L11n336

L11n335

L11n337

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n336 at Knotilus!

Link Presentations

[edit Notes on L11n336's Link Presentations]

Planar diagram presentation	X6172 X16,12,17,11 X8493 X2,18,3,17 X14,6,15,5 X18,7,19,8 X12,16,5,15 X13,20,14,21 X9,13,10,22 X21,11,22,10 X4,19,1,20
Gauss code	{1, -4, 3, -11}, {5, -1, 6, -3, -9, 10, 2, -7}, {-8, -5, 7, -2, 4, -6, 11, 8, -10, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(u-1) (v-1) (w-1) \left(v^2+v w+w^2\right)}{\sqrt{u} v^{3/2} w^{3/2}}$ (db)
Jones polynomial	$-q^7+3 q^6-4 q^5+8 q^4-7 q^3+8 q^2+ q^{-2} -7 q-3 q^{-1} +6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$2 z^4 a^{-2} +z^4 a^{-4} +4 z^2 a^{-2} -z^2 a^{-6} -3 z^2+a^2+3 a^{-2} -2 a^{-4} -2+ a^{-2} z^{-2} -2 a^{-4} z^{-2} + a^{-6} z^{-2}$ (db)
Kauffman polynomial	$2 z^9 a^{-3} +2 z^9 a^{-5} +5 z^8 a^{-2} +8 z^8 a^{-4} +3 z^8 a^{-6} +5 z^7 a^{-1} -z^7 a^{-3} -5 z^7 a^{-5} +z^7 a^{-7} -17 z^6 a^{-2} -33 z^6 a^{-4} -14 z^6 a^{-6} +2 z^6-17 z^5 a^{-1} -16 z^5 a^{-3} -3 z^5 a^{-5} -4 z^5 a^{-7} +15 z^4 a^{-2} +39 z^4 a^{-4} +20 z^4 a^{-6} -4 z^4+3 a z^3+19 z^3 a^{-1} +18 z^3 a^{-3} +6 z^3 a^{-5} +4 z^3 a^{-7} +a^2 z^2-3 z^2 a^{-2} -17 z^2 a^{-4} -9 z^2 a^{-6} +6 z^2-2 a z-6 z a^{-1} -2 z a^{-3} +2 z a^{-5} -a^2- a^{-2} -2 a^{-4} -2 a^{-6} -1-2 a^{-3} z^{-1} -2 a^{-5} z^{-1} + a^{-2} z^{-2} +2 a^{-4} z^{-2} + a^{-6} z^{-2}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 χ 15                   1 -1 13                 2   2 11               2 1   -1 9             6 2     4 7           3 4       1 5         5 4         1 3     1 3 3           1 1     4 5             -1 -1   1 4               3 -3   2                 -2 -5 1                   1

Integral Khovanov Homology (db, data source)

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

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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