L11a204

L11a203

L11a205

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a204 at Knotilus!

Link Presentations

[edit Notes on L11a204's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X16,5,17,6 X18,13,19,14 X14,17,15,18 X22,20,7,19 X20,12,21,11 X12,22,13,21 X6718 X4,15,5,16
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 8, -9, 5, -6, 11, -4, 6, -5, 7, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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