L11a465

L11a464

L11a466

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Link Presentations

[edit Notes on L11a465's Link Presentations]

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

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	$-q^8+4 q^7-9 q^6+13 q^5-16 q^4+19 q^3+ q^{-3} -17 q^2-2 q^{-2} +15 q+6 q^{-1} -9$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^4 a^{-6} -z^2 a^{-6} - a^{-6} +z^6 a^{-4} +2 z^4 a^{-4} +2 z^2 a^{-4} + a^{-4} +z^6 a^{-2} +2 z^4 a^{-2} +a^2 z^2+3 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +2 a^2+3 a^{-2} -2 z^4-5 z^2-2 z^{-2} -5$ (db)
Kauffman polynomial	$z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +7 z^9 a^{-3} +5 z^9 a^{-5} +5 z^8 a^{-2} +12 z^8 a^{-4} +9 z^8 a^{-6} +2 z^8+2 a z^7+z^7 a^{-1} -11 z^7 a^{-3} -2 z^7 a^{-5} +8 z^7 a^{-7} +a^2 z^6-9 z^6 a^{-2} -31 z^6 a^{-4} -18 z^6 a^{-6} +4 z^6 a^{-8} +z^6-5 a z^5-5 z^5 a^{-1} +8 z^5 a^{-3} -7 z^5 a^{-5} -14 z^5 a^{-7} +z^5 a^{-9} -4 a^2 z^4+31 z^4 a^{-4} +15 z^4 a^{-6} -5 z^4 a^{-8} -15 z^4+2 a z^3-4 z^3 a^{-1} -5 z^3 a^{-3} +8 z^3 a^{-5} +6 z^3 a^{-7} -z^3 a^{-9} +6 a^2 z^2+7 z^2 a^{-2} -13 z^2 a^{-4} -7 z^2 a^{-6} +19 z^2+3 a z+7 z a^{-1} +3 z a^{-3} -3 z a^{-5} -2 z a^{-7} -4 a^2-6 a^{-2} +2 a^{-4} +2 a^{-6} -9-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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                     3   3 13                   6 1   -5 11                 7 3     4 9               9 6       -3 7             10 7         3 5           7 9           2 3         8 10             -2 1       5 11               6 -1     1 4                 -3 -3   1 5                   4 -5   1                     -1 -7 1                       1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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