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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X16,8,17,7 X22,18,5,17 X18,14,19,13 X20,9,21,10 X14,20,15,19 X8,21,9,22 X10,16,11,15 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 3q-9/2 - 8q-7/2 + 13q-5/2 - 19q-3/2 + 22q-1/2 - 23q1/2 + 20q3/2 - 16q5/2 + 10q7/2 - 4q9/2 + q11/2

A2 (sl(3)) Invariant:

q-18 + q-16 - q-14 + 3q-12 + q-10 - 2q-8 + 5q-6 - 2q-4 + 2q-2 + 2 - 2q2 + 5q4 - 4q6 + 3q8 - 4q12 + 2q14 - q16

HOMFLY-PT Polynomial:

a-3z-1 + 3a-3z + 2a-3z3 + a-3z5 - 3a-1z-1 - 9a-1z - 9a-1z3 - 4a-1z5 - a-1z7 + 3az-1 + 8az + 8az3 + 3az5 - 2a3z-1 - 5a3z - 3a3z3 + a5z-1 + a5z

Kauffman Polynomial:

- a-6z4 - 4a-5z5 + a-4 - 4a-4z2 + 8a-4z4 - 10a-4z6 - a-3z-1 + 4a-3z - 13a-3z3 + 23a-3z5 - 16a-3z7 + 2a-2 - 4a-2z2 - 6a-2z4 + 24a-2z6 - 16a-2z8 - 3a-1z-1 + 14a-1z - 30a-1z3 + 33a-1z5 - 9a-1z9 + 11z2 - 46z4 + 63z6 - 21z8 - 2z10 - 3az-1 + 15az - 27az3 + 6az5 + 25az7 - 13az9 - 2a2 + 15a2z2 - 42a2z4 + 39a2z6 - 8a2z8 - 2a2z10 - 2a3z-1 + 9a3z - 16a3z3 + 4a3z5 + 8a3z7 - 4a3z9 + 4a4z2 - 11a4z4 + 10a4z6 - 3a4z8 - a5z-1 + 4a5z - 6a5z3 + 4a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                       1 j = 10                     3   j = 8                   7 1   j = 6                 9 3     j = 4               11 7       j = 2             12 9         j = 0           11 12           j = -2         8 11             j = -4       5 11               j = -6     3 8                 j = -8   1 6                   j = -10   2                     j = -12 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, 80]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 80]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[16, 8, 17, 7], X[22, 18, 5, 17], > X[18, 14, 19, 13], X[20, 9, 21, 10], X[14, 20, 15, 19], X[8, 21, 9, 22], > X[10, 16, 11, 15], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -8, 6, -9, 11, -2, 5, -7, 9, -3, 4, -5, > 7, -6, 8, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 3 8 13 19 22 3/2 -q + ---- - ---- + ---- - ---- + ------- - 23 Sqrt[q] + 20 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 16 q + 10 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 -14 3 -10 2 5 2 2 2 4 6 2 + q + q - q + --- + q - -- + -- - -- + -- - 2 q + 5 q - 4 q + 12 8 6 4 2 q q q q q 8 12 14 16 > 3 q - 4 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 80]][a, z]
Out[8]=	3 5 3 1 3 3 a 2 a a 3 z 9 z 3 5 2 z ---- - --- + --- - ---- + -- + --- - --- + 8 a z - 5 a z + a z + ---- - 3 a z z z z 3 a 3 a z a a 3 5 5 7 9 z 3 3 3 z 4 z 5 z > ---- + 8 a z - 3 a z + -- - ---- + 3 a z - -- a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 80]][a, z]
Out[9]=	3 5 -4 2 2 1 3 3 a 2 a a 4 z 14 z 3 a + -- - 2 a - ---- - --- - --- - ---- - -- + --- + ---- + 15 a z + 9 a z + 2 3 a z z z z 3 a a a z a 2 2 3 3 5 2 4 z 4 z 2 2 4 2 13 z 30 z > 4 a z + 11 z - ---- - ---- + 15 a z + 4 a z - ----- - ----- - 4 2 3 a a a a 4 4 4 3 3 3 5 3 4 z 8 z 6 z 2 4 > 27 a z - 16 a z - 6 a z - 46 z - -- + ---- - ---- - 42 a z - 6 4 2 a a a 5 5 5 4 4 4 z 23 z 33 z 5 3 5 5 5 6 > 11 a z - ---- + ----- + ----- + 6 a z + 4 a z + 4 a z + 63 z - 5 3 a a a 6 6 7 10 z 24 z 2 6 4 6 16 z 7 3 7 5 7 > ----- + ----- + 39 a z + 10 a z - ----- + 25 a z + 8 a z - a z - 4 2 3 a a a 8 9 8 16 z 2 8 4 8 9 z 9 3 9 10 > 21 z - ----- - 8 a z - 3 a z - ---- - 13 a z - 4 a z - 2 z - 2 a a 2 10 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 6 3 8 5 11 12 + 12 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 8 11 11 2 4 4 2 6 2 6 3 > ----- + -- + ---- + 9 q t + 11 q t + 7 q t + 9 q t + 3 q t + 2 2 t 2 q t q t 8 3 8 4 10 4 12 5 > 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a80

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