L10a155

L10a154

L10a156

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a155 at Knotilus!

Link Presentations

[edit Notes on L10a155's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X20,14,17,13 X18,8,19,7 X10,18,11,17 X14,9,15,10 X8,15,9,16 X16,20,5,19 X2536 X4,11,1,12
Gauss code	{1, -9, 2, -10}, {5, -4, 8, -3}, {9, -1, 4, -7, 6, -5, 10, -2, 3, -6, 7, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(w-1) \left(u v w^2-3 u v w+2 u v+4 u w-2 u-2 v w^2+4 v w+2 w^2-3 w+1\right)}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db)
Jones polynomial	$q^4-4 q^3+10 q^2-12 q+16-16 q^{-1} +15 q^{-2} -11 q^{-3} +7 q^{-4} -3 q^{-5} + q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^6-3 z^2 a^4-2 a^4+3 z^4 a^2+5 z^2 a^2+a^2 z^{-2} +4 a^2-z^6-3 z^4-7 z^2-2 z^{-2} -6+z^4 a^{-2} +z^2 a^{-2} + a^{-2} z^{-2} +3 a^{-2}$ (db)
Kauffman polynomial	$2 a^3 z^9+2 a z^9+4 a^4 z^8+12 a^2 z^8+8 z^8+3 a^5 z^7+7 a^3 z^7+16 a z^7+12 z^7 a^{-1} +a^6 z^6-7 a^4 z^6-21 a^2 z^6+10 z^6 a^{-2} -3 z^6-8 a^5 z^5-27 a^3 z^5-38 a z^5-15 z^5 a^{-1} +4 z^5 a^{-3} -3 a^6 z^4+4 a^2 z^4-12 z^4 a^{-2} +z^4 a^{-4} -12 z^4+7 a^5 z^3+23 a^3 z^3+17 a z^3+z^3 a^{-1} +3 a^6 z^2+4 a^4 z^2+3 a^2 z^2+8 z^2 a^{-2} +10 z^2-2 a^5 z-6 a^3 z+2 a z+6 z a^{-1} -a^6-2 a^2-5 a^{-2} -7-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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 χ 9                     1 1 7                   4 1 -3 5                 6     6 3               6 4     -2 1             10 6       4 -1           8 8         0 -3         7 8           -1 -5       4 8             4 -7     3 7               -4 -9   1 5                 4 -11   2                   -2 -13 1                     1

Integral Khovanov Homology (db, data source)

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

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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