L11a431

L11a430

L11a432

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Link Presentations

[edit Notes on L11a431's Link Presentations]

Planar diagram presentation	X6172 X14,6,15,5 X8493 X2,16,3,15 X16,7,17,8 X18,10,19,9 X20,11,21,12 X22,20,13,19 X12,14,5,13 X4,17,1,18 X10,21,11,22
Gauss code	{1, -4, 3, -10}, {2, -1, 5, -3, 6, -11, 7, -9}, {9, -2, 4, -5, 10, -6, 8, -7, 11, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(u-1) (v-1) (w-1) \left(v^2 w^2-v^2 w-v w^2+v w-v-w+1\right)}{\sqrt{u} v^{3/2} w^{3/2}}$ (db)
Jones polynomial	$q^7-4 q^6+7 q^5-11 q^4- q^{-4} +16 q^3+4 q^{-3} -17 q^2-7 q^{-2} +18 q+12 q^{-1} -14$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^8 a^{-2} -5 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-8 z^4 a^{-2} +3 z^4 a^{-4} +7 z^4-2 a^2 z^2-3 z^2 a^{-2} +z^2 a^{-4} +4 z^2+a^2+3 a^{-2} - a^{-4} -3+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2}$ (db)
Kauffman polynomial	$2 z^{10} a^{-2} +2 z^{10}+5 a z^9+11 z^9 a^{-1} +6 z^9 a^{-3} +4 a^2 z^8+9 z^8 a^{-2} +8 z^8 a^{-4} +5 z^8+a^3 z^7-16 a z^7-31 z^7 a^{-1} -6 z^7 a^{-3} +8 z^7 a^{-5} -15 a^2 z^6-36 z^6 a^{-2} -8 z^6 a^{-4} +7 z^6 a^{-6} -36 z^6-3 a^3 z^5+11 a z^5+19 z^5 a^{-1} -5 z^5 a^{-3} -6 z^5 a^{-5} +4 z^5 a^{-7} +16 a^2 z^4+28 z^4 a^{-2} -4 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} +40 z^4+2 a^3 z^3+2 z^3 a^{-3} -3 z^3 a^{-5} -3 z^3 a^{-7} -5 a^2 z^2+z^2 a^{-2} +5 z^2 a^{-4} +2 z^2 a^{-6} -7 z^2+a z+3 z a^{-1} +3 z a^{-3} +z a^{-5} -2 a^2-6 a^{-2} -2 a^{-4} -5-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                       1 1 13                     3   -3 11                   4 1   3 9                 7 3     -4 7               9 4       5 5             8 7         -1 3           10 9           1 1         8 12             4 -1       4 6               -2 -3     3 8                 5 -5   1 4                   -3 -7   3                     3 -9 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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