L11a118

L11a117

L11a119

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a118 at Knotilus!

Link Presentations

[edit Notes on L11a118's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X16,10,17,9 X20,11,21,12 X8,21,9,22 X18,7,19,8 X12,19,13,20 X10,16,11,15 X22,17,5,18 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 6, -5, 3, -8, 4, -7, 11, -2, 8, -3, 9, -6, 7, -4, 5, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{3 u v^4-12 u v^3+14 u v^2-7 u v+u+v^5-7 v^4+14 v^3-12 v^2+3 v}{\sqrt{u} v^{5/2}}$ (db)
Jones polynomial	$-\sqrt{q}+\frac{4}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{16}{q^{5/2}}-\frac{22}{q^{7/2}}+\frac{24}{q^{9/2}}-\frac{24}{q^{11/2}}+\frac{20}{q^{13/2}}-\frac{15}{q^{15/2}}+\frac{8}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11} z^{-1} +4 a^9 z+3 a^9 z^{-1} -6 a^7 z^3-8 a^7 z-3 a^7 z^{-1} +3 a^5 z^5+6 a^5 z^3+6 a^5 z+2 a^5 z^{-1} +a^3 z^5-2 a^3 z^3-4 a^3 z-a^3 z^{-1} -a z^3$ (db)
Kauffman polynomial	$-z^6 a^{12}+3 z^4 a^{12}-3 z^2 a^{12}+a^{12}-3 z^7 a^{11}+7 z^5 a^{11}-6 z^3 a^{11}+3 z a^{11}-a^{11} z^{-1} -5 z^8 a^{10}+8 z^6 a^{10}-z^4 a^{10}-4 z^2 a^{10}+2 a^{10}-5 z^9 a^9+2 z^7 a^9+13 z^5 a^9-17 z^3 a^9+12 z a^9-3 a^9 z^{-1} -2 z^{10} a^8-13 z^8 a^8+36 z^6 a^8-28 z^4 a^8+9 z^2 a^8-13 z^9 a^7+15 z^7 a^7+13 z^5 a^7-26 z^3 a^7+15 z a^7-3 a^7 z^{-1} -2 z^{10} a^6-20 z^8 a^6+52 z^6 a^6-44 z^4 a^6+15 z^2 a^6-2 a^6-8 z^9 a^5+z^7 a^5+22 z^5 a^5-26 z^3 a^5+11 z a^5-2 a^5 z^{-1} -12 z^8 a^4+21 z^6 a^4-16 z^4 a^4+5 z^2 a^4-9 z^7 a^3+14 z^5 a^3-10 z^3 a^3+5 z a^3-a^3 z^{-1} -4 z^6 a^2+4 z^4 a^2-z^5 a+z^3 a$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                       1 1 0                     3   -3 -2                   7 1   6 -4                 10 4     -6 -6               12 6       6 -8             12 10         -2 -10           12 12           0 -12         9 13             4 -14       6 11               -5 -16     2 9                 7 -18   1 6                   -5 -20   2                     2 -22 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-5$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-4$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{12}$ $r=-3$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=-2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

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.

L11a117 L11a119