L10a118

L10a117

L10a119

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a118 at Knotilus!

Rich Schwartz' "25" [1]

Two interlaced pentagons.

Two hollow interlaced pentagrams.

Link Presentations

[edit Notes on L10a118's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{-t(1)^4 t(2)^4-t(1)^3 t(2)^3-t(1)^2 t(2)^2-t(1) t(2)-1}{t(1)^2 t(2)^2}$ (db)
Jones polynomial	$-\frac{1}{q^{9/2}}-\frac{1}{q^{29/2}}+\frac{1}{q^{27/2}}-\frac{1}{q^{25/2}}+\frac{1}{q^{23/2}}-\frac{1}{q^{21/2}}+\frac{1}{q^{19/2}}-\frac{1}{q^{17/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-9 (db)
HOMFLY-PT polynomial	$a^{11} z^7+7 a^{11} z^5+15 a^{11} z^3+10 a^{11} z+a^{11} z^{-1} -a^9 z^9-9 a^9 z^7-28 a^9 z^5-35 a^9 z^3-15 a^9 z-a^9 z^{-1}$ (db)
Kauffman polynomial	$a^{19} z+a^{18} z^2+a^{17} z^3-a^{17} z+a^{16} z^4-2 a^{16} z^2+a^{15} z^5-3 a^{15} z^3+a^{15} z+a^{14} z^6-4 a^{14} z^4+3 a^{14} z^2+a^{13} z^7-5 a^{13} z^5+6 a^{13} z^3-a^{13} z+a^{12} z^8-6 a^{12} z^6+10 a^{12} z^4-4 a^{12} z^2+a^{11} z^9-8 a^{11} z^7+22 a^{11} z^5-25 a^{11} z^3+11 a^{11} z-a^{11} z^{-1} +a^{10} z^8-7 a^{10} z^6+15 a^{10} z^4-10 a^{10} z^2+a^{10}+a^9 z^9-9 a^9 z^7+28 a^9 z^5-35 a^9 z^3+15 a^9 z-a^9 z^{-1}$ (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 -8                     1 1 -10                     1 1 -12                 1     1 -14                       0 -16             1 1       0 -18                       0 -20         1 1           0 -22                       0 -24     1 1               0 -26                       0 -28 1 1                   0 -30 1                     1

Integral Khovanov Homology (db, data source)

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

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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