13             1 1 1     1 11           1 3 1       1 9         2 1 1         2 7       1 4 2           1 5     1 1 1             1 3   1 2 1               0 1                       0 -1 1                     1

Integral Khovanov Homology (db, data source)

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

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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