L11a148

L11a147

L11a149

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a148 at Knotilus!

Link Presentations

[edit Notes on L11a148's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X20,11,21,12 X12,6,13,5 X4,20,5,19 X14,18,15,17 X16,8,17,7 X22,16,7,15 X18,14,19,13 X6,21,1,22
Gauss code	{1, -2, 3, -6, 5, -11}, {8, -1, 2, -3, 4, -5, 10, -7, 9, -8, 7, -10, 6, -4, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{2 u^2 v^4-5 u^2 v^3+5 u^2 v^2-2 u^2 v-2 u v^4+6 u v^3-9 u v^2+6 u v-2 u-2 v^3+5 v^2-5 v+2}{u v^2}$ (db)
Jones polynomial	$4 q^{9/2}-\frac{3}{q^{9/2}}-7 q^{7/2}+\frac{6}{q^{7/2}}+11 q^{5/2}-\frac{11}{q^{5/2}}-15 q^{3/2}+\frac{14}{q^{3/2}}-q^{11/2}+\frac{1}{q^{11/2}}+16 \sqrt{q}-\frac{17}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^3 z^5-z^5 a^{-3} -3 a^3 z^3-2 z^3 a^{-3} -2 a^3 z+z a^{-3} + a^{-3} z^{-1} +a z^7+z^7 a^{-1} +4 a z^5+3 z^5 a^{-1} +6 a z^3+5 a z-5 z a^{-1} +2 a z^{-1} -3 a^{-1} z^{-1}$ (db)
Kauffman polynomial	$a^6 z^4-a^6 z^2+z^7 a^{-5} +3 a^5 z^5-3 z^5 a^{-5} -3 a^5 z^3+2 z^3 a^{-5} +a^5 z+4 z^8 a^{-4} +5 a^4 z^6-15 z^6 a^{-4} -4 a^4 z^4+15 z^4 a^{-4} +a^4 z^2-2 z^2 a^{-4} - a^{-4} +5 z^9 a^{-3} +7 a^3 z^7-17 z^7 a^{-3} -9 a^3 z^5+15 z^5 a^{-3} +6 a^3 z^3-3 z^3 a^{-3} -a^3 z-z a^{-3} + a^{-3} z^{-1} +2 z^{10} a^{-2} +7 a^2 z^8+3 z^8 a^{-2} -9 a^2 z^6-26 z^6 a^{-2} +2 a^2 z^4+25 z^4 a^{-2} +2 a^2 z^2-2 z^2 a^{-2} -3 a^{-2} +5 a z^9+10 z^9 a^{-1} -3 a z^7-28 z^7 a^{-1} -12 a z^5+18 z^5 a^{-1} +15 a z^3+z^3 a^{-1} -8 a z-7 z a^{-1} +2 a z^{-1} +3 a^{-1} z^{-1} +2 z^{10}+6 z^8-25 z^6+17 z^4-3$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 12                       1 1 10                     3   -3 8                   4 1   3 6                 7 3     -4 4               8 4       4 2             8 7         -1 0           9 8           1 -2         6 9             3 -4       5 8               -3 -6     2 7                 5 -8   1 4                   -3 -10   2                     2 -12 1                       -1

Integral Khovanov Homology (db, data source)

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

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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