L11n94

L11n93

L11n95

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n94 at Knotilus!

Link Presentations

[edit Notes on L11n94's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X9,14,10,15 X19,22,20,5 X11,21,12,20 X21,11,22,10 X15,19,16,18 X7,17,8,16 X17,9,18,8 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, -8, 9, -3, 6, -5, -2, 11, 3, -7, 8, -9, 7, -4, 5, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(u-1) (v-1) \left(v^2+1\right)}{\sqrt{u} v^{3/2}}$ (db)
Jones polynomial	$q^{9/2}-2 q^{7/2}+2 q^{5/2}-\frac{1}{q^{5/2}}-4 q^{3/2}+\frac{2}{q^{3/2}}+q^{15/2}-q^{13/2}+3 \sqrt{q}-\frac{3}{\sqrt{q}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z a^{-7} + a^{-7} z^{-1} -z^3 a^{-5} -4 z a^{-5} -3 a^{-5} z^{-1} +2 z^3 a^{-3} +6 z a^{-3} +3 a^{-3} z^{-1} -z^5 a^{-1} +a z^3-4 z^3 a^{-1} +2 a z-5 z a^{-1} +a z^{-1} -2 a^{-1} z^{-1}$ (db)
Kauffman polynomial	$-z^9 a^{-1} -z^9 a^{-3} -4 z^8 a^{-2} -2 z^8 a^{-4} -2 z^8-a z^7+2 z^7 a^{-1} +3 z^7 a^{-3} -z^7 a^{-5} -z^7 a^{-7} +21 z^6 a^{-2} +12 z^6 a^{-4} -z^6 a^{-8} +10 z^6+5 a z^5+9 z^5 a^{-1} +6 z^5 a^{-3} +8 z^5 a^{-5} +6 z^5 a^{-7} -29 z^4 a^{-2} -18 z^4 a^{-4} +3 z^4 a^{-6} +5 z^4 a^{-8} -13 z^4-7 a z^3-19 z^3 a^{-1} -18 z^3 a^{-3} -14 z^3 a^{-5} -8 z^3 a^{-7} +13 z^2 a^{-2} +8 z^2 a^{-4} -5 z^2 a^{-6} -5 z^2 a^{-8} +5 z^2+4 a z+11 z a^{-1} +14 z a^{-3} +10 z a^{-5} +3 z a^{-7} -2 a^{-2} +2 a^{-6} + a^{-8} -a z^{-1} -2 a^{-1} z^{-1} -3 a^{-3} z^{-1} -3 a^{-5} z^{-1} - a^{-7} z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                         0 12                 1 1 1   1 10               2 1       -1 8             2 2 1       1 6           2 3 1         0 4         3 2 1           2 2       2 4 1             1 0     1 2 1               0 -2   1 2                   1 -4   1                     -1 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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