L11a57

L11a56

L11a58

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Link Presentations

[edit Notes on L11a57's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(t(1)-1) (t(2)-1) \left(t(2)^2-3 t(2)+1\right) \left(t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$-4 q^{9/2}+\frac{3}{q^{9/2}}+8 q^{7/2}-\frac{7}{q^{7/2}}-13 q^{5/2}+\frac{12}{q^{5/2}}+17 q^{3/2}-\frac{16}{q^{3/2}}+q^{11/2}-\frac{1}{q^{11/2}}-20 \sqrt{q}+\frac{18}{\sqrt{q}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^5 z+a^5 z^{-1} +z^5 a^{-3} -3 a^3 z^3+2 z^3 a^{-3} -6 a^3 z+z a^{-3} -3 a^3 z^{-1} -z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +9 a z^3-7 z^3 a^{-1} +9 a z-5 z a^{-1} +4 a z^{-1} -2 a^{-1} z^{-1}$ (db)
Kauffman polynomial	$-a^2 z^{10}-z^{10}-3 a^3 z^9-8 a z^9-5 z^9 a^{-1} -3 a^4 z^8-11 a^2 z^8-10 z^8 a^{-2} -18 z^8-a^5 z^7+3 a^3 z^7+5 a z^7-10 z^7 a^{-1} -11 z^7 a^{-3} +11 a^4 z^6+44 a^2 z^6+10 z^6 a^{-2} -8 z^6 a^{-4} +51 z^6+4 a^5 z^5+17 a^3 z^5+42 a z^5+47 z^5 a^{-1} +14 z^5 a^{-3} -4 z^5 a^{-5} -14 a^4 z^4-47 a^2 z^4+3 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} -38 z^4-6 a^5 z^3-32 a^3 z^3-60 a z^3-44 z^3 a^{-1} -8 z^3 a^{-3} +2 z^3 a^{-5} +7 a^4 z^2+19 a^2 z^2-2 z^2 a^{-2} -2 z^2 a^{-4} +12 z^2+4 a^5 z+18 a^3 z+26 a z+15 z a^{-1} +3 z a^{-3} -a^4-3 a^2- a^{-2} -2-a^5 z^{-1} -3 a^3 z^{-1} -4 a z^{-1} -2 a^{-1} z^{-1}$ (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                   5 1   -4 6                 8 3     5 4               9 5       -4 2             11 8         3 0           9 11           2 -2         7 9             -2 -4       5 9               4 -6     2 7                 -5 -8   1 5                   4 -10   2                     -2 -12 1                       1

Integral Khovanov Homology (db, data source)

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

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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