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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X14,17,15,18 X16,7,17,8 X22,15,7,16 X18,13,19,14 X6,20,1,19 X20,12,21,11 X12,6,13,5 X4,21,5,22

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 4q-17/2 - 8q-15/2 + 13q-13/2 - 17q-11/2 + 19q-9/2 - 19q-7/2 + 15q-5/2 - 12q-3/2 + 6q-1/2 - 3q1/2 + q3/2

A2 (sl(3)) Invariant:

q-28 - 2q-26 + q-24 - q-22 - 3q-20 + 3q-18 - 3q-16 + 3q-14 + q-12 + q-10 + 6q-8 - q-6 + 4q-4 - 1 + q2 - q4

HOMFLY-PT Polynomial:

2az + 3az3 + az5 - 2a3z-1 - 7a3z - 7a3z3 - 4a3z5 - a3z7 + 3a5z-1 + 4a5z - a5z3 - 3a5z5 - a5z7 - a7z-1 + 2a7z3 + a7z5

Kauffman Polynomial:

- 2z2 + 3z4 - z6 + 3az - 8az3 + 9az5 - 3az7 - 2a2z2 - a2z4 + 8a2z6 - 4a2z8 - 2a3z-1 + 8a3z - 8a3z3 + 2a3z5 + 6a3z7 - 4a3z9 + 3a4 - a4z2 - 10a4z4 + 13a4z6 - 3a4z8 - 2a4z10 - 3a5z-1 + 5a5z + 6a5z3 - 24a5z5 + 24a5z7 - 10a5z9 + 3a6 - 18a6z4 + 21a6z6 - 7a6z8 - 2a6z10 - a7z-1 + 2a7z3 - 5a7z5 + 8a7z7 - 6a7z9 + a8 - 6a8z4 + 13a8z6 - 8a8z8 - 3a9z3 + 11a9z5 - 7a9z7 - a10z2 + 6a10z4 - 4a10z6 + a11z3 - 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                 8 2     j = -4               8 5       j = -6             11 7         j = -8           9 9           j = -10         8 10             j = -12       5 9               j = -14     3 8                 j = -16   1 5                   j = -18   3                     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, 149]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 149]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[14, 17, 15, 18], > X[16, 7, 17, 8], X[22, 15, 7, 16], X[18, 13, 19, 14], X[6, 20, 1, 19], > X[20, 12, 21, 11], X[12, 6, 13, 5], X[4, 21, 5, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 10, -8}, > {5, -1, 2, -3, 9, -10, 7, -4, 6, -5, 4, -7, 8, -9, 11, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 4 8 13 17 19 19 15 12 -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 2 -24 -22 3 3 3 3 -12 -10 6 -1 + q - --- + q - q - --- + --- - --- + --- + q + q + -- - 26 20 18 16 14 8 q q q q q q -6 4 2 4 > q + -- + q - q 4 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 149]][a, z]
Out[8]=	3 5 7 -2 a 3 a a 3 5 3 3 3 5 3 ----- + ---- - -- + 2 a z - 7 a z + 4 a z + 3 a z - 7 a z - a z + z z z 7 3 5 3 5 5 5 7 5 3 7 5 7 > 2 a z + a z - 4 a z - 3 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 149]][a, z]
Out[9]=	3 5 7 4 6 8 2 a 3 a a 3 5 2 3 a + 3 a + a - ---- - ---- - -- + 3 a z + 8 a z + 5 a z - 2 z - z z z 2 2 4 2 10 2 3 3 3 5 3 7 3 9 3 > 2 a z - a z - a z - 8 a z - 8 a z + 6 a z + 2 a z - 3 a z + 11 3 4 2 4 4 4 6 4 8 4 10 4 5 > a z + 3 z - a z - 10 a z - 18 a z - 6 a z + 6 a z + 9 a z + 3 5 5 5 7 5 9 5 11 5 6 2 6 > 2 a z - 24 a z - 5 a z + 11 a z - a z - z + 8 a z + 4 6 6 6 8 6 10 6 7 3 7 5 7 > 13 a z + 21 a z + 13 a z - 4 a z - 3 a z + 6 a z + 24 a z + 7 7 9 7 2 8 4 8 6 8 8 8 3 9 > 8 a z - 7 a z - 4 a z - 3 a z - 7 a z - 8 a z - 4 a z - 5 9 7 9 4 10 6 10 > 10 a z - 6 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 8 1 3 1 5 3 8 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 9 8 10 9 9 11 7 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 L11a149

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