L11a415

L11a414

L11a416

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a415 at Knotilus!

Link Presentations

[edit Notes on L11a415's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X18,10,19,9 X16,8,17,7 X22,16,11,15 X20,14,21,13 X14,22,15,21 X10,18,5,17 X8,20,9,19 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, 3, -8}, {11, -2, 6, -7, 5, -4, 8, -3, 9, -6, 7, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{-t(1) t(3)^4+t(1) t(2) t(3)^4-2 t(2) t(3)^4+t(3)^4+t(1) t(2)^2 t(3)^3-2 t(2)^2 t(3)^3+2 t(1) t(3)^3-3 t(1) t(2) t(3)^3+4 t(2) t(3)^3-t(3)^3-t(1) t(2)^2 t(3)^2+2 t(2)^2 t(3)^2-2 t(1) t(3)^2+4 t(1) t(2) t(3)^2-4 t(2) t(3)^2+t(3)^2+t(1) t(2)^2 t(3)-2 t(2)^2 t(3)+2 t(1) t(3)-4 t(1) t(2) t(3)+3 t(2) t(3)-t(3)-t(1) t(2)^2+t(2)^2+2 t(1) t(2)-t(2)}{\sqrt{t(1)} t(2) t(3)^2}$ (db)
Jones polynomial	$-q^8+3 q^7-7 q^6+11 q^5-14 q^4+17 q^3+ q^{-3} -15 q^2-2 q^{-2} +14 q+6 q^{-1} -9$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^4 a^{-6} -2 z^2 a^{-6} - a^{-6} z^{-2} -2 a^{-6} +z^6 a^{-4} +3 z^4 a^{-4} +6 z^2 a^{-4} +4 a^{-4} z^{-2} +8 a^{-4} +z^6 a^{-2} +z^4 a^{-2} +a^2 z^2-4 z^2 a^{-2} -5 a^{-2} z^{-2} +2 a^2-8 a^{-2} -2 z^4-4 z^2+2 z^{-2}$ (db)
Kauffman polynomial	$z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +6 z^9 a^{-3} +4 z^9 a^{-5} +5 z^8 a^{-2} +9 z^8 a^{-4} +7 z^8 a^{-6} +3 z^8+2 a z^7+2 z^7 a^{-1} -8 z^7 a^{-3} -2 z^7 a^{-5} +6 z^7 a^{-7} +a^2 z^6-17 z^6 a^{-2} -30 z^6 a^{-4} -18 z^6 a^{-6} +3 z^6 a^{-8} -7 z^6-5 a z^5-16 z^5 a^{-1} -9 z^5 a^{-3} -13 z^5 a^{-5} -14 z^5 a^{-7} +z^5 a^{-9} -4 a^2 z^4+27 z^4 a^{-2} +44 z^4 a^{-4} +22 z^4 a^{-6} -5 z^4 a^{-8} +6 z^4+2 a z^3+21 z^3 a^{-1} +35 z^3 a^{-3} +30 z^3 a^{-5} +12 z^3 a^{-7} -2 z^3 a^{-9} +5 a^2 z^2-33 z^2 a^{-2} -33 z^2 a^{-4} -12 z^2 a^{-6} -7 z^2+a z-16 z a^{-1} -33 z a^{-3} -21 z a^{-5} -5 z a^{-7} -2 a^2+20 a^{-2} +17 a^{-4} +4 a^{-6} +6+5 a^{-1} z^{-1} +9 a^{-3} z^{-1} +5 a^{-5} z^{-1} + a^{-7} z^{-1} -5 a^{-2} z^{-2} -4 a^{-4} z^{-2} - a^{-6} z^{-2} -2 z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     2   2 13                   5 1   -4 11                 6 2     4 9               8 5       -3 7             9 6         3 5           8 10           2 3         6 7             -1 1       4 9               5 -1     2 5                 -3 -3     4                   4 -5 1 2                     -1 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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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