L11a520

L11a519

L11a521

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a520 at Knotilus!

Link Presentations

[edit Notes on L11a520's Link Presentations]

Planar diagram presentation	X8192 X18,8,19,7 X16,6,17,5 X10,16,11,15 X14,3,15,4 X4,11,5,12 X2,20,3,19 X20,9,21,10 X12,13,7,14 X22,18,13,17 X6,21,1,22
Gauss code	{1, -7, 5, -6, 3, -11}, {2, -1, 8, -4, 6, -9}, {9, -5, 4, -3, 10, -2, 7, -8, 11, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(u-1) (v-1) (w-1) \left(u v w^2-u v w+u v-u w^2+2 u w-u-v w^2+2 v w-v+w^2-w+1\right)}{u v w^{3/2}}$ (db)
Jones polynomial	$-q^5+5 q^4-13 q^3+23 q^2-31 q+37-36 q^{-1} +33 q^{-2} -23 q^{-3} +15 q^{-4} -6 q^{-5} + q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^4 z^4+a^4 z^{-2} -2 a^2 z^6-z^6 a^{-2} -4 a^2 z^4-2 z^4 a^{-2} -2 a^2 z^2-2 z^2 a^{-2} -2 a^2 z^{-2} -a^2+z^8+4 z^6+7 z^4+4 z^2+ z^{-2} +1$ (db)
Kauffman polynomial	$7 a^2 z^{10}+7 z^{10}+17 a^3 z^9+36 a z^9+19 z^9 a^{-1} +15 a^4 z^8+22 a^2 z^8+21 z^8 a^{-2} +28 z^8+6 a^5 z^7-27 a^3 z^7-67 a z^7-21 z^7 a^{-1} +13 z^7 a^{-3} +a^6 z^6-27 a^4 z^6-72 a^2 z^6-32 z^6 a^{-2} +5 z^6 a^{-4} -81 z^6-6 a^5 z^5+6 a^3 z^5+26 a z^5-13 z^5 a^{-3} +z^5 a^{-5} +11 a^4 z^4+41 a^2 z^4+21 z^4 a^{-2} -2 z^4 a^{-4} +53 z^4+a^3 z^3-a z^3+3 z^3 a^{-1} +5 z^3 a^{-3} +2 a^4 z^2-2 a^2 z^2-6 z^2 a^{-2} -10 z^2-a^3 z-a z+a^4+a^2+1+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     4   4 7                   9 1   -8 5                 14 4     10 3               17 9       -8 1             20 14         6 -1           18 19           1 -3         15 18             -3 -5       10 20               10 -7     5 13                 -8 -9   1 10                   9 -11   5                     -5 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-2$ ${\mathbb Z}^{20}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{15}$ $r=-1$ ${\mathbb Z}^{18}\oplus{\mathbb Z}_2^{18}$ ${\mathbb Z}^{18}$ $r=0$ ${\mathbb Z}^{19}\oplus{\mathbb Z}_2^{18}$ ${\mathbb Z}^{20}$ $r=1$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{17}$ ${\mathbb Z}^{17}$ $r=2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{14}$ ${\mathbb Z}^{14}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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