L11a257

L11a256

L11a258

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a257 at Knotilus!

Link Presentations

[edit Notes on L11a257's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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