L11a499

L11a498

L11a500

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Link Presentations

[edit Notes on L11a499's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(1)-1) (t(2)-1) (t(3)-1) \left(2 t(3)^2-3 t(3)+2\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db)
Jones polynomial	$-q^8+3 q^7-7 q^6+12 q^5-16 q^4+19 q^3-17 q^2+16 q-11+7 q^{-1} -2 q^{-2} + q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^6 a^{-2} +z^6 a^{-4} +z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2-z^2 a^{-2} +6 z^2 a^{-4} -2 z^2 a^{-6} -4 z^2+2 a^2+4 a^{-4} -2 a^{-6} -4+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2}$ (db)
Kauffman polynomial	$z^5 a^{-9} -2 z^3 a^{-9} +3 z^6 a^{-8} -5 z^4 a^{-8} +6 z^7 a^{-7} -13 z^5 a^{-7} +11 z^3 a^{-7} -4 z a^{-7} +8 z^8 a^{-6} -21 z^6 a^{-6} +27 z^4 a^{-6} -14 z^2 a^{-6} +4 a^{-6} +5 z^9 a^{-5} -4 z^7 a^{-5} -9 z^5 a^{-5} +21 z^3 a^{-5} -8 z a^{-5} +z^{10} a^{-4} +14 z^8 a^{-4} -46 z^6 a^{-4} +62 z^4 a^{-4} -36 z^2 a^{-4} +8 a^{-4} +8 z^9 a^{-3} -14 z^7 a^{-3} +9 z^5 a^{-3} -3 z^3 a^{-3} +z^{10} a^{-2} +9 z^8 a^{-2} +a^2 z^6-26 z^6 a^{-2} -4 a^2 z^4+27 z^4 a^{-2} +6 a^2 z^2-16 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -4 a^2+3 z^9 a^{-1} +2 a z^7-2 z^7 a^{-1} -4 a z^5-11 z^3 a^{-1} +4 a z+8 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +3 z^8-3 z^6-7 z^4+12 z^2+2 z^{-2} -7$ (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                 7 2     5 9               9 5       -4 7             10 7         3 5           7 9           2 3         9 10             -1 1       6 11               5 -1     1 5                 -4 -3   1 6                   5 -5   1                     -1 -7 1                       1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{9}$ $r=1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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