L10a117

L10a116

L10a118

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a117 at Knotilus!

Link Presentations

[edit Notes on L10a117's Link Presentations]

Planar diagram presentation	X12,1,13,2 X2,13,3,14 X14,3,15,4 X4,11,5,12 X20,9,11,10 X16,5,17,6 X18,7,19,8 X6,17,7,18 X8,19,9,20 X10,15,1,16
Gauss code	{1, -2, 3, -4, 6, -8, 7, -9, 5, -10}, {4, -1, 2, -3, 10, -6, 8, -7, 9, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{u^4 \left(-v^3\right)-u^3 v^4+2 u^3 v^3-2 u^3 v^2-2 u^2 v^3+3 u^2 v^2-2 u^2 v-2 u v^2+2 u v-u-v}{u^2 v^2}$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{5}{q^{15/2}}+\frac{6}{q^{17/2}}-\frac{6}{q^{19/2}}+\frac{5}{q^{21/2}}-\frac{4}{q^{23/2}}+\frac{2}{q^{25/2}}-\frac{1}{q^{27/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$z^5 a^{11}+4 z^3 a^{11}+4 z a^{11}+a^{11} z^{-1} -z^7 a^9-5 z^5 a^9-7 z^3 a^9-3 z a^9-a^9 z^{-1} -z^7 a^7-6 z^5 a^7-11 z^3 a^7-6 z a^7$ (db)
Kauffman polynomial	$-z^3 a^{17}+z a^{17}-2 z^4 a^{16}+z^2 a^{16}-3 z^5 a^{15}+3 z^3 a^{15}-2 z a^{15}-3 z^6 a^{14}+3 z^4 a^{14}-z^2 a^{14}-3 z^7 a^{13}+6 z^5 a^{13}-5 z^3 a^{13}+2 z a^{13}-2 z^8 a^{12}+4 z^6 a^{12}-2 z^4 a^{12}+2 z^2 a^{12}-z^9 a^{11}+2 z^7 a^{11}-2 z^5 a^{11}+7 z^3 a^{11}-5 z a^{11}+a^{11} z^{-1} -3 z^8 a^{10}+11 z^6 a^{10}-10 z^4 a^{10}+3 z^2 a^{10}-a^{10}-z^9 a^9+4 z^7 a^9-5 z^5 a^9+5 z^3 a^9-4 z a^9+a^9 z^{-1} -z^8 a^8+4 z^6 a^8-3 z^4 a^8-z^2 a^8-z^7 a^7+6 z^5 a^7-11 z^3 a^7+6 z a^7$ (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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                     1 1 -8                   1 1 0 -10                 2     2 -12               2 1     -1 -14             3 2       1 -16           3 2         -1 -18         3 3           0 -20       2 3             1 -22     2 3               -1 -24     2                 2 -26 1 2                   -1 -28 1                     1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-8$ $i=-6$ $r=-10$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{2}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

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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