L11a22

L11a21

L11a23

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a22 at Knotilus!

Link Presentations

[edit Notes on L11a22's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X18,10,19,9 X22,14,5,13 X20,16,21,15 X16,20,17,19 X14,22,15,21 X8,18,9,17 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 4, -2, 11, -3, 5, -8, 6, -7, 9, -4, 7, -6, 8, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(t(1)-1) (t(2)-1) \left(3 t(2)^2-4 t(2)+3\right)}{\sqrt{t(1)} t(2)^{3/2}}$ (db)
Jones polynomial	$12 q^{9/2}-11 q^{7/2}+7 q^{5/2}-5 q^{3/2}+q^{21/2}-3 q^{19/2}+6 q^{17/2}-8 q^{15/2}+11 q^{13/2}-13 q^{11/2}+2 \sqrt{q}-\frac{1}{\sqrt{q}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^5 a^{-3} -z^5 a^{-5} -z^5 a^{-7} +z^3 a^{-1} -2 z^3 a^{-3} -2 z^3 a^{-7} +z^3 a^{-9} +2 z a^{-1} -2 z a^{-3} +2 z a^{-5} -3 z a^{-7} +z a^{-9} + a^{-1} z^{-1} - a^{-3} z^{-1} + a^{-5} z^{-1} -2 a^{-7} z^{-1} + a^{-9} z^{-1}$ (db)
Kauffman polynomial	$-z^{10} a^{-6} -z^{10} a^{-8} -2 z^9 a^{-5} -6 z^9 a^{-7} -4 z^9 a^{-9} -2 z^8 a^{-4} -2 z^8 a^{-6} -5 z^8 a^{-8} -5 z^8 a^{-10} -2 z^7 a^{-3} +3 z^7 a^{-5} +21 z^7 a^{-7} +13 z^7 a^{-9} -3 z^7 a^{-11} -2 z^6 a^{-2} +11 z^6 a^{-6} +28 z^6 a^{-8} +18 z^6 a^{-10} -z^6 a^{-12} -z^5 a^{-1} -8 z^5 a^{-5} -32 z^5 a^{-7} -14 z^5 a^{-9} +9 z^5 a^{-11} +4 z^4 a^{-2} +3 z^4 a^{-4} -23 z^4 a^{-6} -44 z^4 a^{-8} -19 z^4 a^{-10} +3 z^4 a^{-12} +3 z^3 a^{-1} +7 z^3 a^{-3} +12 z^3 a^{-5} +20 z^3 a^{-7} +8 z^3 a^{-9} -4 z^3 a^{-11} -z^2 a^{-2} +17 z^2 a^{-6} +27 z^2 a^{-8} +10 z^2 a^{-10} -z^2 a^{-12} -3 z a^{-1} -5 z a^{-3} -7 z a^{-5} -8 z a^{-7} -3 z a^{-9} - a^{-2} -4 a^{-6} -7 a^{-8} -3 a^{-10} + a^{-1} z^{-1} + a^{-3} z^{-1} + a^{-5} z^{-1} +2 a^{-7} z^{-1} + a^{-9} z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 22                       1 -1 20                     2   2 18                   4 1   -3 16                 4 2     2 14               7 4       -3 12             6 4         2 10           6 7           1 8         5 6             -1 6       2 6               4 4     3 5                 -2 2   1 4                   3 0   1                     -1 -2 1                       1

Integral Khovanov Homology (db, data source)

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

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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