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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X20,5,21,6 X14,8,15,7 X16,14,17,13 X8,16,1,15 X22,17,9,18 X18,21,19,22 X6,9,7,10 X4,19,5,20

Gauss Code:

{{1, -2, 3, -11, 4, -10, 5, -7}, {10, -1, 2, -3, 6, -5, 7, -6, 8, -9, 11, -4, 9, -8}}

Jones Polynomial:

- q-19/2 + 3q-17/2 - 7q-15/2 + 12q-13/2 - 15q-11/2 + 17q-9/2 - 18q-7/2 + 14q-5/2 - 11q-3/2 + 6q-1/2 - 3q1/2 + q3/2

A2 (sl(3)) Invariant:

q-28 - q-26 + 2q-24 - 3q-20 + 2q-18 - 3q-16 + 3q-14 + 2q-12 + q-10 + 5q-8 - 2q-6 + 3q-4 - 1 + q2 - q4

HOMFLY-PT Polynomial:

2az + 3az3 + az5 - a3z-1 - 4a3z - 6a3z3 - 4a3z5 - a3z7 + a5z-1 - 3a5z - 6a5z3 - 4a5z5 - a5z7 + 3a7z + 3a7z3 + a7z5

Kauffman Polynomial:

- 2z2 + 3z4 - z6 + 3az - 8az3 + 9az5 - 3az7 - 4a2z4 + 9a2z6 - 4a2z8 - a3z-1 + 5a3z - 12a3z3 + 11a3z5 + a3z7 - 3a3z9 + a4 + 6a4z2 - 22a4z4 + 24a4z6 - 8a4z8 - a4z10 - a5z-1 - a5z + a5z3 - 5a5z5 + 11a5z7 - 7a5z9 + 13a6z2 - 30a6z4 + 27a6z6 - 10a6z8 - a6z10 - 2a7z + a7z3 + a7z5 + 2a7z7 - 4a7z9 + 7a8z2 - 10a8z4 + 10a8z6 - 6a8z8 - 2a9z3 + 7a9z5 - 5a9z7 - 2a10z2 + 5a10z4 - 3a10z6 - a11z + 2a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                       1 j = 2                     2   j = 0                   4 1   j = -2                 7 2     j = -4               8 5       j = -6             10 6         j = -8           7 8           j = -10         8 10             j = -12       5 8               j = -14     2 7                 j = -16   1 5                   j = -18   2                     j = -20 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 320]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 320]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[20, 5, 21, 6], > X[14, 8, 15, 7], X[16, 14, 17, 13], X[8, 16, 1, 15], X[22, 17, 9, 18], > X[18, 21, 19, 22], X[6, 9, 7, 10], X[4, 19, 5, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 4, -10, 5, -7}, > {10, -1, 2, -3, 6, -5, 7, -6, 8, -9, 11, -4, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 3 7 12 15 17 18 14 11 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q 6 3/2 > ------- - 3 Sqrt[q] + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -26 2 3 2 3 3 2 -10 5 2 3 -1 + q - q + --- - --- + --- - --- + --- + --- + q + -- - -- + -- + 24 20 18 16 14 12 8 6 4 q q q q q q q q q 2 4 > q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 320]][a, z]
Out[8]=	3 5 a a 3 5 7 3 3 3 5 3 -(--) + -- + 2 a z - 4 a z - 3 a z + 3 a z + 3 a z - 6 a z - 6 a z + z z 7 3 5 3 5 5 5 7 5 3 7 5 7 > 3 a z + a z - 4 a z - 4 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 320]][a, z]
Out[9]=	3 5 4 a a 3 5 7 11 2 4 2 a - -- - -- + 3 a z + 5 a z - a z - 2 a z - a z - 2 z + 6 a z + z z 6 2 8 2 10 2 3 3 3 5 3 7 3 > 13 a z + 7 a z - 2 a z - 8 a z - 12 a z + a z + a z - 9 3 11 3 4 2 4 4 4 6 4 8 4 > 2 a z + 2 a z + 3 z - 4 a z - 22 a z - 30 a z - 10 a z + 10 4 5 3 5 5 5 7 5 9 5 11 5 6 > 5 a z + 9 a z + 11 a z - 5 a z + a z + 7 a z - a z - z + 2 6 4 6 6 6 8 6 10 6 7 3 7 > 9 a z + 24 a z + 27 a z + 10 a z - 3 a z - 3 a z + a z + 5 7 7 7 9 7 2 8 4 8 6 8 8 8 > 11 a z + 2 a z - 5 a z - 4 a z - 8 a z - 10 a z - 6 a z - 3 9 5 9 7 9 4 10 6 10 > 3 a z - 7 a z - 4 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 7 1 2 1 5 2 7 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 7 16 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 8 8 10 7 8 10 6 8 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 4 t + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 q t q t q t q t q t q t q t q t 2 t 2 2 2 4 3 > --- + t + 2 q t + q t 2 q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a320

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