L11a273

L11a272

L11a274

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a273 at Knotilus!

Link Presentations

[edit Notes on L11a273's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X6,9,7,10 X18,7,19,8 X22,19,9,20 X20,15,21,16 X16,21,17,22 X8,17,1,18 X4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -5, 6, -10}, {5, -1, 2, -3, 11, -4, 8, -9, 10, -6, 7, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{-2 u^3 v^4+2 u^3 v^3-u^3 v^2-u^2 v^5+3 u^2 v^4-5 u^2 v^3+3 u^2 v^2-u^2 v-u v^4+3 u v^3-5 u v^2+3 u v-u-v^3+2 v^2-2 v}{u^{3/2} v^{5/2}}$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{6}{q^{13/2}}-\frac{9}{q^{15/2}}+\frac{11}{q^{17/2}}-\frac{11}{q^{19/2}}+\frac{10}{q^{21/2}}-\frac{8}{q^{23/2}}+\frac{5}{q^{25/2}}-\frac{3}{q^{27/2}}+\frac{1}{q^{29/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{13} \left(-z^3\right)-3 a^{13} z-a^{13} z^{-1} +3 a^{11} z^5+13 a^{11} z^3+14 a^{11} z+3 a^{11} z^{-1} -2 a^9 z^7-11 a^9 z^5-19 a^9 z^3-12 a^9 z-2 a^9 z^{-1} -a^7 z^7-5 a^7 z^5-7 a^7 z^3-3 a^7 z$ (db)
Kauffman polynomial	$-z^4 a^{18}+z^2 a^{18}-3 z^5 a^{17}+4 z^3 a^{17}-z a^{17}-4 z^6 a^{16}+4 z^4 a^{16}-4 z^7 a^{15}+3 z^5 a^{15}+2 z^3 a^{15}-2 z a^{15}-4 z^8 a^{14}+7 z^6 a^{14}-7 z^4 a^{14}+4 z^2 a^{14}-a^{14}-3 z^9 a^{13}+6 z^7 a^{13}-5 z^5 a^{13}+3 z^3 a^{13}-3 z a^{13}+a^{13} z^{-1} -z^{10} a^{12}-4 z^8 a^{12}+25 z^6 a^{12}-40 z^4 a^{12}+24 z^2 a^{12}-3 a^{12}-6 z^9 a^{11}+25 z^7 a^{11}-39 z^5 a^{11}+35 z^3 a^{11}-18 z a^{11}+3 a^{11} z^{-1} -z^{10} a^{10}-2 z^8 a^{10}+22 z^6 a^{10}-35 z^4 a^{10}+20 z^2 a^{10}-3 a^{10}-3 z^9 a^9+14 z^7 a^9-23 z^5 a^9+23 z^3 a^9-13 z a^9+2 a^9 z^{-1} -2 z^8 a^8+8 z^6 a^8-7 z^4 a^8+z^2 a^8-z^7 a^7+5 z^5 a^7-7 z^3 a^7+3 z a^7$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                       1 1 -8                     2 1 -1 -10                   3     3 -12                 3 2     -1 -14               6 3       3 -16             5 3         -2 -18           6 6           0 -20         4 5             1 -22       4 6               -2 -24     2 5                 3 -26   1 3                   -2 -28   2                     2 -30 1                       -1

Integral Khovanov Homology (db, data source)

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

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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