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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X18,9,19,10 X20,15,21,16 X16,19,17,20 X8,17,9,18 X4,21,1,22 X12,6,13,5 X10,4,11,3 X22,12,5,11 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

q-15/2 - 4q-13/2 + 9q-11/2 - 14q-9/2 + 20q-7/2 - 24q-5/2 + 22q-3/2 - 21q-1/2 + 15q1/2 - 9q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-22 + 2q-20 - 2q-18 - q-16 + q-14 - 6q-12 + 4q-10 + q-8 + 4q-6 + 6q-4 - 2q-2 + 5 - 3q2 + 2q6 - 2q8 + q10

HOMFLY-PT Polynomial:

- a-1z - 2a-1z3 - a-1z5 - 2az-1 + 3az3 + 3az5 + az7 + 3a3z-1 + 2a3z + 3a3z3 + 3a3z5 + a3z7 - a5z-1 - a5z - 2a5z3 - a5z5

Kauffman Polynomial:

a-3z3 - a-3z5 - 2a-2z2 + 5a-2z4 - 4a-2z6 + 2a-1z - 8a-1z3 + 12a-1z5 - 8a-1z7 - 3z2 + z4 + 9z6 - 9z8 + 2az-1 + az - 13az3 + 20az5 - 4az7 - 6az9 - 3a2 + a2z2 - 10a2z4 + 27a2z6 - 14a2z8 - 2a2z10 + 3a3z-1 - 3a3z + a3z3 + 15a3z7 - 12a3z9 - 3a4 + 5a4z2 - 19a4z4 + 31a4z6 - 12a4z8 - 2a4z10 + a5z-1 - 2a5z + 2a5z5 + 7a5z7 - 6a5z9 - a6 + 2a6z2 - 11a6z4 + 16a6z6 - 7a6z8 - 5a7z3 + 9a7z5 - 4a7z7 - a8z2 + 2a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                       1 j = 6                     3   j = 4                   6 1   j = 2                 9 3     j = 0               12 6       j = -2             12 11         j = -4           12 10           j = -6         8 12             j = -8       6 12               j = -10     3 8                 j = -12   1 6                   j = -14   3                     j = -16 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, 45]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 45]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[18, 9, 19, 10], X[20, 15, 21, 16], > X[16, 19, 17, 20], X[8, 17, 9, 18], X[4, 21, 1, 22], X[12, 6, 13, 5], > X[10, 4, 11, 3], X[22, 12, 5, 11], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 9, -7}, {8, -1, 2, -6, 3, -9, 10, -8, 11, -2, 4, -5, 6, -3, > 5, -4, 7, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 4 9 14 20 24 22 21 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 15 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 2 -16 -14 6 4 -8 4 6 2 2 5 - q + --- - --- - q + q - --- + --- + q + -- + -- - -- - 3 q + 20 18 12 10 6 4 2 q q q q q q q 6 8 10 > 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 45]][a, z]
Out[8]=	3 5 3 5 -2 a 3 a a z 3 5 2 z 3 3 3 5 3 z ---- + ---- - -- - - + 2 a z - a z - ---- + 3 a z + 3 a z - 2 a z - -- + z z z a a a 5 3 5 5 5 7 3 7 > 3 a z + 3 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 45]][a, z]
Out[9]=	3 5 2 4 6 2 a 3 a a 2 z 3 5 2 -3 a - 3 a - a + --- + ---- + -- + --- + a z - 3 a z - 2 a z - 3 z - z z z a 2 3 3 2 z 2 2 4 2 6 2 8 2 z 8 z 3 3 3 > ---- + a z + 5 a z + 2 a z - a z + -- - ---- - 13 a z + a z - 2 3 a a a 4 5 7 3 4 5 z 2 4 4 4 6 4 8 4 z > 5 a z + z + ---- - 10 a z - 19 a z - 11 a z + 2 a z - -- + 2 3 a a 5 6 12 z 5 5 5 7 5 6 4 z 2 6 4 6 > ----- + 20 a z + 2 a z + 9 a z + 9 z - ---- + 27 a z + 31 a z + a 2 a 7 6 6 8 6 8 z 7 3 7 5 7 7 7 8 > 16 a z - a z - ---- - 4 a z + 15 a z + 7 a z - 4 a z - 9 z - a 2 8 4 8 6 8 9 3 9 5 9 2 10 > 14 a z - 12 a z - 7 a z - 6 a z - 12 a z - 6 a z - 2 a z - 4 10 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	11 1 3 1 6 3 8 6 12 12 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 8 12 12 10 12 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 6 t + 9 q t + 3 q t + 6 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a45

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