L11a149

L11a148

L11a150

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a149 at Knotilus!

Link Presentations

[edit Notes on L11a149's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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