L11a542

L11a541

L11a543

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a542 at Knotilus!

Link Presentations

[edit Notes on L11a542's Link Presentations]

Planar diagram presentation	X6172 X12,6,13,5 X8493 X2,16,3,15 X16,7,17,8 X22,20,15,19 X14,22,11,21 X20,14,21,13 X18,10,19,9 X10,12,5,11 X4,17,1,18
Gauss code	{1, -4, 3, -11}, {10, -2, 8, -7}, {2, -1, 5, -3, 9, -10}, {4, -5, 11, -9, 6, -8, 7, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(u-1) (w-1) (x-1) (2 v w x+v (-w)-v x+v-w x+w+x-2)}{\sqrt{u} \sqrt{v} w x}$ (db)
Jones polynomial	$25 q^{9/2}-27 q^{7/2}+20 q^{5/2}-17 q^{3/2}+\frac{1}{q^{3/2}}-q^{19/2}+5 q^{17/2}-11 q^{15/2}+17 q^{13/2}-24 q^{11/2}+8 \sqrt{q}-\frac{4}{\sqrt{q}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^7 a^{-3} +z^7 a^{-5} -z^5 a^{-1} +3 z^5 a^{-3} +2 z^5 a^{-5} -z^5 a^{-7} -2 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} -z^3 a^{-7} +2 z a^{-3} -3 z a^{-5} +z a^{-7} +2 a^{-1} z^{-1} -4 a^{-3} z^{-1} +2 a^{-5} z^{-1} + a^{-1} z^{-3} -3 a^{-3} z^{-3} +3 a^{-5} z^{-3} - a^{-7} z^{-3}$ (db)
Kauffman polynomial	$z^5 a^{-11} +5 z^6 a^{-10} -4 z^4 a^{-10} +11 z^7 a^{-9} -15 z^5 a^{-9} +6 z^3 a^{-9} +13 z^8 a^{-8} -16 z^6 a^{-8} +3 z^4 a^{-8} +z^2 a^{-8} +8 z^9 a^{-7} +6 z^7 a^{-7} -31 z^5 a^{-7} +20 z^3 a^{-7} + a^{-7} z^{-3} -3 z a^{-7} -2 a^{-7} z^{-1} +2 z^{10} a^{-6} +22 z^8 a^{-6} -43 z^6 a^{-6} +14 z^4 a^{-6} +3 z^2 a^{-6} -3 a^{-6} z^{-2} +4 a^{-6} +13 z^9 a^{-5} -5 z^7 a^{-5} -35 z^5 a^{-5} +36 z^3 a^{-5} +3 a^{-5} z^{-3} -11 z a^{-5} -3 a^{-5} z^{-1} +2 z^{10} a^{-4} +15 z^8 a^{-4} -32 z^6 a^{-4} +9 z^4 a^{-4} +4 z^2 a^{-4} -6 a^{-4} z^{-2} +7 a^{-4} +5 z^9 a^{-3} +4 z^7 a^{-3} -30 z^5 a^{-3} +32 z^3 a^{-3} +3 a^{-3} z^{-3} -11 z a^{-3} -3 a^{-3} z^{-1} +6 z^8 a^{-2} -9 z^6 a^{-2} +3 z^2 a^{-2} -3 a^{-2} z^{-2} +4 a^{-2} +4 z^7 a^{-1} -10 z^5 a^{-1} +10 z^3 a^{-1} + a^{-1} z^{-3} -3 z a^{-1} -2 a^{-1} z^{-1} +z^6-2 z^4+z^2$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 20                       1 1 18                     4   -4 16                   7 1   6 14                 10 4     -6 12               14 7       7 10             13 12         -1 8           14 12           2 6         10 17             7 4       7 10               -3 2     3 12                 9 0   1 5                   -4 -2   3                     3 -4 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=2$ ${\mathbb Z}^{17}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{14}$ $r=3$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=4$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{14}$ $r=5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=8$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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