L11a80

L11a79

L11a81

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Link Presentations

[edit Notes on L11a80's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{t(1) t(2)^5-2 t(2)^5-4 t(1) t(2)^4+8 t(2)^4+9 t(1) t(2)^3-11 t(2)^3-11 t(1) t(2)^2+9 t(2)^2+8 t(1) t(2)-4 t(2)-2 t(1)+1}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$q^{11/2}-4 q^{9/2}+10 q^{7/2}-16 q^{5/2}+20 q^{3/2}-23 \sqrt{q}+\frac{22}{\sqrt{q}}-\frac{19}{q^{3/2}}+\frac{13}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -3 a^3 z^3+8 a z^3-9 z^3 a^{-1} +2 z^3 a^{-3} +a^5 z-5 a^3 z+8 a z-9 z a^{-1} +3 z a^{-3} +a^5 z^{-1} -2 a^3 z^{-1} +3 a z^{-1} -3 a^{-1} z^{-1} + a^{-3} z^{-1}$ (db)
Kauffman polynomial	$z^4 a^{-6} +a^5 z^7-4 a^5 z^5+4 z^5 a^{-5} +6 a^5 z^3-4 a^5 z+a^5 z^{-1} +3 a^4 z^8-10 a^4 z^6+10 z^6 a^{-4} +11 a^4 z^4-8 z^4 a^{-4} -4 a^4 z^2+4 z^2 a^{-4} - a^{-4} +4 a^3 z^9-8 a^3 z^7+16 z^7 a^{-3} -4 a^3 z^5-23 z^5 a^{-3} +16 a^3 z^3+13 z^3 a^{-3} -9 a^3 z-4 z a^{-3} +2 a^3 z^{-1} + a^{-3} z^{-1} +2 a^2 z^{10}+8 a^2 z^8+16 z^8 a^{-2} -39 a^2 z^6-24 z^6 a^{-2} +42 a^2 z^4+6 z^4 a^{-2} -15 a^2 z^2+4 z^2 a^{-2} +2 a^2-2 a^{-2} +13 a z^9+9 z^9 a^{-1} -25 a z^7-6 a z^5-33 z^5 a^{-1} +27 a z^3+30 z^3 a^{-1} -15 a z-14 z a^{-1} +3 a z^{-1} +3 a^{-1} z^{-1} +2 z^{10}+21 z^8-63 z^6+46 z^4-11 z^2$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     3   3 8                   7 1   -6 6                 9 3     6 4               11 7       -4 2             12 9         3 0           11 12           1 -2         8 11             -3 -4       5 11               6 -6     3 8                 -5 -8   1 6                   5 -10   2                     -2 -12 1                       1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=0$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{12}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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