L11n41

L11n40

L11n42

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n41 at Knotilus!

Link Presentations

[edit Notes on L11n41's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X9,18,10,19 X8493 X5,14,6,15 X15,20,16,21 X17,22,18,5 X21,16,22,17 X19,10,20,11 X2,12,3,11
Gauss code	{1, -11, 5, -3}, {-6, -1, 2, -5, -4, 10, 11, -2, 3, 6, -7, 9, -8, 4, -10, 7, -9, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(u-1) (v-1) \left(v^4-v^3+v^2-v+1\right)}{\sqrt{u} v^{5/2}}$ (db)
Jones polynomial	$-\frac{7}{q^{9/2}}+\frac{5}{q^{7/2}}-\frac{5}{q^{5/2}}+\frac{2}{q^{3/2}}+\frac{1}{q^{19/2}}-\frac{2}{q^{17/2}}+\frac{5}{q^{15/2}}-\frac{6}{q^{13/2}}+\frac{6}{q^{11/2}}-\frac{1}{\sqrt{q}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^9 z^{-1} -z^5 a^7-3 z^3 a^7-z a^7+a^7 z^{-1} +z^7 a^5+5 z^5 a^5+8 z^3 a^5+6 z a^5+2 a^5 z^{-1} -z^5 a^3-4 z^3 a^3-5 z a^3-2 a^3 z^{-1}$ (db)
Kauffman polynomial	$-z^2 a^{12}+a^{12}-2 z^3 a^{11}+z a^{11}-z^6 a^{10}+z^4 a^{10}-2 z^2 a^{10}-3 z^7 a^9+10 z^5 a^9-12 z^3 a^9+4 z a^9+a^9 z^{-1} -3 z^8 a^8+9 z^6 a^8-6 z^4 a^8+3 z^2 a^8-3 a^8-z^9 a^7-3 z^7 a^7+23 z^5 a^7-27 z^3 a^7+8 z a^7+a^7 z^{-1} -5 z^8 a^6+18 z^6 a^6-14 z^4 a^6+2 z^2 a^6-z^9 a^5-z^7 a^5+18 z^5 a^5-26 z^3 a^5+12 z a^5-2 a^5 z^{-1} -2 z^8 a^4+8 z^6 a^4-7 z^4 a^4-2 z^2 a^4+3 a^4-z^7 a^3+5 z^5 a^3-9 z^3 a^3+7 z a^3-2 a^3 z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 0                   1 1 -2                 1   -1 -4               4 1   3 -6             3 3     0 -8           4 2       2 -10       1 3 3         1 -12       4 4           0 -14     2 3             1 -16   1 4               -3 -18   1                 1 -20 1                   -1

Integral Khovanov Homology (db, data source)

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

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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