L11n137

L11n136

L11n138

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n137 at Knotilus!

Link Presentations

[edit Notes on L11n137's Link Presentations]

Planar diagram presentation	X8192 X18,11,19,12 X10,4,11,3 X2,17,3,18 X12,5,13,6 X6718 X16,10,17,9 X13,20,14,21 X15,22,16,7 X4,20,5,19 X21,14,22,15
Gauss code	{1, -4, 3, -10, 5, -6}, {6, -1, 7, -3, 2, -5, -8, 11, -9, -7, 4, -2, 10, 8, -11, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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