L11a425

L11a424

L11a426

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Link Presentations

[edit Notes on L11a425's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{2 u v^2 w^2-3 u v^2 w+u v^2-5 u v w^2+7 u v w-3 u v+3 u w^2-5 u w+2 u-2 v^2 w^2+5 v^2 w-3 v^2+3 v w^2-7 v w+5 v-w^2+3 w-2}{\sqrt{u} v w}$ (db)
Jones polynomial	$-q^8+4 q^7-9 q^6+14 q^5-18 q^4+21 q^3-19 q^2+17 q-11+7 q^{-1} -2 q^{-2} + q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^4 a^{-6} -z^2 a^{-6} - a^{-6} +z^6 a^{-4} +2 z^4 a^{-4} +3 z^2 a^{-4} +2 a^{-4} +z^6 a^{-2} +z^4 a^{-2} +a^2 z^2+a^2 z^{-2} + a^{-2} z^{-2} +2 a^2+ a^{-2} -2 z^4-4 z^2-2 z^{-2} -4$ (db)
Kauffman polynomial	$z^5 a^{-9} -z^3 a^{-9} +4 z^6 a^{-8} -5 z^4 a^{-8} +z^2 a^{-8} +8 z^7 a^{-7} -13 z^5 a^{-7} +7 z^3 a^{-7} -2 z a^{-7} +9 z^8 a^{-6} -14 z^6 a^{-6} +9 z^4 a^{-6} -5 z^2 a^{-6} +2 a^{-6} +5 z^9 a^{-5} +4 z^7 a^{-5} -21 z^5 a^{-5} +17 z^3 a^{-5} -4 z a^{-5} +z^{10} a^{-4} +16 z^8 a^{-4} -39 z^6 a^{-4} +37 z^4 a^{-4} -18 z^2 a^{-4} +4 a^{-4} +8 z^9 a^{-3} -7 z^7 a^{-3} -6 z^5 a^{-3} +5 z^3 a^{-3} +z^{10} a^{-2} +10 z^8 a^{-2} +a^2 z^6-25 z^6 a^{-2} -4 a^2 z^4+21 z^4 a^{-2} +6 a^2 z^2-7 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -4 a^2-2 a^{-2} +3 z^9 a^{-1} +2 a z^7-z^7 a^{-1} -4 a z^5-3 z^5 a^{-1} -4 z^3 a^{-1} +4 a z+6 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +3 z^8-3 z^6-6 z^4+11 z^2+2 z^{-2} -7$ (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                     3   3 13                   6 1   -5 11                 8 3     5 9               10 6       -4 7             11 8         3 5           9 11           2 3         8 10             -2 1       5 11               6 -1     2 6                 -4 -3   1 6                   5 -5   1                     -1 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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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