L11a414

L11a413

L11a415

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Link Presentations

[edit Notes on L11a414's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{t(1) t(2)^2 t(3)^4-2 t(2)^2 t(3)^4-t(1) t(2) t(3)^4+t(2) t(3)^4-t(1) t(2)^2 t(3)^3+2 t(2)^2 t(3)^3-t(1) t(3)^3+2 t(1) t(2) t(3)^3-2 t(2) t(3)^3+t(3)^3+t(1) t(2)^2 t(3)^2-2 t(2)^2 t(3)^2+2 t(1) t(3)^2-2 t(1) t(2) t(3)^2+2 t(2) t(3)^2-t(3)^2-t(1) t(2)^2 t(3)+t(2)^2 t(3)-2 t(1) t(3)+2 t(1) t(2) t(3)-2 t(2) t(3)+t(3)+2 t(1)-t(1) t(2)+t(2)-1}{\sqrt{t(1)} t(2) t(3)^2}$ (db)
Jones polynomial	$-q^9+3 q^8-6 q^7+9 q^6-11 q^5+12 q^4-11 q^3+10 q^2-6 q+5- q^{-1} + q^{-2}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^8 a^{-4} -2 z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -11 z^4 a^{-2} +14 z^4 a^{-4} -4 z^4 a^{-6} +z^4-21 z^2 a^{-2} +18 z^2 a^{-4} -5 z^2 a^{-6} +5 z^2-17 a^{-2} +13 a^{-4} -3 a^{-6} +7-5 a^{-2} z^{-2} +4 a^{-4} z^{-2} - a^{-6} z^{-2} +2 z^{-2}$ (db)
Kauffman polynomial	$z^{10} a^{-2} +z^{10} a^{-4} +z^9 a^{-1} +5 z^9 a^{-3} +4 z^9 a^{-5} -z^8 a^{-2} +6 z^8 a^{-4} +8 z^8 a^{-6} +z^8-3 z^7 a^{-1} -16 z^7 a^{-3} -3 z^7 a^{-5} +10 z^7 a^{-7} -13 z^6 a^{-2} -32 z^6 a^{-4} -17 z^6 a^{-6} +9 z^6 a^{-8} -7 z^6-4 z^5 a^{-1} -z^5 a^{-3} -23 z^5 a^{-5} -20 z^5 a^{-7} +6 z^5 a^{-9} +36 z^4 a^{-2} +37 z^4 a^{-4} +3 z^4 a^{-6} -13 z^4 a^{-8} +3 z^4 a^{-10} +18 z^4+20 z^3 a^{-1} +39 z^3 a^{-3} +33 z^3 a^{-5} +9 z^3 a^{-7} -4 z^3 a^{-9} +z^3 a^{-11} -39 z^2 a^{-2} -20 z^2 a^{-4} +3 z^2 a^{-6} +5 z^2 a^{-8} -21 z^2-19 z a^{-1} -35 z a^{-3} -19 z a^{-5} -2 z a^{-7} +z a^{-9} +22 a^{-2} +13 a^{-4} - a^{-8} +11+5 a^{-1} z^{-1} +9 a^{-3} z^{-1} +5 a^{-5} z^{-1} + a^{-7} z^{-1} -5 a^{-2} z^{-2} -4 a^{-4} z^{-2} - a^{-6} z^{-2} -2 z^{-2}$ (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 19                       1 -1 17                     2   2 15                   4 1   -3 13                 5 2     3 11               6 4       -2 9             6 5         1 7           7 8           1 5         3 4             -1 3       4 8               4 1     1 2                 -1 -1     4                   4 -3 1 1                     0 -5 1                       1

Integral Khovanov Homology (db, data source)

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

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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