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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X16,5,17,6 X18,13,19,14 X14,17,15,18 X22,20,7,19 X20,12,21,11 X12,22,13,21 X6718 X4,15,5,16

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 3q-17/2 - 6q-15/2 + 9q-13/2 - 12q-11/2 + 13q-9/2 - 13q-7/2 + 10q-5/2 - 8q-3/2 + 4q-1/2 - 2q1/2 + q3/2

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

az-1 + 4az + 4az3 + az5 - 3a3z-1 - 10a3z - 10a3z3 - 5a3z5 - a3z7 + 2a5z-1 + a5z - 4a5z3 - 4a5z5 - a5z7 + 2a7z + 3a7z3 + a7z5

Kauffman Polynomial:

1 - 4z2 + 4z4 - z6 - az-1 + 4az - 7az3 + 7az5 - 2az7 + 3a2 - 9a2z2 + 6a2z4 + 3a2z6 - 2a2z8 - 3a3z-1 + 12a3z - 12a3z3 + 4a3z5 + 3a3z7 - 2a3z9 + 3a4 - 3a4z2 - 5a4z4 + 6a4z6 - a4z8 - a4z10 - 2a5z-1 + 4a5z + 3a5z3 - 17a5z5 + 14a5z7 - 5a5z9 + 6a6z2 - 16a6z4 + 11a6z6 - 3a6z8 - a6z10 - 3a7z + 6a7z3 - 7a7z5 + 5a7z7 - 3a7z9 + 2a8z2 - 3a8z4 + 6a8z6 - 4a8z8 + 6a9z5 - 4a9z7 - 2a10z2 + 6a10z4 - 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                     1   j = 0                   3 1   j = -2                 5 1     j = -4               6 4       j = -6             7 4         j = -8           6 6           j = -10         6 7             j = -12       3 6               j = -14     3 6                 j = -16   1 4                   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, 204]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 204]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[16, 5, 17, 6], > X[18, 13, 19, 14], X[14, 17, 15, 18], X[22, 20, 7, 19], X[20, 12, 21, 11], > X[12, 22, 13, 21], X[6, 7, 1, 8], X[4, 15, 5, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 4, -10}, > {10, -1, 2, -3, 8, -9, 5, -6, 11, -4, 6, -5, 7, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 3 6 9 12 13 13 10 8 -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 4 3/2 > ------- - 2 Sqrt[q] + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -26 -24 -20 3 -16 3 -12 -10 4 -6 -1 + q - q + q - q + --- - q + --- + q + q + -- - q + 18 14 8 q q q 2 -2 4 > -- - q - q 4 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 204]][a, z]
Out[8]=	3 5 a 3 a 2 a 3 5 7 3 3 3 - - ---- + ---- + 4 a z - 10 a z + a z + 2 a z + 4 a z - 10 a z - z z z 5 3 7 3 5 3 5 5 5 7 5 3 7 5 7 > 4 a z + 3 a z + a z - 5 a z - 4 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 204]][a, z]
Out[9]=	3 5 2 4 a 3 a 2 a 3 5 7 11 1 + 3 a + 3 a - - - ---- - ---- + 4 a z + 12 a z + 4 a z - 3 a z - a z - z z z 2 2 2 4 2 6 2 8 2 10 2 3 > 4 z - 9 a z - 3 a z + 6 a z + 2 a z - 2 a z - 7 a z - 3 3 5 3 7 3 11 3 4 2 4 4 4 > 12 a z + 3 a z + 6 a z + 2 a z + 4 z + 6 a z - 5 a z - 6 4 8 4 10 4 5 3 5 5 5 7 5 > 16 a z - 3 a z + 6 a z + 7 a z + 4 a z - 17 a z - 7 a z + 9 5 11 5 6 2 6 4 6 6 6 8 6 10 6 > 6 a z - a z - z + 3 a z + 6 a z + 11 a z + 6 a z - 3 a z - 7 3 7 5 7 7 7 9 7 2 8 4 8 > 2 a z + 3 a z + 14 a z + 5 a z - 4 a z - 2 a z - a z - 6 8 8 8 3 9 5 9 7 9 4 10 6 10 > 3 a z - 4 a z - 2 a z - 5 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 5 1 2 1 4 3 6 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 6 6 7 6 6 7 4 6 t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 3 t + -- + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q t q t q 2 2 2 4 3 > t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a204

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