L10a161

L10a160

L10a162

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a161 at Knotilus!

Link Presentations

[edit Notes on L10a161's Link Presentations]

Planar diagram presentation	X8192 X14,4,15,3 X10,20,11,19 X18,10,19,9 X20,12,13,11 X12,14,7,13 X16,6,17,5 X2738 X4,16,5,15 X6,18,1,17
Gauss code	{1, -8, 2, -9, 7, -10}, {8, -1, 4, -3, 5, -6}, {6, -2, 9, -7, 10, -4, 3, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{t(1)^2 t(2)^2 t(3)^3-t(1) t(2)^2 t(3)^3-t(1)^2 t(2) t(3)^3-t(1)^2 t(3)^2+t(1) t(2)^2 t(3)^2-t(2)^2 t(3)^2+t(1)^2 t(2) t(3)^2-t(1) t(2) t(3)^2+t(1)^2 t(3)+t(2)^2 t(3)-t(1) t(3)+t(1) t(2) t(3)-t(2) t(3)+t(1)+t(2)-1}{t(1) t(2) t(3)^{3/2}}$ (db)
Jones polynomial	$q^{11}-2 q^{10}+3 q^9-4 q^8+5 q^7-4 q^6+5 q^5-3 q^4+3 q^3-q^2+q$ (db)
Signature	6 (db)
HOMFLY-PT polynomial	$z^6 a^{-8} +5 z^4 a^{-8} +7 z^2 a^{-8} + a^{-8} z^{-2} +3 a^{-8} -z^8 a^{-6} -7 z^6 a^{-6} -17 z^4 a^{-6} -18 z^2 a^{-6} -2 a^{-6} z^{-2} -9 a^{-6} +z^6 a^{-4} +6 z^4 a^{-4} +11 z^2 a^{-4} + a^{-4} z^{-2} +6 a^{-4}$ (db)
Kauffman polynomial	$z^9 a^{-5} +z^9 a^{-7} +z^8 a^{-4} +4 z^8 a^{-6} +3 z^8 a^{-8} -5 z^7 a^{-5} -z^7 a^{-7} +4 z^7 a^{-9} -7 z^6 a^{-4} -23 z^6 a^{-6} -11 z^6 a^{-8} +5 z^6 a^{-10} +5 z^5 a^{-5} -11 z^5 a^{-7} -12 z^5 a^{-9} +4 z^5 a^{-11} +17 z^4 a^{-4} +43 z^4 a^{-6} +9 z^4 a^{-8} -14 z^4 a^{-10} +3 z^4 a^{-12} +6 z^3 a^{-5} +20 z^3 a^{-7} +6 z^3 a^{-9} -6 z^3 a^{-11} +2 z^3 a^{-13} -17 z^2 a^{-4} -32 z^2 a^{-6} -3 z^2 a^{-8} +9 z^2 a^{-10} -2 z^2 a^{-12} +z^2 a^{-14} -9 z a^{-5} -9 z a^{-7} +7 a^{-4} +11 a^{-6} +3 a^{-8} -2 a^{-10} +2 a^{-5} z^{-1} +2 a^{-7} z^{-1} - a^{-4} z^{-2} -2 a^{-6} z^{-2} - a^{-8} z^{-2}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 23                     1 1 21                   2 1 -1 19                 1     1 17               3 2     -1 15             2 1       1 13           2 3         1 11         3 2           1 9       2 4             2 7     1 1               0 5   1 3                 2 3                       0 1 1                     1

Integral Khovanov Homology (db, data source)

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

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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