L10a75

L10a74

L10a76

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a75 at Knotilus!

Link Presentations

[edit Notes on L10a75's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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