L10a63

L10a62

L10a64

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a63 at Knotilus!

Link Presentations

[edit Notes on L10a63's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X20,10,7,9 X2738 X4,20,5,19 X14,5,15,6 X16,11,17,12 X18,13,19,14 X12,17,13,18 X6,15,1,16
Gauss code	{1, -4, 2, -5, 6, -10}, {4, -1, 3, -2, 7, -9, 8, -6, 10, -7, 9, -8, 5, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{u^2 v^4-2 u^2 v^3+2 u^2 v^2-2 u^2 v+u^2-2 u v^4+5 u v^3-5 u v^2+5 u v-2 u+v^4-2 v^3+2 v^2-2 v+1}{u v^2}$ (db)
Jones polynomial	$\frac{11}{q^{9/2}}-\frac{12}{q^{7/2}}+\frac{10}{q^{5/2}}+q^{3/2}-\frac{9}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{3}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{9}{q^{11/2}}-3 \sqrt{q}+\frac{5}{\sqrt{q}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^3 a^7-2 z a^7-a^7 z^{-1} +2 z^5 a^5+7 z^3 a^5+8 z a^5+3 a^5 z^{-1} -z^7 a^3-5 z^5 a^3-10 z^3 a^3-9 z a^3-2 a^3 z^{-1} +z^5 a+3 z^3 a+2 z a$ (db)
Kauffman polynomial	$-z^4 a^{10}+z^2 a^{10}-3 z^5 a^9+3 z^3 a^9-5 z^6 a^8+6 z^4 a^8-3 z^2 a^8+a^8-6 z^7 a^7+10 z^5 a^7-10 z^3 a^7+5 z a^7-a^7 z^{-1} -4 z^8 a^6+2 z^6 a^6+6 z^4 a^6-9 z^2 a^6+3 a^6-z^9 a^5-10 z^7 a^5+32 z^5 a^5-35 z^3 a^5+16 z a^5-3 a^5 z^{-1} -7 z^8 a^4+14 z^6 a^4-3 z^4 a^4-6 z^2 a^4+3 a^4-z^9 a^3-7 z^7 a^3+29 z^5 a^3-32 z^3 a^3+15 z a^3-2 a^3 z^{-1} -3 z^8 a^2+6 z^6 a^2+z^4 a^2-3 z^2 a^2-3 z^7 a+10 z^5 a-10 z^3 a+4 z a-z^6+3 z^4-2 z^2$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                     1 -1 2                   2   2 0                 3 1   -2 -2               6 2     4 -4             5 4       -1 -6           7 5         2 -8         5 6           1 -10       4 6             -2 -12     2 5               3 -14   1 4                 -3 -16   2                   2 -18 1                     -1

Integral Khovanov Homology (db, data source)

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

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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